MCAT Physics and Math Review
Chapter 10: Mathematics
Going to the grocery store is not so different from solving an MCAT multiple choice question. You begin the process by determining how much of each item you need in the near future. Once you know what you need, you check what you already have in order to determine the quantity you need to buy to reach your goal. When you get to the store, you compare that amount to the containers on the shelves. Often they won’t match exactly. Say, for example, that you need a total of 16 ounces of peas for a recipe. You already have five at home, so you only need 11 more. Packages of peas, however, may only come in 10- or 16-ounce packages. At that point, you choose the best one for your needs—the 16-ounce package; better to have a little extra than to run short!
If you’ve ever shopped in an international grocery store, this process can become even more elaborate because the packaging sizes and currency may not match the units with which you are familiar. You may have never taken the time to consider how intensive one’s critical thinking must be to efficiently navigate the grocery store, but recognize that it’s the same process you need to use on Test Day. First, figure out what you want (what is the question looking for), what you have (information in a passage, question stem, or outside knowledge), and what’s needed (calculations and critical thinking), and then make a decision (by matching your answer, eliminating wrong answer choices, or guessing strategically). In this chapter, we’ll be focusing on the calculations and critical thinking of mathematics. The math required for the MCAT is on the level of precalculus. You won’t need any derivatives or integrals on Test Day, but rapid application of arithmetic, exponent and logarithm rules, trigonometry, statistics, and graphical analysis may be necessary to navigate the MCAT efficiently. In this chapter, you won’t see very much new content, but consider this an opportunity to hone your mathematics skills.
10.1 Arithmetic and Significant Figures
The MCAT often uses numbers that aren’t particularly “nice” looking, especially considering that calculators cannot be used on the test. However, the testmakers also know that calculators aren’t allowed, so even the most complex math still has to be solvable in a reasonable amount of time. We reconcile these two opposing concepts by using a few Test Day tricks: scientific notation, which can help us narrow down the exponent of our answer choice and often gives the answer directly; and judicious estimation, which will differentiate between otherwise similar answers. While significant figures won’t lead us to an answer in the way that the other MCAT skills will, it is a testable topic on the MCAT.
It is essential to use judicious rounding and math strategies on Test Day to get through the MCAT efficiently. This is particularly true in the Chemical and Physical Foundations of Biological Systems section. Do not try to solve for an exact answer—do only as much as you need to be able to choose the right answer choice!
Scientific notation is a style of writing numbers that takes advantage of powers of ten. In scientific notation, a number is written with a significand and an exponent. This is much easier to conceptualize with an example. Consider the number 217. The math using this number can be somewhat cumbersome. By transforming it into scientific notation (2.17 × 102), the number becomes easier to manipulate because the power of 10 has been pulled out. In this case, 2.17 is the significand (also called the coefficient or mantissa), and 102 is the exponent.
The significand must be a number with an absolute value in the range [1,10). This means that it is any real number between –10 and –1 (not including –10) or between 1 and 10 (not including 10). By extension, the significand cannot begin with a 0, nor can it begin with two digits before the decimal point. The exponent, on the other hand, can be any whole number—positive, negative, or 0.
If at any time your calculations are not in scientific notation, consider adjusting them. While there is a small time investment converting to scientific notation, the time saved on subsequent calculations usually makes up for—and often exceeds—this time investment. This is especially true for questions in which the answers differ by powers of ten. The only exception to maintaining scientific notation is in the calculation of square roots, which are discussed later in this chapter.
Significant figures provide an indication of our certainty of a measurement, and help us to avoid exceeding that certainty when performing calculations. Significant figures are determined by the precision of the instrument being used for measurement. For example, imagine that you are measuring the width of a block of wood with a ruler. The ruler has markings for centimeters and millimeters; you could state with confidence the width of the block in millimeters—say, 55 millimeters.
However, on this ruler, there are no markings smaller than millimeters; you’d be forced to estimate where within the interval between two millimeter markings the block reaches—say 55.2 millimeters. You cannot be 100 percent confident about this decimal, but some information is better than none, and writing it down lets you know that you were confident about the first two digits.
Significant figures are important because they give an indication of the accuracy of a measurement. Inaccurate measurements can bias research or lead to faulty conclusions. When presented with data, look for accuracy of the measurements in two ways: identifying the number of significant digits in a number, and looking for error margins or statistical significance in graphs. These latter topics are discussed in Chapter 12 of MCAT Physics and Math Review.
