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# 8.5: Similar and Congruent Figures

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Mathematical Floor

Mrs. Gilman brought a small group of students over to look at this tile floor in the hallway of the art museum.

“You see, there is even math in the floor,” she said, smiling. Mrs. Gilman is one of those teachers who loves to point out every place where math can be found.

“Okay, I get it,” Jesse started. “I see the squares.”

“There is a lot more math than just squares,” Mrs. Gilman said, walking away with a huge smile on her face.

“She frustrates me sometimes,” Kara whispered, staring at the floor. “Where is the math besides the squares?”

“I think she is talking about the size of the squares,” Hannah chimed in. “See? There are two different sizes.”

“Actually there are three different sizes, and there could be more that I haven’t found yet,” Jesse said.

“Remember when we learned about comparing shapes that are alike and aren’t alike? It has to do with proportions or something like that,” Hannah chimed in again.

All three students stopped talking and began looking at the floor again.

“Oh yeah, congruent and similar figures, but which are which?”Kara asked.

What is the difference between congruent and similar figures? This lesson will teach you all about congruent and similar figures. When you are all finished with this lesson, you will have a chance to study the floor again and see if you can find the congruent and the similar figures.

What You Will Learn

By the end of this lesson you will be able to demonstrate the following skills:

• Recognize congruence.
• Find unknown measures of congruent figures.
• Recognize similarity.
• Check for similarity between given figures

Teaching Time

I. Recognize Congruence

In the last lesson we began using the word “congruent.” We talked about congruent lines and congruent angles. The word congruent means exactly the same. Sometimes, you will see this symbol \begin{align*}\cong\end{align*}.

In this lesson, we are going to use the word congruent to compare figures.

Congruent figures have exactly the same size and shape. They have congruent sides and congruent angles. Here are some pairs of congruent figures.

Compare the figures in each pair. They are exactly the same! If you’re not sure, imagine that you could cut out one figure and place it on top of the other. If they match exactly, they are congruent.

How can we recognize congruence?

We test for congruency by comparing each side and angle of two figures to see if all aspects of both are the same. If the sides are the same length and the angles are equal, the figures are congruent. Each side and angle of one figure corresponds to a side or angle in the other. We call these corresponding parts. For instance, the top point of one triangle corresponds to the top point of the other triangle in a congruent pair.

It is not always easy to see the corresponding parts of two figures. One figure may be rotated differently so that the corresponding parts appear to be in different places. If you’re not sure, trace one figure and place it on top of the other to see if you can make them match. Let’s see if we can recognize some congruent figures.

Example

Which pair of figures below is congruent?

Let’s analyze one pair at a time to see if we can find any corresponding angles and sides that are congruent.

The figures in the first pair appear to be the same shape, if we rotate one \begin{align*}180^\circ\end{align*} so they both point up. Now we can see all of the corresponding parts, such as the angle at the top and the two long sides on the left and right. This is not enough to go on, however. We need to compare the measures of the angles and the lengths of the sides. If any one set of corresponding parts doesn’t match, the figures cannot be congruent.

We only know the measure of one angle in the first two figures. We can compare these angles if they are corresponding parts. They are, because if we rotate one figure these angles are in the same place at the top of each figure. Now compare their measures. The angle in the first figure is \begin{align*}45^\circ\end{align*}. The corresponding angle in the second figure is \begin{align*}55^\circ\end{align*}. Because the angles are different, these two figures are not congruent. Let’s look at the next pair.

The two triangles in the second pair seem to have corresponding parts: a long base and a wide angle at the top. We need to know whether any of these corresponding parts are congruent, however. We know the measure of the top angle in each figure: it is \begin{align*}110^\circ\end{align*} in both. These figures might be congruent, but we need to see if the sides are congruent to be sure (as we said, similar figures also have congruent angles, but their sides are different lengths). We know the measure of each triangle’s base: one is 2 inches and the other is 4 inches. These sides are not congruent, so the triangles are not congruent. Remember, every side and every angle must be the same in order for figures to be congruent.

That leaves the last pair. Can you find the corresponding parts? If we rotate the second figure \begin{align*}180^\circ\end{align*}, we have two shapes that look like the letter \begin{align*}L\end{align*}. Now compare the corresponding sides. The bottom side of each is 8 cm, the long left side of each is 8 cm, two sides are 6 cm, and two sides are 2 cm. All of the angles in both figures are \begin{align*}90^\circ\end{align*}. Because every side and angle in one figure corresponds to a congruent side and angle in the second, these two figures are congruent.

8J. Lesson Exercises

Answer true or false for each question

1. Congruent figures have the same number of sides and angles.
2. Congruent figures can have one pair of angles with the same measure, but not all angles have the same measure.
3. Congruent figures can be different sizes as long as the angle measures are the same.

Discuss your answers with a friend. Be sure you understand why each answer is true or false.

II. Find Unknown Measures of Congruent Figures

We know that congruent figures have exactly the same angles and sides. That means we can use the information we have about one figure in a pair of congruent figures to find the measure of a corresponding angle or side in the other figure. Let’s see how this works. Take a look at the congruent figures below.

We have been told these two parallelograms are congruent.

Can you find the corresponding parts?

If not, trace one parallelogram and place it on top of the other. Rotate it until the parts correspond.

Which sides and angles correspond?

We can see that side \begin{align*}AB\end{align*} corresponds to side \begin{align*}PQ\end{align*}. Because they are congruent, we write.

\begin{align*}AB \cong PQ.\end{align*}

What other sides are congruent? Let’s write them out.

\begin{align*}AB &\cong PQ\\ BC &\cong QR\\ AD &\cong PS\\ DC &\cong SR\end{align*}

We can also write down the corresponding angles, which we know must be congruent because the figures are congruent.

\begin{align*}\angle A &\cong \angle P && \angle D \cong \angle S\\ \angle B &\cong \angle Q && \angle C \cong \angle R\end{align*}

Now that we understand all of the corresponding relationships in the two figures, we can use what we know about one figure to find the measure of a side or angle in the second figure.

Can we find the length of side \begin{align*}AB\end{align*}?

We do not know the length of \begin{align*}AB\end{align*}. However, we do know that it is congruent to \begin{align*}PQ\end{align*}, so if we can find the length of \begin{align*}PQ\end{align*} then it will be the same for \begin{align*}AB.\end{align*} Since \begin{align*}PQ\end{align*} is 7 centimeters, \begin{align*}AB\end{align*} must also be 7 centimeters long.

Now let’s look at the angles. Can we find the measure of \begin{align*}\angle C\end{align*}?

It corresponds to \begin{align*}\angle R\end{align*}, but we do not know the measure of \begin{align*}\angle R\end{align*} either. Well, we do know the measures of two of the angles in the first parallelogram: \begin{align*}70^\circ\end{align*} and \begin{align*}110^\circ\end{align*}. If we had three, we could subtract from \begin{align*}360^\circ\end{align*} to find the fourth, because all quadrilaterals have angles that add up to \begin{align*}360^\circ\end{align*}. We do not know the measure of \begin{align*}\angle B\end{align*}, but this time we do know the measure of its corresponding angle, \begin{align*} \angle Q\end{align*}. These two angles are congruent, so we know that \begin{align*}\angle B\end{align*} must measure \begin{align*}70^\circ\end{align*}. Now we know three of the angles in the first figure, so we can subtract to find the measure of \begin{align*}\angle C\end{align*}.

\begin{align*}360 - (70 + 110 + 70) &= \angle C\\ 360 - 250 &= \angle C\\ 110^\circ &= \angle C\end{align*}

We were able to combine the given information from both figures because we knew that they were congruent.

Yes and the more you work on puzzles like this one the easier they will become.

8K. Lesson Exercises

1. What is the measure of \begin{align*}\angle M\end{align*}? (The two triangles are congruent.)

Take a few minutes to check your answer with a friend. Correct any errors and then continue with the next section.

III. Recognize Similarity

Some figures look identical except they are different sizes. The angles even look the same. When we have figures that are proportional to each other, we call these figures similar figures. Similar figures have the same angle measures but different side lengths.

What is an example of similar figures?

Squares are similar shapes because they always have four \begin{align*}90^\circ\end{align*} angles and four equal sides, even if the lengths of their sides differ. Other shapes can be similar too, if their angles are equal.

Let’s look at some pairs of similar shapes.

Notice that in each pair the figures look the same, but one is smaller than the other. Since they are not the same size, they are not congruent. However, they have the same angles, so they are similar.

IV. Check for Similarity between Given Figures

Unlike congruent figures, similar figures are not exactly the same. They do have corresponding features, but only their corresponding angles are congruent; the corresponding sides are not. Thus when we are dealing with pairs of similar figures, we should look at the angles rather than the sides. In similar figures, the angles are congruent, even if the sides are not.

Notice that one angle in each pair of figures corresponds to an angle in the other figure. They have the same shape but not the same size. Therefore they are similar.

Let’s find the corresponding angles in similar figures.

Example

List the corresponding angles in the figures below.

Angles \begin{align*}G\end{align*} and \begin{align*}W\end{align*} are both right angles, so they correspond to each other. Imagine you can turn the figures to line up the right angles. You might even trace the small figure so that you can place it on top of the larger one.

How do the angles line up?

Angles \begin{align*}H\end{align*} and \begin{align*}X\end{align*} correspond to each other. So do angles \begin{align*}I\end{align*} and \begin{align*}Y\end{align*} and angles \begin{align*}J\end{align*} and \begin{align*}Z\end{align*}. Now we can name these two quadrilaterals: \begin{align*}GHIJ\end{align*} is similar to \begin{align*}WXYZ\end{align*}.

As we’ve said, the sides in similar figures are not congruent. They are proportional, however. Proportions have the same ratio. Look at \begin{align*}GHIJ\end{align*} and \begin{align*}WXYZ\end{align*} again. We can write each pair of sides as a proportion.

\begin{align*}\frac{GH}{WX},\frac{HI}{XY},\frac{IJ}{YZ},\frac{GJ}{WZ}\end{align*}

The sides from one figure are on the top, and the proportional sides of the other figure are on the bottom.

Example

List all of the pairs of corresponding sides in the figures below as proportions.

Try lining up the figures by their angles. It may help to trace one figure and rotate it until it matches the other.

Which sides are proportional?

\begin{align*}OP\end{align*} and \begin{align*}RS\end{align*} are the shortest sides in each figure. They are proportional, so we write

\begin{align*}\frac{OP}{RS}\end{align*}

Now that we’ve got one pair, let’s do the same for the rest.

\begin{align*}\frac{NO}{QR}, \frac{MP}{TS}, \frac{MN}{TQ}\end{align*}

Now let’s use what we have learned to check for similarity between figures.

Example

Which pair of figures below is similar?

For figures to be similar, we know that the angles must be congruent and the sides must exist in proportional relationships to each other. Let’s check each pair one at a time.

We only know some of the angles in each triangle in the first pair. They both have a \begin{align*}50^{\circ}\end{align*} angle, so that’s a good start. All three angles must be congruent, however, so let’s solve for the missing angle in each angle. Remember, the sum of the three angles is always \begin{align*}180^{\circ}\end{align*} for a triangle.

\begin{align*}&\text{Triangle 1} && \text{Triangle 2}\\ &50 + 60 + \text{angle} \ 3 = 180 && 50 + 80 + \text{angle} \ 3 = 180\\ &110 + \text{angle} \ 3 = 180 && 130 + \text{angle} \ 3 = 180\\ &\text{angle} \ 3 = 180 - 110 && \text{angle} \ 3 = 180 - 130 \\ &\text{angle} \ 3 = 70^{\circ} && \text{angle} \ 3 = 50^{\circ}\end{align*}

The angles in the first triangle are \begin{align*}50^{\circ}\end{align*}, \begin{align*}60^{\circ}\end{align*}, and \begin{align*}70^{\circ}\end{align*}. The angles in the second triangle are \begin{align*}50^{\circ}\end{align*}, \begin{align*}50^{\circ}\end{align*}, and \begin{align*}80^{\circ}\end{align*}. These triangles are not similar because their angle measures are different.

Let’s move on to the next pair.

This time we know side lengths, not angles. We need to check whether each set of corresponding sides is proportional. First, let’s write out the pairs of proportional corresponding sides

\begin{align*}\frac{6}{3}, \frac{6}{3}, \frac{4}{1}\end{align*}

The proportions show side lengths from the large triangle on the top and its corresponding side in the small triangle on the bottom. The pairs of sides must have the same proportion in order for the triangles to be similar. We can test whether the three proportions above are the same by dividing each. If the quotient is the same, the pairs of sides must exist in the same proportion to each other.

\begin{align*}\frac{6}{3} = 2\\ \frac{6}{3} = 2\\ \frac{4}{1} = 4\end{align*}

When we divide, only two pairs of sides have the same proportion (2). The third pair of sides does not exist in the same proportion as the other two, so these triangles cannot be similar.

That leaves the last pair. We have been given the measures of some of the angles. If all of the corresponding angles are congruent, then these two figures are similar. We know the measure of three angles in each figure. In fact, they are all corresponding angles. Therefore the one unknown angle in the first figure corresponds to the unknown angle in the second figure.

As we know, the four angles in a quadrilateral must have a sum of \begin{align*}360^{\circ}\end{align*}. Therefore the unknown angle in each figure must combine with the other three to have this sum. Because the three known angles are the same for both figures, we don’t even need to solve for the fourth to know that it will be the same in both figures. These two figures are similar because their angle measures are all congruent.

Now let’s use what we have learned to solve the problem in the introduction.

## Real Life Example Completed

The Mathematical Floor

Here is the original problem once again. Reread it and then answer the questions at the end of this passage.

Mrs. Gilman brought a small group of students over to look at this tile floor in the hallway of the art museum.

“You see, there is even math in the floor,” she said, smiling. Mrs. Gilman is one of those teachers who loves to point out every place where math can be found.

“Okay, I get it,” Jesse started. “I see the squares.”

“There is a lot more math than just squares,” Mrs. Gilman said, walking away with a huge smile on her face.

“She frustrates me sometimes,” Kara whispered, staring at the floor. “Where is the math besides the squares?”

“I think she is talking about the size of the squares,” Hannah chimed in. “See? There are two different sizes.”

“Actually there are three different sizes, and there could be more that I haven’t found yet,” Jesse said.

“Remember when we learned about comparing shapes that are alike and aren’t alike? It has to do with proportions or something like that,” Hannah chimed in again.

All three students stopped talking and began looking at the floor again.

“Oh yeah, congruent and similar figures, but which are which?” Kara asked.

The students are working on which figures in the floor pattern are congruent and which ones are similar.

The congruent figures are exactly the same. We can say that the small dark brown squares are congruent because they are just like each other. They have the same side lengths. What is one other pair of congruent squares?

The similar figures compare squares of different sizes. You can see that the figures are squares, so they all have 90 degree angles. The side lengths are different, but because the angles are congruent, we can say that they have the same shape, but not the same size. This makes them similar figures.

The small dark brown square is similar to the large dark brown square. The small dark brown square is also similar to the square created by the ivory colored tile. There is a relationship between the different squares. Are there any more comparisons? Make a few notes in your notebook.

## Vocabulary

Congruent
having exactly the same shape and size. All side lengths and angle measures are the same.
Similar
having the same shape but not the same size. All angle measures are the same, but side lengths are not.

## Time to Practice

Directions: Tell whether the pairs of figures below are congruent, similar, or neither.

Directions: Name the corresponding parts to those given below.

7. \begin{align*}\angle R\end{align*}

8. \begin{align*}MN\end{align*}

9. \begin{align*}\angle O\end{align*}

Directions: Use the relationships between congruent figures to find the measure of \begin{align*}g\end{align*}. Show your work.

10.

Directions: Use the relationships between congruent figures to find the measure of \begin{align*}\angle T\end{align*}. Show your work.

11.

Directions: Answer each of the following questions.

12. Triangles \begin{align*}ABC\end{align*} and \begin{align*}DEF\end{align*} are congruent. If the measure of angle \begin{align*}A\end{align*} is \begin{align*}58^{\circ}\end{align*}, what is the measure of angle \begin{align*}D\end{align*} if it corresponds to angle \begin{align*}A\end{align*}?

13. True or false. If triangles \begin{align*}DEF\end{align*} and \begin{align*}GHI\end{align*} are similar, then the side lengths may be different but the angle measures are the same.

14. True or false. Similar figures have exactly the same size and shape.

15. True or false. Congruent figures are exactly the same in every way.

16. Triangles \begin{align*}LMN\end{align*} and \begin{align*}HIJ\end{align*} are similar. If this is true, then the side lengths are the same, true or false.

17. What is a proportion?

18. True or false. To figure out if two figures are similar, see if their side lengths form a proportion.

19. Define similar figures

20. Define congruent figures.

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A recent birthday present (my 36th) was Elias Wegert's Visual complex functions. On the sides of the pages are strips -- excerpts from the critical strip of the Riemann zeta function with nontrivial roots. He wrote a paper about the stochastic period of 2$\pi$, which is easily observable in the below image as the yellow diagonals:
He's got the images of the functions up on a functions gallery on the website associated with the book. I thought I couldn't see what was going on so I used the gimp to prune the white regions from the images and to glue the remainders together. So, this led to a mathoverflow question after I asked Matt McIrvin to take a Fourier transform -- which he did with a set discretely supported on the roots of the Riemann zeta function, and a post on the n-category cafe concerning Freeman Dyson's speculation that the roots might form a quasicrystal. There are some obvious questions:
• What about other L-functions? The best way of performing such calculations is to use sage and its implementation of the Dokhitser L functions calculator.
• What is the local phase at each root (how much is each root rotated?) -- and this lives on U(1).

Bounty: I am offering a bottle of whisky to anyone who can successfully translate Jacob Lurie's Higher Topos Theory (and/or the stacks project) into the analytic combinatorical language of the manner of Flajolet's Analytic Combinatorics.

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# Electric Field: Required Charge

APPH12-4DLJ01

The magnitude of the electric field $39\text{ cm}$ from the center of a charged sphere (with a radius of $14\text{ cm}$) is $560\text{ N/C}$.

What is the charge on the sphere?