In the situation we just described only the first two digits would be considered significant because we know that they were measured accurately. We can hold on to the third digit during calculations, but by the time we reach a final answer, we need to reduce the answer to an appropriate number of significant figures. To determine the number of significant figures in a number:
· Count all numbers between the first nonzero digit on the left and the last nonzero digit on the right. Any digit between these two markers (including 0) is significant.
· Any zeroes to the left of the first nonzero digit are considered leading zeroes and are not significant.
· If there are zeroes to the right of the last nonzero digit and there is a decimal point in the number, then those zeroes are significant figures. If there is no decimal point, they are not significant. For example, 3490 has three significant figures, while 3490.0 has five.
· For measurements, the last digit is usually an estimation and is not considered significant (as in the example above).
Scientific notation can clarify significant figures when it contains a decimal point. When converting between standard numbers and scientific notation, be sure to maintain the number of significant figures. 100.0 is written in scientific notation as 1.000 × 102 while 100 is written as 1 × 102because the trailing zeroes in the first example are significant while in the second example they are not.
Math with Significant Figures
Significant figure estimations are most important in the laboratory sciences, particularly analytical chemistry. For multiplication and division, maintain as many digits as possible throughout the calculations so that there is very little rounding error, then round to the number of significant digits that is the same as the least number of significant digits in any of the factors, divisors, or dividends. With addition and subtraction, decimal points are maintained rather than maintaining significant figures. The convention for decimal points is the same as for significant figures: the answer may have only as many decimal digits as the initial number with the fewest decimal digits.
Most Test Day math (and, by extension, this Kaplan MCAT Review series) neglects significant figures in the answer choices. These calculations are only necessary when specified by the question stem or passage.
Determine the volume of a cylinder with a radius that is measured as 7.45 m and a height of 8.323 m. (Note: Use 3.14159 as π, and round the answer to the correct number of significant digits.)
Because all of the factors are multiplied, the answer should have the same number of significant digits as the factor with the fewest number of significant digits. In this case, that is the radius, which has only two significant digits (remember that, in the case of measurements, the last digit is an estimate and is not considered significant). Therefore, the correct answer is 1500 or 1.5 × 103.
On Test Day, much of your math will be determined by the answer choices provided. If the answer choices are very close together, there will be minimal opportunity for rounding; when they are far apart, rough estimations are all that are necessary. While estimation of addition and subtraction are relatively simple rounding choices, we’ll review a few tricks for multiplication and division.
Consider the following multiplication problem: (3.17 × 104) × (4.53 × 105). To three significant digits, the answer to this multiplication problem is 1.44 × 1010, but this precise calculation is beyond the scope of mental math. However, even if the answer choices are close, it is generally acceptable to round to one decimal place, or (3.2 × 104) × (4.5 × 105). When rounding numbers in multiplication, keep in mind whether the rounded number is larger or smaller than the original number. If one number is rounded up, it is best to round the other number down slightly to compensate. Even with this rounding, the answer still comes out as 1.44 × 1010.
If the answer choices are very far apart—differing by, say, powers of ten—we can adjust the numbers so that one contains only one significant digit, further simplifying the math. In this example, the calculations could be adjusted to (3 × 104) × (4.5 × 105), or 1.35 × 1010. This represents an error of 6.25%, which is still close enough to choose the correct answer for most questions on Test Day.
Let’s also consider division as an avenue for estimation. While in our multiplication example we adjusted each number in an opposite direction, with division we are attempting to make proportional adjustments in the same direction. Consider the following example:
When rounding numbers to be multiplied, round one number up and one number down to compensate. When rounding numbers to be divided, round both numbers in the same direction to compensate.
Estimate the value of 15.4 ÷ 3.80.
Estimations in division should be made by shifting both numbers in the same direction. It is often easier to adjust the divisor first to simplify calculations. If we round the divisor up to 4, we should round the dividend up accordingly. In this case, it makes sense to round the dividend up to 16—which is not only a whole number, but also a multiple of 4. Our estimate is 16 ÷ 4 = 4. Note that, despite this very rough adjustment, we are still very close to the true value of 4.05.
MCAT Concept Check 10.1:
1. Describe the process for converting a number into scientific notation. What values are possible for the significand?
2. Highlight or circle the significant digits in the following numbers:
3. When rounding two numbers containing decimals, in which direction(s) should each number go for multiplication? For division?