A

$9.5\times10^{-5}\text{ C}$

B

$2.4\times10^{-6}\text{ C}$

C

$2.4\times10^{-8}\text{ C}$

D

$9.5\times10^{-9}\text{ C}$


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Research

# Risk of neonatal drug withdrawal after intrauterine co-exposure to opioids and psychotropic medications: cohort study

BMJ 2017; 358 (Published 02 August 2017) Cite this as: BMJ 2017;358:j3326
1. Krista F Huybrechts, assistant professor of medicine1,
2. Brian T Bateman, associate professor of anesthesiology1 2,
3. Rishi J Desai, instructor in medicine1,
4. Sonia Hernandez-Diaz, professor of epidemiology3,
5. Kathryn Rough, postdoctoral research fellow1 3,
6. Helen Mogun, analyst1,
7. Leslie S Kerzner, assistant professor of pediatrics4,
8. Jonathan M Davis, professor of pediatrics5,
9. Megan Stover, fellow in maternal-fetal medicine and genetics6,
10. Devan Bartels, resident physician7,
11. Jennifer Cottral, resident physician7,
12. Elisabetta Patorno, assistant professor of medicine1
1. 1Division of Pharmacoepidemiology and Pharmacoeconomics, Department of Medicine, Brigham and Women’s Hospital and Harvard Medical School, Boston, MA, USA
2. 2Department of Anesthesiology, Perioperative and Pain Medicine, Brigham and Women’s Hospital and Harvard Medical School, Boston, MA, USA
3. 3Department of Epidemiology, Harvard T.H. Chan School of Public Health, Boston, MA, USA
4. 4Department of Pediatrics, Massachusetts General Hospital for Children and Harvard Medical School, Boston, MA, USA
5. 5Department of Pediatrics, The Floating Hospital at Tufts Medical Center and the Tufts Clinical and Translational Science Institute, Tufts University , Boston, MA, USA
6. 6Department of Obstetrics and Gynecology, Tufts Medical Center and Tufts University School of Medicine, Boston, MA, USA
7. 7Department of Anesthesia, Critical Care, and Pain Medicine, Massachusetts General Hospital and Harvard Medical School, Boston, MA, USA
1. Correspondence to: K F Huybrechts khuybrechts{at}bwh.harvard.edu
• Accepted 5 July 2017

## Abstract

Objectives To assess the impact of in utero co-exposure to psychotropic medications and opioids on the incidence and severity of neonatal drug withdrawal.

Design Observational cohort study.

Setting Nationwide sample of pregnancies in publicly insured women in the US, nested in the Medicaid Analytic eXtract (2000-10).

Participants 201 275 pregnant women with public insurance who were exposed to opioids around the time of delivery and their liveborn infants.

Interventions In utero exposure to psychotropic medications, in particular antidepressants, atypical antipsychotics, benzodiazepines, gabapentin, and non-benzodiazepine hypnotics (Z drugs), with prescriptions filled within the same time window as prescriptions for opioids.

Main outcome measure Diagnosis of neonatal drug withdrawal in infants exposed in utero to opioids and psychotropic medications compared with opioids alone.

Results The absolute risk for neonatal drug withdrawal ranged from 1.0% in infants exposed in utero to prescription opioids alone to 11.4% for those exposed to opioids co-prescribed with gabapentin. Among neonates exposed in utero to prescription opioids, the relative risk adjusted for propensity score was 1.34 (95% confidence interval 1.22 to 1.47) with concomitant exposure to antidepressants, 1.49 (1.35 to 1.63) with benzodiazepines, 1.61 (1.26 to 2.06) with gabapentin, 1.20 (0.95 to 1.51) with antipsychotics, and 1.01 (0.88 to 1.15) with Z drugs. In utero exposure to two or more psychotropic medications along with opioids was associated with a twofold increased risk of withdrawal (2.05, 1.77 to 2.37). The severity of the withdrawal seemed increased in neonates exposed to both opioids and psychotropic medications compared with opioids alone.

Conclusions During pregnancy, the use of psychotropic medications in addition to prescription opioids is common, despite a lack of safety data. The current findings suggest that these drugs could further increase the risk and severity of neonatal drug withdrawal.

## Introduction

Prescriptions for opioid analgesics in the US have increased more than threefold over the past two decades,12 and this rise also extends to prescriptions to pregnant women. About 14-22% of pregnancies in the US are complicated by exposure to prescription opioid medications.34 A strong increase in the prescription of opioids has also been observed in most European countries (such as Germany, Italy, Spain, Scandinavia, UK, the Netherlands) and Australia, though the number of prescriptions and defined daily doses remains much lower than in the US and Canada.5 Data from a population based registry in Norway showed that 6% of pregnant women filled at least one opioid prescription in 2004-06.6 Neonatal abstinence syndrome (NAS) is a well recognized complication of intrauterine exposure to opioid analgesics, with manifestations ranging from difficulties with feeding and sleeping to more severe complications including impaired thermoregulation, seizures, failure to thrive, and respiratory distress.7 From 2000 to 2012, hospital charges for this syndrome increased from about $190m (about £150m; €170m) to$1.5bn.8 In the US, about every 25 minutes an infant is born with signs of drug withdrawal.89

Although neonatal drug withdrawal occurs most commonly in the context of antepartum illicit or prescription opioid use, other classes of medication—in particular psychotropic medications—can also cause signs of withdrawal. It is often difficult to determine whether the signs of withdrawal in the newborn are primarily the result of intrauterine exposure to opioids, other psychotropic medications (including nicotine), or a combination of both. The label for all antipsychotics was updated in 2011 to include the potential risk for withdrawal in newborns whose mothers were treated with these drugs during the third trimester of pregnancy, based on 69 cases reported in FDA’s adverse event reporting system.10 Withdrawal has also been described in infants exposed antenatally to barbiturates, benzodiazepines, and antidepressants,7 and recently there have been reports of severe withdrawal from gabapentin.11 Psychotropic medications are often used by pregnant women who are also receiving an opioid,121314 raising concerns about an increase in the incidence and severity of neonatal drug withdrawal from potential drug-drug interactions.14 Whereas intrauterine exposure to psychotropic medications has been used as a risk stratification factor,14 the risks of neonatal drug withdrawal in those exposed to opioids during the pregnancy with or without co-exposure to individual and combinations of psychotropic medications have not previously been examined. Little is therefore known about how this risk varies by the type of psychotropic medication or about the risks associated with psychotropic polypharmacy.

We assessed whether the risk of neonatal drug withdrawal in newborns exposed in utero to one or more psychotropic medications and opioid analgesics is greater and more severe than in those exposed to opioid analgesics alone in a cohort of publicly insured mother-infant dyads.

## Methods

### Data source and study cohort

The study cohort was nested in the nationwide Medicaid Analytic eXtract (MAX) for 2000-10. The MAX includes demographic and insurance enrollment information, medical visits, and admissions to hospital, inpatient or outpatient diagnoses and procedures, and prescriptions filled on an outpatient basis. We included all pregnancies in women aged 12-55 who were linked to a liveborn infant (further details are available elsewhere15). Women were required to be eligible for Medicaid from 180 days before delivery through to one month after delivery. Infants were required to have the same coverage through Medicaid during the first month after birth or until death if this occurred sooner. All women were required to have filled at least one outpatient prescription for an opioid analgesic (see table A in appendix for a list of those included) during the 45 days before delivery. We chose this window to increase the likelihood of opioid exposure around the time of delivery.

### Psychotropic medications

Women who also filled a prescription for psychotropic medication during the 45 days before delivery were considered as exposed. We chose this window to ensure intrauterine exposure to both opioids and psychotropic medications near or at the time of delivery, which is relevant for neonatal drug withdrawal. We distinguished between five different exposure groups: antidepressants, atypical antipsychotics (hereafter antipsychotics), gabapentin (included because of its mood stabilizing properties), benzodiazepines, and non-benzodiazepine hypnotics (hereafter Z drugs) (table B in the appendix). In secondary analyses, we also examined the risk associated with exposure to one or more of the psychotropic medications found to be associated with an increased risk of withdrawal in the primary analyses. The reference group consisted of women who received opioids but who did not fill an outpatient prescription for the psychotropic medication of interest during the 180 days before delivery.

### Neonatal drug withdrawal

Neonatal drug withdrawal was defined based on the presence of ICD-9 (international classification of diseases, ninth revision) code 779.5x in maternal or infant claims from delivery to one month after delivery. We analyzed the maternal record as Medicaid claims are sometimes recorded under the mother before the infant’s eligibility has been processed.11 The outcome definition was validated in the subset of study participants who received their obstetrical care in the Partners HealthCare system. We randomly selected a sample of cases of neonatal drug withdrawal (n=50) that occurred in the absence of exposure to prescription opioids (for example, withdrawal from heroin, illicit opioids, methadone) and all cases of withdrawal that occurred with known exposure to prescription opioids (n=7) and retrieved their electronic medical records at the delivery hospital. Two reviewers (DB and JC), blinded to the exposure status, adjudicated these cases; discrepancies were resolved by consensus (four of 50 cases with prescription opioids, none of the seven cases without prescription opioids). We estimated a positive predictive value of 91% (95% confidence interval 82% to 97%) overall and 100% (65% to 100%) in the subset of infants with intrauterine exposure to prescribed opioids.

### Covariates

The specific compound, form, and dose of opioid are important potential confounding variables. We expressed the dose as the total amount of prescription opioids accumulated during the last 45 days of pregnancy in oral morphine equivalents (see table A in appendix).161718 Additional covariates assessed as potential confounders included calendar year, demographics (age, race/ethnicity), obstetric characteristics (multiple gestation, preterm delivery, pre-eclampsia), treatment indications (pain, psychiatric diagnoses), indicators of substance abuse or dependence, concomitant medications, and general markers of disease burden (intensity of healthcare use). We also assessed diagnoses of alcohol or drug abuse or dependence, tobacco smoking, methadone, and use of buprenorphine, naloxone, and naltrexone. Other than the psychotropic exposures of interest, we considered the concomitant use of other anxiolytics, barbiturates, other hypnotics, stimulants, primidone, phenobarbital, and pregabalin, all of which have been associated with withdrawal in adults.19 General markers of disease burden included the obstetric morbidity index,20 number of distinct prescriptions, number of distinct diagnoses, and number of outpatient visits, hospital admissions, and emergency room visits.21 All variables were assessed from 180 days before delivery through to delivery, except for measures of intensity of healthcare use, which were assessed from 180 to 90 days before delivery to avoid these measures being affected by early awareness of possible late pregnancy complications (table C in the appendix shows a complete list of covariates).

### Data analyses

We described baseline characteristics in women exposed and unexposed to different psychotropic medications and estimated the absolute risk of neonatal drug withdrawal for each exposure group, as well as the unadjusted risk ratio with its 95% confidence interval for each exposure contrast. In the first adjusted analysis, we accounted for the specific opioid compound (that is, generic name and form (immediate release or extended release) as listed in table A in the appendix) and dose expressed in morphine equivalents, which is expected to be the most important confounder. In the second adjusted analysis, we also accounted for additional potential confounders, as described above (table C in the appendix). For each of the two levels of analysis, we used propensity scores as a summary confounding score estimated using logistic regression given the large number of prespecified variables. We included only main effects in the models. We restricted the population to observations within a propensity score range that was common to both exposed and unexposed women and created 25 equally sized strata based on the distribution among the women exposed to psychotropics.22 The unexposed observations (that is, exposed only to opioid) were weighted with the distribution of the treated among propensity score strata. We assessed balance of baseline characteristics in this weighted population, which considers potential confounding variables, using the standardized mean difference. An absolute standardized mean difference over 0.1 was considered as evidence of residual imbalance. Adjusted effects were estimated with generalized linear models (SAS Proc Genmod with a weight statement and log link function).

### Sensitivity analyses

As we might not have accounted for all potential confounders available in our data using prespecified covariates, we conducted high dimensional propensity score analyses that included 200 empirically defined covariates, in addition to the investigator defined covariates, to account for potential residual confounding.23 We then varied the exposure definition for propensity score adjusted analyses by expanding the exposure window for both opioids and psychotropics from 45 to 90 days before delivery to evaluate the potential risk associated with exposure earlier in pregnancy and by requiring two or more prescriptions for a psychotropic medication to be filled during the 90 days before delivery, at least one of which had to be filled during the last 45 days to reduce the likelihood of misclassification of exposure. Subclasses of antidepressant medications (selective serotonin reuptake inhibitors (SSRIs), selective norepinephrine reuptake inhibitors, tricyclic antidepressants, bupropion, other antidepressants) were examined as well as the most commonly used SSRIs (citalopram, escitalopram, fluoxetine, paroxetine, sertraline); this was feasible given the large number of women taking antidepressants in late pregnancy.

### Exploratory analyses: severity of withdrawal

As we did not have Finnegan scores24 available, we explored whether proxies for the severity of withdrawal symptoms in neonates with in utero exposure to one or more of the psychotropic medications were different compared with those exposed only to opioids. Proxy severity indicators included a recorded diagnosis in the first 30 days of life indicating feeding difficulties (ICD-9 779.3x), respiratory symptoms (ICD-9 769.xx-770.xx), or signs of seizures (ICD-9 779.0x and 780.3x). We also evaluated a composite outcome defined as the presence of any of these three severity indicators and the neonate’s length of hospital stay. Adjusted analyses used the same set of covariates and analytic approach as the main analyses.

### Patient involvement

No patients were involved in setting the research question or the outcome measures, nor were they involved in developing plans for design or implementation of the study. No patients were asked to advise on interpretation or writing up of results. There are no plans to disseminate the results of the research to study participants or the relevant patient community.

## Results

The cohort included 201 275 pregnancies in women with exposure to prescription opioids in the 45 days before delivery. Among those, 14 183 women (7.0%) also filled a prescription for an antidepressant, 993 (0.5%) for an antipsychotic, 5361 (2.7%) for a benzodiazepine, 501 (0.3%) for gabapentin, and 10 105 (5.0%) for a Z drug. Women exposed to psychotropic medications differed from those who were not exposed on most of the baseline characteristics. Exposed women were older, were more likely to be white, had more pain and psychiatric diagnoses, used more other psychotropic medications, more commonly had diagnoses of substance abuse or dependence, and had a higher burden of comorbidity. They also tended to have a higher cumulative dose of opioids during the last 45 days of their pregnancy, which was most prominent for users of benzodiazepines (mean dose 1352 mg v 291 mg oral morphine equivalents) and gabapentin (1169 mg v 326 mg). The difference in baseline characteristics was less pronounced for women exposed to Z drugs (table 1, fig 1, and tables D-J in the appendix).

Table 1

Selected cohort characteristics for women exposed to gabapentin and prescription opioids during last 45 days of pregnancy and women exposed to prescription opioids only: unadjusted, adjusted for propensity score (PS), and adjusted for high dimensional propensity score (hdPS). Medicaid Analytic eXtract, 2000-10*. Figures are number (percentage) unless stated otherwise and standardized mean difference (SMD)

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Fig 1 Absolute standardized mean difference illustrating balance in baseline characteristics before and after adjustment for high dimensional propensity score. SMD <0.1 indicates characteristics are well balanced

The absolute risk for neonatal drug withdrawal was substantially higher among women who were co-exposed to opioids and psychotropic medications than among women exposed to opioids alone. Absolute risk of withdrawal in neonates ranged from 2.3% among women exposed to Z drugs to 11.4% among women exposed to gabapentin. The risk among women exposed to prescription opioids alone was around 1%, with small variations given the different exclusions made in each exposure contrast (table 2).

Table 2

Absolute risk of neonatal drug withdrawal (neonatal abstinence syndrome) after intrauterine exposure to both opioids and psychotropic medications versus opioids alone. Medicaid Analytic eXtract, 2000-10

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The unadjusted relative risk was 3.48 (95% confidence interval 3.15 to 3.84) for antidepressants, 5.42 (4.28 to 6.85) for antipsychotics, 7.43 (6.71 to 8.23) for benzodiazepines, 9.08 (7.09 to 11.63) for gabapentin, and 1.87 (1.63 to 2.13) for Z drugs. The strength of the associations was much attenuated after adjustment for specific opioid compound, form (that is, extended versus immediate release), and cumulative dose (fig 2, table K in the appendix).

Fig 2 Adjusted relative risk of neonatal drug withdrawal according to maternal exposure to psychotropic medications in addition to prescription opioids, according to level of adjustment for confounding. Medicaid Analytic eXtract, 2000-10

In fully adjusted analyses, the baseline characteristics were well balanced (fig 1, table 1, and tables D-J in the appendix). The risk of withdrawal was increased for antidepressants (1.34, 95% confidence interval 1.22 to 1.47), benzodiazepines (1.49, 1.35 to 1.63), and gabapentin (1.61, 1.26 to 2.06), and, to a lesser degree, for antipsychotics (1.20, 0.95 to 1.51), but not for Z drugs (1.01, 0.88 to 1.15). Exposure to any single additional psychotropic medication (antidepressants or benzodiazepines or gabapentin) was associated with a 37% increased risk of withdrawal (1.37, 1.26 to 1.49) in adjusted analyses. Exposure to two or more of these medications was associated with a doubling of the risk (2.05, 1.77 to 2.37; fig 2, table K in the appendix)

### Sensitivity analyses

Our findings were corroborated in analyses adjusted for high dimensional propensity score, which showed an increased risk for antidepressants, benzodiazepines, and gabapentin. The results for any one or multiple psychotropics were upheld as well. We found no association for antipsychotics and Z drugs with relative risks of 1.03 and 1.00, respectively (fig 2, table K in the appendix). The balance in baseline characteristics was further improved with high dimensional propensity score stratification compared with propensity score stratification, in particular for antipsychotics and gabapentin (fig 1). Extension of the exposure window to 90 days before delivery and a requirement for two or more prescriptions for a psychotropic medication to be filled during the 90 days before delivery did not change the findings either (table L in the appendix). All antidepressant subclasses and specific SSRIs seemed to be associated with a small to modest increase in risk, but some of the associations were imprecisely estimated (table M in the appendix).

### Exploratory analyses

Among infants with a diagnosis of drug withdrawal, intrauterine exposure to antidepressants, benzodiazepines, or gabapentin in addition to opioids seemed to be associated with an increase in the severity of symptoms of withdrawal assessed with proxy indicators of severity, compared with exposure to opioids alone, after adjustment for opioid dose and other potential confounding variables (1.16, 95% confidence interval 1.06 to 1.27). The effect was most pronounced for seizures (1.68, 1.03 to 2.72; table 3). The mean length of stay in hospital was increased by one day, from 13 to 14 days (mean difference 1.13 days, 95% confidence interval −0.03 to 2.28; P<0.06).

Table 3

Relative risk of severe neonatal drug withdrawal according to maternal exposure to psychotropic medications and prescription opioids versus prescription opioids alone. Medicaid Analytic eXtract, 2000-10

View this table:

## Discussion

### Statement of principal findings

In this cohort of 201 275 pregnancies in women who used prescription opioids close to delivery, we observed a 30-60% increase in the risk of neonatal drug withdrawal associated with co-exposure to antidepressants, benzodiazepines, and gabapentin. Exposure to two or more of these psychotropic medications was associated with a doubling of the risk. We found no significant increase in risk with atypical antipsychotics and Z drugs (non-benzodiazepine hypnotics). The severity of the withdrawal symptoms seemed to increase with concurrent exposure to psychotropic medications and opioids.

### Comparison with other studies

Psychotropic medications, notably benzodiazepines and SSRIs, have been identified as exposures that increase the incidence and severity of neonatal drug withdrawal in studies of pregnant women with addiction. In a study of 450 women who misused drugs and were prescribed substitute methadone, maternal use of benzodiazepines significantly increased the likelihood of neonatal drug withdrawal requiring treatment (odds ratio 1.73, 95% confidence interval 1.17 to 2.55) in univariate analyses. Maternal use of SSRIs was associated with a small non-significant increase in risk (1.30, 0.63 to 2.70).25 Similarly, a small study in an urban drug treatment center found that nearly twice as many newborns (83% v 42%) of women exposed to methadone and other illicit substances including benzodiazepines (n=19) received pharmacotherapy for more severe neonatal drug withdrawal than newborns born to women receiving methadone only (n=18). There was no difference, however, in the number of days of pharmacotherapy required to treat the withdrawal, indicating that while there was a difference in the incidence of withdrawal, there was no difference in severity among affected infants.26 In contrast, a study of infants with drug withdrawal born at Boston Medical Center in 2003-09 found that maternal use of prescribed methadone for opiate addiction and benzodiazepines (n=56) compared with methadone alone (n=158) increased the length of stay for infants with drug withdrawal by six days. Maternal use of methadone and SSRIs (n=51) increased the length of stay by four and a half days.27 Based on these findings, intrauterine exposure to psychotropic medications has been used as a risk stratification factor in a study examining the risk of neonatal drug withdrawal with exposure to prescription opioids.14 We built on this work by examining the absolute and relative risk and the severity of withdrawal with co-exposure to specific psychotropics and combinations of psychotropic medications.

### Strengths and limitations

The strengths of this study relate to both internal and external validity. The large size of the exposed groups allows for the estimation of the effect sizes with good precision, reducing the risk of random error. The richness of the source data allows for adjustment for a broad range of (proxies for) potential confounding variables, reducing the risk of confounding bias. As all exposure and covariate information is recorded before the occurrence of the withdrawal, there is no concern for differential recording. The study cohort is drawn from a nationwide sample of publicly insured pregnant women, and almost half of all pregnant women in the US are publicly insured. As there is no reason to believe that the underlying biological relations would be different in this study population compared with other populations of pregnant women, we should be able to generalize from our observations to the broader experience of pregnancies inside the US and elsewhere, where the use of prescription opioids is also on the rise.28

Several limitations should also be noted. Given the nature of the data, meconium or cord toxicology is not available to document fetal exposure. As several of the opioids included in the study can be used on an as needed basis, their use might not lead to relevant fetal exposure. This results in a lower overall risk of neonatal withdrawal than that seen in women with substance use disorders or addiction or in women in medication assisted treatment programs, as has previously been documented.14 It should be noted, however, that our objective was to assess the risk of withdrawal in neonates of women who were co-prescribed opioid analgesics and other psychotropics. The fact that in many of these women fetal exposure could be minimal and will not lead to withdrawal is an important finding in itself. ICD-9 779.5x is the main code used to record drug withdrawal syndrome in the infant of an opioid exposed mother and does not enable us to distinguish a synergistic effect between opioids and psychotropics (that is, an increase in the risk of opioid withdrawal) from an independent effect of psychotropics (that is, risk of withdrawal from psychotropic medications themselves). In clinical practice, however, it is also challenging to differentiate whether the withdrawal symptoms are attributable to opioids, other medications, or a combination of the two, making the management of these infants challenging, as illustrated in the case reported in the appendix. Although we observed a doubling of the risk in infants who had been exposed to two or more psychotropic medications in addition to prescription opioids, we cannot exclude the possibility that pediatricians are more likely to diagnose withdrawal in infants with known exposure to polypharmacy. It should be acknowledged that in some cases the constellation of neonatal symptoms after in utero exposure to SSRIs could be better explained by potential toxicity associated with increased serotonin concentrations rather than withdrawal (attributable to a relative hyposerotonergic state).7 We were not able to evaluate the risk of withdrawal in infants of women on maintenance treatment with buprenorphine or methadone because such treatment is not captured comprehensively in our data, but this is unlikely to confound the analyses given our restriction to a population with prescription opioids. Our data source did not contain a quantitative measure of severity such as the Finnegan score.24 We therefore explored the effect of exposure on the severity of withdrawal using indirect proxies, which are less sensitive to changes in severity than a direct quantitative measure and which might actually reflect other co-morbidities. Likewise, length of stay as documented in healthcare use data is an imperfect measure of severity as it can be affected by unrelated medical conditions or social concerns such as the need to arrange for foster care before discharge. In some parts of the US, infants are discharged home while they are still being weaned off opioids or other medications, which would result in an underestimation of the severity of withdrawal. The relatively short hospital length of stay and the one day increase because of concomitant exposure to psychotropics should be interpreted in that context. Unfortunately, information on medications used to treat the neonate’s withdrawal symptoms in hospital was not available. We also did not have information on breastfeeding, which could reduce the severity of withdrawal.29

Other limitations are those characteristic of drug safety studies that use large administrative databases or nationwide registries. Dispensing of a medication does not necessarily imply that it was actually taken or was taken as prescribed. Given that we observed an increased risk for several of the exposure groups, this is less of a concern as misclassification of the exposure is expected to bias towards the null. Despite our focus on prescription opioids, the possibility remains that the level of exposure could have been underestimated if some women also took opioids illicitly or overestimated if some women, for example, sold or passed on their opioids and therefore did not take them. In that regard it is reassuring that we did not observe an increased risk for infants of women exposed to antipsychotics as the likelihood of abuse is high in this group of more severely ill women. As gabapentin is often used off label as an analgesic, its use could be associated with more severe pain and illicit opioid use, which could lead to residual confounding. Sensitivity analyses, however, indicate that the strength of the association between unmeasured illicit opioid use and both gabapentin exposure and neonatal drug withdrawal would have to be unrealistically high for confounding to fully explain the observed association, especially as we adjusted for pain indications and indicators of substance abuse and dependence (fig B in the appendix). Some (milder) cases might have been missed if physicians did not associate an ICD-9 diagnosis with symptoms such as crying, irritability, jitteriness, and restlessness. As long as the specificity of the outcome definition is high (as indicated by the results of our internal validation study) and the sensitivity is non-differential between exposed and unexposed pregnancies, however, the relative risk is unbiased. Finally, we were unable to examine the risk associated with the first generation typical antipsychotics given the small number of exposed pregnancies (n=119).

### Interpretation and clinical implications

In conclusion, our findings suggest that among women using prescription opioids during pregnancy, co-exposure to antidepressants, benzodiazepines, and gabapentin might be associated with an increased risk of drug withdrawal in the neonate. The findings for gabapentin are consistent with earlier observations of symptoms of withdrawal in the adult population after abrupt cessation of high dose gabapentin3031 and observations in a recently published case series of 19 infants exposed to both opioids and gabapentin prenatally.32 This is especially concerning as gabapentin is being prescribed more often for control of pain. It has also been suggested that gabapentin is sometimes used illicitly to enhance the “high” associated with methadone treatment.

Chronic pain management is a serious global health priority, with chronic pain affecting around 20% of the population. Women are affected more often than men. The use of prescription opioids has grown significantly over the past two decades, including among pregnant women. The increased risks of neonatal drug withdrawal in those born to women concomitantly using opioids and antidepressants, benzodiazepines, or gabapentin suggest that clinicians should be cautious in prescribing these medications together in late pregnancy and in prescribing psychotropic medications to women with known or suspected illicit opioid use. Because pain and mental health conditions often occur together, co-exposure will be unavoidable in many instances. In those cases, our findings might be helpful in risk stratification for exposed infants. Our findings also imply that it will be important for neonatologists and pediatricians to rethink treatment protocols for infants born to women who were prescribed multiple drugs during their pregnancy.

#### What is already known on this topic

• Neonatal drug withdrawal is common; in the US about every 25 minutes an infant is born with signs of drug withdrawal

• Neonatal drug withdrawal is a well recognized complication of intrauterine exposure to illicit or prescription opioids, but other psychotropic medications can also cause signs of withdrawal

• Psychotropic medications are often co-prescribed with opioids in pregnancy, and the use of both has increased significantly, raising concerns about an increase in the incidence and severity of neonatal drug withdrawal from potential drug-drug interactions, but these risks are not well understood

• This large population based cohort study showed a 30-60% increase in the risk of neonatal drug withdrawal associated with co-exposure to antidepressants, benzodiazepines, and gabapentin compared with opioids alone; no significant increase in risk was observed for atypical antipsychotics and Z drugs (non-benzodiazepine hypnotics)

• In utero exposure to psychotropic polypharmacy along with opioids is associated with a twofold increased risk of neonatal withdrawal

• Withdrawal symptoms seem to be more severe in neonates with concurrent exposure to opioids and psychotropic medications compared with opioids alone

## Footnotes

• We thank Mengdong He and Sara Dejene, who helped with the implementation of the outcome validation study.

• Contributors: KFH conceptualized and designed the study, carried out the analyses, and drafted the initial manuscript. BTB, RJD, SH-D, KR, and EP conceptualized and designed the study, critically reviewed the results of analyses, and reviewed and revised the manuscript. HM carried out the analyses and reviewed and revised the manuscript. LSK, JMD, and MS critically reviewed the results of analyses and reviewed and revised the manuscript. DB and JC adjudicated the neonatal withdrawal cases in the outcome validation study, critically reviewed the analyses results, reviewed and revised the manuscript. All authors approved the final manuscript as submitted. KFH is guarantor.

• Funding: This study was supported by an R01 grant (R01 MH100216) from the National Institute of Mental Health. KFH was supported by a career development grant K01MH099141 from the National Institute of Mental Health. BTB was supported by a career development grant K08HD075831 from the National Institute of Child Health and Human Development. JMD was supported by R01 (R01DA032889-04) and R21 (R21DA041706-01) grants from the National Institute of Drug Abuse. The funding organization had no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review, or approval of the manuscript; and decision to submit the manuscript for publication.

• Competing interests: All authors have completed the ICMJE uniform disclosure form at www.icmje.org/coi_disclosure.pdf and declare: SHD has consulted for AstraZeneca and UCB for unrelated topics and is co-investigator of a study funded by Eli Lilly and Pfizer, unrelated to the topic of this manuscript; KFH and BTB are co-investigators of a study funded by Eli Lilly and Pfizer, unrelated to the topic of this manuscript; BTB is the principal investigator of a study funded by Baxalta, unrelated to the topic of this manuscript; EP is co-principal investigator of a study funded by Boehringer Ingelheim and site principal investigator of a study funded by GlaxoSmithKline, both unrelated to the topic of this manuscript; no financial relationships with any companies that might have an interest in the submitted work in the previous 3 years; no other relationships or activities that could appear to have influenced the submitted work.

• Ethical approval: The research was approved by the institutional review board of Brigham and Women’s Hospital, which granted a waiver of informed consent (IRB 2013P001741).

• Data sharing: No additional data available.

• Transparency: The lead author affirms that the manuscript is an honest, accurate, and transparent account of the study being reported; that no important aspects of the study have been omitted; and that any discrepancies from the study as planned (and, if relevant, registered) have been explained.

This is an Open Access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc/4.0/.

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# How do I find output size of a network?

The input to a convolutional layer of a neural network is an image of size $$128\times 128\times 3.$$

$$40$$ convolutional filters of size $$5\times 5$$ are applied to it. Would you get an output? If no then why not? If yes, what will be the output size?

Yes, and it would be $$124 \times 124 \times40$$.
$$(dim_x - filterdim_x + 1) \times (dim_y - filterdim_y + 1) \times filters$$
$$(128 - 5 + 1) \times (128 - 5 + 1) \times 40 = 124 \times 124 \times40$$


text:
# If the latitude of station Gregg in Problem 19.21 was 45°22’58.6430” and the azimuth of the line was

If the latitude of station Gregg in Problem 19.21 was 45°22’58.6430” and the azimuth of the line was 110°33’03.8”what are the geodetic and mark-to-mark distances for this observation? (Use the GRS80 ellipsoid.)

Compute the back azimuth of a line 5863 m long in the east-west direction at a mean latitude of 45°01’32.0654”,whose forward azimuth is 88°16’33.2” from north. (Use an average radius for the Earth of 6,371,000 m.)


text: All input voltage appears across the diode. The purpose of this assignment is for students to completely understand what they have done and can summarize and analyze. C . There are two types of rectifiers, viz.- half wave rectifier and full wave rectifier. But what is its purpose? The University of Technology, Jamaica Electronic Devices 1 Rectification and smoothing circuits Monique Hepburn 1001296. In this lab, you’ll build a series of circuits, each of which will come closer to that goal. A full-wave rectifier converts the whole of the input waveform to one of constant polarity (positive or negative) at its output. Georectification is the process of adding coordinate data to a raster layer. This lab studies diodes. Rectification is a process of converting alternating current into direct current by using a diode. Laboratory 6: Power Supply Filter (smoothing capacitor) and Voltage Regulator IC Aims and Objectives ... dc voltage waveform by full wave rectification. Lab Report . L N $_\mathrm{2}$ can cause severe skin and eye "burns". A rectifier is a circuit which converts ac to dc current. Apparatus/Materials: Oscilloscope,… When superposing the two maps, putting the map of Montreal on top in the table of content and giving it transparency is a good idea as it will facilitate the rectification process. In addition the gap between the output pulses of diode current makes it more difficult to eliminate the AC ripple that remains after the rectification process. The following components should be contained for each experiment, along with any additional material required by your instructor. […] 0 C. However the results obtained are not in agreement with the solvent mixture supplied. This lab uses liquid nitrogen (LN $_\mathrm{2}$), which is very cold: approximately $77\,\mathrm{K}$. where . Apparatus/Materials: Oscilloscope, 0-12 volts A.C. power supply, 4 diodes, 56 and 120 resistors, 220 F, 500 F and 1200 F capacitors.. TEET 3145 Circuit Analysis II Lab and EENG 2111 Circuit Analysis Lab . The second message will inform you about a lack of information about the projection of the second map, this is fine as it is a goal of this lab to add this information. Diodes are widely used as a rectifier. Name of Experiment: Full Wave Rectification (using bridge rectifier) Theory: Rectification is a process by which alternating voltage is converted into a direct voltage. Aim: To design and simulate a Half Wave Rectifier … The conversion of AC into DC is called Rectification. Rectifier Wikipedia. It consists of one or more semiconductor devices or vacuum tubes. It can be done by manually entering x and y data or by lining up the raster with already rectified raster or vector data. ... Understanding Full Wave Rectification VS Half Wave Rectification ... Capacitor 2 - Basic Filtering - Electronics Lab - … • Title and date • Introduction (purpose, reaction) • Physical data (including calculations) • Procedure outline Need to report the video? Three-phase AC and 3-phase full-wave rectifier output. In any case of rectification—single-phase or polyphase—the amount of AC voltage mixed with the rectifier’s DC output is called ripple voltage. In half wave rectification, either the positive or negative half of the AC wave is passed, while the other half is blocked. Laboratory 3: Diodes and Rectification Report ELECTRICAL SYSTEMS (ELEN1000) STUDENT NUMBER: 19764023 NAME: Ethan These are called axis-aligned anisotropic Gaussian filters. The design starts with a discussion of transformer, rectification, and smoothing. By developing Eq. Writing an Organic Synthesis Lab Report . Semiconducting diode performs this work effectively. Assume the diode to be ideal i.e., V f = 0, R r = Infinite, R f = 0. View Assignment - Lab 3.pdf from ELEN 1000 at Curtin University. BACK TO TOP. Learn more about distillation here. Three sinusoidal signals have been generated in MATLAB, the third of which was a sum of the other two. It is used to separate liquids from nonvolatile solids or in the separation of two or more liquids having different boiling points. HALF-WAVE & FULL-WAVE RECTIFICATION Objectives: To recognize a half-wave rectified sinusoidal voltage. These are the three main functions that are used to set up an oscilloscope. Electronic Devices can convert AC power into DC power with high efficiency. Rates of 50-100 Hz are sufficient to … Because only one half of the input waveform reaches the output, it is very inefficient if used for power transfer. Peak Inverse Voltage Peak inverse voltage (PIV) across the diode: a parameter, which defines the choice of the diode. Consider the given circuit. is the frequency (in Hz) of the applied voltage or current and . Bridge rectifier . Full Wave And ... with smoothing' 'DIODE Half Wave Rectifier Lab Report Virtual April 20th, 2018 - Thus The Diode In half wave rectification of a single-phase supply, either the positive or negative half of the AC wave is passed, while the other half is blocked. Rectification Report Area: 316 Equipment Tag ID: 316-01-P001A Equipment Description: Hot Oil Circulation Pump B Before Corrective Action Problem: High vibration in pump causing tripping, which is directly proportional to product temperature. (10) In Eq. 2 . Independent variable for the response data y, specified as a column vector.If you do not provide x, methods that require x assume x = 1:length(y).Specify x data when y is not sorted or uniformly spaced. 3 7-Calculate the D C output voltage and compare with the measured value .C. OBJECTIVE To investigate rectification and smoothing circuits.. PRELAB/THEORY Half-wave rectification The diagram below shows a half wave rectifier. The diagram in the figure below shows the full-wave rectification of three-phase AC. The use of these controls is described in the following sections of this lab. is the capacitance in farads. Title: RECTIFICATION AND SMOOTHING OF AN A.C. VOLTAGE Aim: To investigate the rectification and smoothing of an alternating (a.c.) voltage. AFTER COMPLETING THIS LAB, YOU SHOULD When using a smoothing capacitor, the time between the peaks is much greater for a half wave rectifier than for a full wave rectifier. Half-wave Rectifier with Smoothing Capacitor For example 6V RMS AC is rectified to full wave DC of about 4.6V RMS (1.4V is lost in the bridge rectifier), with smoothing this increases to almost the peak value giving 1.4 × 4.6 = 6.4V smooth DC. Hence based on the results obtained it is possible that instead of cyclohexane and toluene, the mixture used for the experiment was toluene and hexane or methanol and toluene. Now you are required to obtain a constant ... available at the end of this lab script. Filter the image with anisotropic Gaussian smoothing kernels. The reverse operation is performed by the inverter.. Mean By Rectification A Plus Topper. * Make some research on Three Phase Full Wave Full Controlled Rectifier. If x is not uniform and you do not specify method, lowess is used.If you specify a smoothing method that requires x to be sorted, the function automatically sorts the x data. What do you mean by rectification? Figure 3.12b & c Half-wave rectifier with smoothing capacitor. Diode as a Rectifier: A rectifier is a device that converts alternating current to direct current. Title: RECTIFICATION AND SMOOTHING OF AN A.C. VOLTAGE Aim: To investigate the rectification and smoothing of an alternating (a.c.) voltage. f . 3. BEFORE COME TO THIS LAB, YOU SHOULD * Read the technical background on the report. As with resistance, reactance obeys Ohm’s law: Impedance . Formal lab report writing is a clear and consistent description that you make while you record information. If rectification and smoothing with a low-pass filter is performed with hardware prior to sampling and storing data in the computer, the sampling rate could be drastically reduced because of the reduced bandwidth of the linear envelope. imgaussfilt allows the Gaussian kernel to have different standard deviations along row and column dimensions. (7), using Eqs. Components of a Laboratory Notebook. Lab Report Solayman EWU. developed across the resistor. With this lab you will understand the characteristics of three phase full wave full controlled rectifier, measure the voltage and current values. Ripple Voltage. Half wave rectification is not much effective in producing DC output from a 50Hz or 60Hz AC input. The goal of this lab is to learn to Georectify a raster to a given coordinate system. Specify a 2-element vector for sigma when using anisotropic filters. 1-Connect the circuit shown in figure 3.2-Sketch the output voltage waveform .3-Measure the peak voltage of the input waveform .Vp(in) =----- 4-Measure the peak voltage of the output waveform .Vp(out) =----- 5-Calculate the peak voltage of the output waveform and compare with the You will find the relationship between the voltage and current in a diode, and study temperature effects, rectification, nonlinear phenomena, and frequency doubling. A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction. Half Wave With Capacitive Filter YouTube. (10), P D represents the losses in the rectifier ( R D is the equivalent resistance of the rectifier). In half wave rectification, either the positive or negative half of the AC wave is passed, while the other half is blocked. Further pages examine the choice of components and design ideas for voltage regulation circuits. To understand the effect of a reservoir capacitor upon the rectified waveform and its mean value. The boiling point of cyclohexane is 80.7 0 C and the boiling point of toluene is 110.6. Distillation, the process involving the conversion of a liquid into vapor that is subsequently condensed back to liquid form. lab, which has been detailed in this report, explored the use of MATLAB to generate sinusoidal signals. Throughout these pages I will aim to provide a thourough and detailed explanation, with "rules of thumb"; and more detailed calculations for those who wish to use them. (a) Set the load resistance to … The rectification ratio ( ), also known as rectification efficiency, is expressed by DC LD P PP = + (7) where PVIDC DC DC= (8) PVILLL= (9) 2PRIDDL= . Theory: Rectification is the process of converting alternating (ac) voltage to a direct (dc) voltage. Video Tutorial . To understand the term Zmean value as applied to a rectified waveform. BASIC ELECTRONICS - RECTIFICATION AND FILTERING PURPOSE Suppose that you wanted to build a simple DC electronic power supply, which operated off of an AC input (e.g., something you might plug into a standard outlet). Diodes and Diode Circuits TLT-8016 Basic Analog Circuits 2005/2006 8 Half - Wave Rectifier with Smoothing Capacitor Figure 3.12a Half-wave rectifier with smoothing capacitor. Smoothing is not perfect due to the capacitor voltage falling a little as it discharges, giving a small ripple voltage. In this lab we address about the characteristics of half wave diode rectifier circuits. The phasor addition theorem has been employed to calculate the resulting amplitude and phase of the . Introduction to Oscilloscopes Lab Experiment Instrument Controls The controls of a typical oscilloscope can be grouped into three major categories: vertical, horizontal, and trigger. 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# Foot area

Which animal will leave deeper footstep: an elephant weighing 5 tons with a total foot area of 0.5 square meters, or a gazelle weighing 10 kg and a foot area of 50 cm square?

Result

x = (Correct answer is: G)

#### Solution:

$m_{1}=5 \ t=5 \cdot \ 1000 \ kg=5000 \ kg \ \\ S_{1}=0.5 \ \text{m}^2 \ \\ \ \\ m_{2}=10 \ \text{kg} \ \\ S_{2}=50 \ cm^2=50 / 10000 \ m^2=0.005 \ m^2 \ \\ \ \\ g=10 \ \text{m/s}^2 \ \\ \ \\ p_{1}=m_{1} \cdot \ g/S_{1}=5000 \cdot \ 10/0.5=100000 \ \text{Pa} \ \\ p_{2}=m_{2} \cdot \ g/S_{2}=10 \cdot \ 10/0.005=20000 \ \text{Pa} \ \\ \ \\ p_{2}>p_{1} \ \\ \ \\ x=G$

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# Worksheet on Decimal Word Problems | Decimal Word Problems with Answers

Worksheet on Decimal Word Problems will help the students to explore their knowledge of decimal word problems. Solve all the problems to learn the depth concept of Decimal Word Problems. Know the definition, properties, different operations performed on decimals by visiting our website. We have given the complete decimal concepts along with examples. Check out the Decimal Word Problems Worksheet and know the various strategies to solve problems in an easy way.

## What are Decimals?

The decimal is a number that has the whole number and the fractional part separated by a decimal point. The point between the whole number and fractions part is known as the decimal point. For example, 24.9 is the decimal number where 24 is a whole number and 9 is the fractional part after the decimal. We have included addition, subtraction, multiplication, and division decimal word problems below.

### Word Problems on Decimals

Have a look at all the problems given below and get a grip on all types of problems on decimals.

1. Anshika scored 521 marks out of 600 in the final examination. How many marks did she lose?

Solution:

Given that Anshika scored 521 marks out of 600 in the final examination.
So, the total marks are 600.
She scored 521 marks.
To get the marks she loses in an examination, we have to subtract 521 marks from 600.
Now, subtract 521 from 600.
600 – 521 = 79 marks.

Anshika lost 79 marks in the final examination.

2. Arun had 0.24 liters of cold drink. Kiran had 0.68 liters more cold drink. How much cold drink did they have together?

Solution:

Given that Arun had 0.87 liters of cold drink. Kiran had 0.92 liters more cold drink.
Arun = 0.24 litres of cold drink
Kiran = 0.68 liters more cold drink
To find the cold drink they have together we have to add both cold drink liters.
So, add 0.24 liters of cold drink and 0.68 liters more cold drink
0.24 liters of cold drink + 0.68 liters more cold drink = 0.92 liters of cold drink.

They have 0.92 liters of cold drink together.

3. The weight of a baby lion was 169.24 kg. After two years, his weight increased by 107.64 kg. Find the weight of the lion after two years?

Solution:

Given that the weight of a baby lion was 169.24 kg. After two years, his weight increased by 107.64 kg.
The lion’s present age is 169.24 kg.
After two years, his weight increased by 107.64 kg.
Now, add both weights to get the lion’s age after two years.
So, add 169.24 kg and 107.64 kg
169.24 kg + 107.64 kg = 276.88 kg

After two years, the lion weight is 276.88 kg.

4. Ganesh had a rope of 54.28 m. He cut the rope into two pieces. If the length of one piece was 13.26 m, what was the length of the other piece?

Solution:

Given that Ganesh had a rope of 54.28 m. He cut the rope into two pieces.
The total length of the rope is 54.28 m.
The rope is cut into two pieces.
If the length of one piece was 13.26 m, let the other piece’s length is x.
Add two pieces 13.26 m and x m.
13.26 m + x m = 54.28 m
x = 54.28 m – 13.26 m
x = 41.02 m

The length of the other piece is 41.02 m.

5. Each side of a regular polygon is 21.7 m and its perimeter is 108.5 m. Find the number of sides of the polygon?

Solution:

Given that each side of a regular polygon is 21.7 m and its perimeter is 108.5 m.
The perimeter of a polygon is the sum of all sides of a polygon.
To find the number of sides of the polygon, divide the perimeter with each side of a regular polygon is 21.7 m.
Divide 108.5 m with 21.7 m
108.5 m/21.7 m = 5.

The number of sides of the polygon is 5.

6. Fariha took 2.4 minutes to complete the race and Anil took 2.1 minutes to complete the race. Who won the race?

Solution:

Given that Fariha took 2.4 minutes to complete the race and Anil took 2.1 minutes to complete the race.
To find the winner of the race, we need to compare the time taken by each of them. The one who traveled in less time is the winner of the race.
Fariha took 2.4 minutes.
Anil took 2.1 minutes.
2.1 minutes is less time compared to 2.4 minutes.
Therefore, Anil took less time to complete the race.

Anil won the race.

7. The annual rainfall received by Arunachal Pradesh is 261.5 cm and that by Assam is 297.4 cm. Who received less rainfall?

Solution:

Given that the annual rainfall received by Arunachal Pradesh is 261.5 cm and that by Assam is 297.4 cm.
To find the less rainfall received, we need to compare the rainfall in two states. The place where the less cm rainfall occurred will be the answer.
Arunachal Pradesh = 261.5 cm rainfall
Assam = 297.4 cm rainfall
261.5 cm rainfall is less time compared to 297.4 cm rainfall.

8. Rishu’s height is 134.51 cm. She stands on a tool of height 9.40 cm. What is the combined height now?

Solution:

Given that Rishu’s height is 134.51 cm. She stands on a tool of height 9.40 cm.
To find the combined height, add both given heights.
Add 134.51 cm and 9.40 cm
134.51 cm + 9.40 cm = 143.91 cm.

The combined height is 143.91 cm.

9. The milkman delivers 6.03 liter of milk to a house in the morning and 3.230 liters in the evening. What is the total quantity of milk delivered by the milkman?

Solution:

Given that the milkman delivers 6.03 liter of milk to a house in the morning and 3.230 liters in the evening.
To get the total quantity of milk delivered by the milkman, combine the milk he delivers in the morning and also in the evening.
Milkman delivers 6.03 liter of milk in the morning
Milkman delivers 3.230 liters of milk in the evening
Add 6.03 liter of milk and 3.230 liters of milk.
6.03 liter of milk + 3.230 liters of milk = 9.26 liters of milk

The total quantity of milk delivered by the milkman = 9.26 liters of milk.

10. Rebecca‘s kite is flying at a height of 28.3 m and Shelly’s at a height of 32.6 m from the ground. Whose kite is flying high and by how much?

Solution:

Given that Rebecca‘s kite is flying at a height of 28.3 m and Shelly’s at a height of 32.6 m from the ground.
To find whose kite is flying high, compare the two heights. The largest height of the kite considered to be flying high.
Rebecca‘s kite is flying at a height of 28.3 m
Shelly’s at a height of 32.6 m
32.6 m is greater than the 28.3 m.
Shelly’s kite is flying high.
To find how much the kite is flying high, subtract the heights.
Subtract 28.3 m from 32.6 m.
32.6 m – 28.3 m = 4.3 m

Shelly’s kite is flying high of 4.3 m than Rebecca‘s kite.

11. A bike travels 39.2 km in 7 hours. How much distance will it travel in 1 hour?

Solution:

Given that a bike travels 39.2 km in 7 hours.
Now, to find the distance it will travel in 1 hour, we have to divide the 39.2 km by 7 hours.
Divide 39.2 km by 7 hours.
39.2 km/7 hours = 5.6km/hr

The bike travels 5.6km in one hour.

12. Ron jogged 3.3 km, Mike jogged 4.8 times more distance than Ron. Find the distance covered by Mike?

Solution:

Given that Ron jogged 3.3 km, Mike jogged 4.8 times more distance than Ron.
The distance covered by Mike, multiply the Ron jogged km with the Mike jogged 4.8 times more distance than Ron.
The distance covered by Ron 3.3 kilometers.
Mike jogged 4.8 times more distance than Ron.
Multiply 3.3 kilometers and 4.8 kilometers.
The distance jogged is 3.3 kilometers × 4.8 kilometers = 15.84 kilometers.

The distance covered by mike is 15.84 kilometers.

13. The daily consumption of milk in a house is 4.36 liters. How much milk will be consumed in 30 days?

Solution:

Given that the daily consumption of milk in a house is 4.36 liters.
To find the milk that will be consumed in 30 days, we need to multiply 30 with 4.36 liters.
Multiply 4.36 liters with 30.
4.36 liters × 30 = 130.8 liters

130.8 liters of milk will be consumed in 30 days.

14. A tin contains 13.4 liters of oil. How many such tin contains 80.4 liters of oil?

Solution:

Given that a tin contains 13.4 liters of oil.
To find the number of tins that contain 80.4 liters of oil, we need to divide the 80.4 liters of oil by 13.4 liters of oil.
Divide 80.4 liters of oil by 13.4 liters of oil
80.4 liters of oil/13.4 liters of oil = 6.

6 tins contain 80.4 liters of oil.

15. Find the cost of 58.3 m cloth if the cost of 1 m cloth is $44.80. Solution: Given that the cost of 1 m cloth is$44.80
To find the cost of 58.3 m cloth, multiply 58.3 m cloth with $44.80. Multiply 58.3 m cloth with$44.80
58.3 m cloth × $44.80 =$2611.84

The cost of 58.3 m cloth is $2611.84. 16. Shruti bought a bag for$387.05. She gave the shopkeeper 2 notes of $200. How much money will she get back? Solution: Given that Shruti bought a bag for$387.05. She gave the shopkeeper 2 notes of $200. she gave 2 notes of 200 Add 2 notes cost = 200 + 200 = 400. Now, subtract$400 from $387.05 to get the amount she gets back from the shopkeeper. Subtract$400 from $387.05.$400 – $387.05 =$12.95

Shruti gets back $12.95 from the shopkeeper. 17. A tailor needs 46.36 m of cloth for the shirts and 56.90 m for trousers. How much cloth does the tailor need in all? Solution: Given that a tailor needs 46.36 m of cloth for the shirts and 56.90 m for trousers. The tailor needs all will get by adding the cloth for the shirts and cloth for the trousers. 46.36 m of cloth for the shirts 56.90 m of cloth for trousers. Add 46.36 m and 56.90 m 46.36 m + 56.90 m = 103.26 m The tailor needs 103.26 m cloth in total. 18. A spool of thread has a thread measuring 97.60 m. If 53.44 m thread has been cut, what length of thread is still left in the spool? Solution: Given that a spool of thread has a thread measuring 97.60 m. If 53.44 m thread has been cut, subtract the 53.44 m thread from the thread measuring 97.60 m. The total amount of thread is 97.60 m. The thread that has been cut is 53.44 m. Subtract 53.44 m from 97.60 m. 97.60 m – 53.44 m = 44.16 m. The length of thread is still left in the spool is 44.16 m. 19. The cost of a chair is$3056.94. Tania wants to buy 7 chairs for her house. How much money will she pay to the shopkeeper?

Solution:

Given that the cost of a chair is $3056.94. Tania wants to buy 7 chairs for her house. To find the money she will pay to the shopkeeper, multiply the number of chairs by the cost of the chair. The cost of a chair is$3056.94.
The number of chairs is 7.
Multiply $3056.94 with 7.$3056.94 × 7 = $21398.58 She will pay$21398.58 to the shopkeeper.

20. David has a jug full of milk. He pours the complete milk in 5 glasses, each glass of capacity 0.6 l. How much milk was there in the jug?

Solution:

Given that David has a jug full of milk. He pours the complete milk in 5 glasses, each glass of capacity 0.6 l.
To find the total milk in the jug, multiply 5 glasses of milk with 0.6 l capacity.
The total number of glasses = 5.
Each glass capacity is 0.6 l
Multiply 0.6 l with 5.
0.6 l × 5 = 3 l.

The total milk in the jug is 3 l.

21. Find the area of a square whose side is 2.50 m.

Solution:

Given that the side of a square is 2.50 m.
To find the area of a square, multiply 2.50 m with 2.50 m.
2.50 m × 2.50 m = 6.25 m.

The area of a square whose side is 2.50 m is 6.25 m.

22. The weight of 1 bag of sugar is 11.4 kg. What is the weight of 14 such bags?

Solution:

Given that the weight of 1 bag of sugar is 11.4 kg.
To find the weight of 14 such bags, multiply the weight of 1 bag of sugar with 14 bags.
Multiply the weight of 1 bag of sugar 11.4 kg with 14.
11.4 kg × 14 = 159.6 kg

The weight of 14 such bags is 159.6 kg.

23. A vehicle covers a distance of 10.2 km in 3.4 liters of petrol. How much distance will it cover in 1 liter of petrol?

Solution:

Given that a vehicle covers a distance of 10.2 km in 3.4 liters of petrol.
To find the distance that covers 1 liter of petrol, divide the total distance by given liters of petrol.
The total distance covered is 10.2 km.
The total liters of petrol is 3.4 liters.
Divide 10.2 km by 3.4 liters.
10.2 km/3.4 liters = 3 km/liter.

3 km covered in 1 liter of petrol.

24. Ron has 5.60 l of juice. He pours it into 8 glasses equally. How much juice is there in each glass?

Solution:

Given that Ron has 5.60 l of juice. He pours it into 8 glasses equally.
The total amount of juice is 5.60 l.
The number of glasses is 8.
To find out the juice in each glass, divide the 5.60 l by 8.
divide the 5.60 l by 8
5.60 l/8 = 0.7 l.

0.7 l juice is there in each glass.

25. Shelly has a ribbon of length 44.44 m. She cuts it into 11 equal parts. What is the length of each equal part?

Solution:

Given that Shelly has a ribbon of length 44.44 m. She cuts it into 11 equal parts.
To find the length of each equal part, divide the length of the ribbon by the number of equal parts.
The ribbon length is 44.44 m.
The number of equal parts is 11.
Divide 44.44 m by 11.
44.44 m/11 = 4.4 m.

4.4 m is the length of each equal part.

26. The cost of 4 pens is $49.24. What is the cost of 1 pen? Solution: Given that the cost of 4 pens is$49.24.
To find the cost of each pen, divide the total amount by the number of pens.
The total cost of the pens is $49.24. The total number of pens is 4. Divide$49.24 by 4.
$49.24/4 =$12.31

The cost of 1 pen is \$12.31.

27. The weight of a box is 141.029 kg. What will be the weight of 23 such boxes?

Solution:

Given that the weight of a box is 141.029 kg.
To find the weight of 23 such boxes, multiply 141.029 kg by 23.
Divide 141.029 kg by 23.
141.029 kg × 23 = 3243.667 kg.

The weight of 23 such boxes is 3243.667 kg.

28. Sonia has 18.48 l of juice. She pours it into 8 jars equally. How much juice is there in each jar?

Solution:

Given that Sonia has 18.48 l of juice. She pours it into 8 jars equally.
To find the amount of juice in each jar, divide the total liters of juice by the number of jars.
The total liters of juice is 18.48 l.
The number of jars is 8.
Divide 18.48 l of juice by 8 jars.
18.48 l/8 = 2.31 liters

2.31 liters of juice present in each jar.


text:
## Lesson Instructions.

Students are kindly requested to follow the instruction: 1. Please watch the videos before attending the online class.2. Write down the points you did not understand.3. Please go through the ppt slide by slide and must take note. 4. Attend all interactive activities in this topic. If you do not complete your interactive activities. 5. You will be given chance to ask questions during online class. Please ask questions. 6. You must attend all practice quizzes and video quizzes. Once you have competed the task mark the same as completed.

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