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### F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

### Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won \$1 200. What were the assets of the players at the beginning of the evening?

### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

# Mixing Lemonade

##### Stage: 3 Challenge Level:

Lots of great solutions were submitted to this problem, using a wide variety of approaches. The problem prompted you to use fractions, ratios, percentages, and graphs. In addition, you could investigate and consider which methods worked most effectively in different situations.

The first part of the task is to determine the mixture with the stronger tasting lemonade -

the first glass has $200$ml of lemon juice, and $300ml$ water
the second glass has $100$ml of lemon juice, and $200$ml water

What is the best way to compare these two mixtures? There are several different ways, as shown by the different explanations submitted.

Mahir, from Saltus Grammar School converted the values given so that there is the same amount of water in each glass:

In the first glass, there is $300$ml water, and in the second, there is $200$ml. Therefore, to equate the amounts of water, he multiplied everything in the first glass by two, and everything in the second glass by three. Each glass now has $600$ml water, and so can now be compared. Note that it is "ok" to multiply everything in each glass by a certain number as it is the proportions or relative amounts we are interested in, rather than the absolute amounts.

For the first glass, we now have:

$400$ml lemon juice and $600$ml water

For the second glass, we now have:

$300$ml lemon juice and $600$ml water

From this, we can now tell that the mixture in the first glass must taste stronger: for the same amount of water, there is more lemon juice.

Jonathan, from Wilson's School, used a similar method. However, he instead made the amounts of lemon juice equal, and then saw which glass had more water. The glass with less water for the same amount of juice will be stronger, as the juice is less diluted.

Another related method, used by many people was to use ratios, fractions and/or percentages. Will K. from Wilson's School gave a lovely explanation:

Glass 1 would be stronger as it has a simplified ratio of lemon juice: water of $2:3$, and so is $\frac{2}{5}$ lemon juice, or $40$%, whereas glass $2$ has a simplified ratio of $1:2$ for lemon juice: water, and so is $\frac{1}{3}$ lemon juice, or $33.\dot{3}$ % lemon juice, and so is weaker than glass $1$.

The strategy is to work out the ratio (e.g. $40$ml lemon juice to $120$ml water would be $40:120$ lemon juice: water), simplify it (e.g. $40:120$ simplifies to $4:12$, which simplifies to $1:3$), and then turn that into a fraction ($1:3$ would be $\frac{1}{4}$ as $1 +3 = 4$ and so the $1$ part that is lemon juice is $\frac{1}{4}$ of the drink), and then turn that into a percentage ($\frac{1}{4}$ into $25$%). The glass with the highest percentage being lemon juice (or the lowest percentage being water) would be the strongest glass of lemonade.

Will expressed the strength of the lemonade as a percentage, as these can be easily compared. Sharumilan, also from Wilson's School converted the fractions so that they had a common denominator. In this way, they can be more easily compared. In fact, this is the same as converting to percentages; percentages are fractions with a denominator of $100$! Here is Sharumilan's answer:

The first glass has the stronger tasting lemonade because $\frac{2}{5}$ ($\frac{6}{15}$) of it is lemon juice while only $\frac{1}{3}$ ($\frac{5}{15}$) of the lemonade in the second glass is lemon juice.

What about a more visual approach? Iona, from Whitby Maths Club compared the two solutions by drawing these graphs. Dominic, from Wilson's School, also suggested a graphical approach:

If you wanted to set it out in a pie chart you could easily see which mixture was sronger. The graph with the biggest chunk of lemon in it would be the strongest.

Another suggestion made by several people would be to convert the amounts so that there are equivalent amounts of water or lemon juice, as described above. Then, you could draw a glass and visually see which is the stronger mixture.

Dulan, from Wilson's School, suggested a very nice method, which can display graphically multiple different mixtures:
He thought that a graph could be constructed with $x$ and $y$ axes. On the $x$ axis could be "amount of lemon juice", and on the $y$ axis, "amount of water". Try constructing this for yourself.
What does it mean if two mixtures have the same $x$ coordinate, but different $y$ coordinates?
What if they have the same $y$ coordinates, but different $x$ coordinates?
You should be able to construct straight lines from the origin to the various points representing different mixtures and compare their strengths.
Along each line the strength of all of the mixtures is the same as the proportions do not change.

We have now seen different examples of approaches to this problem. Do the different methods always work? Which method is most efficient?

Nathan, from Wilson's School noted that different methods are more efficient in different situations:

The ratio or fraction strategy would be more efficient for difficult fractions e.g. $\frac{300}{430}$ and $\frac{290}{560}$ but using a mental method would be more efficient for numbers like $\frac{30}{50}$ and $\frac{40}{50}$, as you can see that $\frac{40}{50}$ has a more concentrated supply of lemonade.

Several people had their own preference of method, depending on what they felt most comfortable using.

Sharumilan explained this, and also examined the combination of the different mixtures:

I always use fractions because it seems easier for me and the graph helped me to get the fractions quickly so it could actually be a mix of both.

The strength of combined mixtures always has to be the same as or in between the two mixtures. I have said "the same as" because if the two mixtures are identical, the combined mixture has to be the same as well.

Try this out for yourself, with some squash, or cordial...

Thank you very much to everyone who submitted solutions. There were many correct solutions, and so we could not mention them all. Well done!

If you enjoyed this problem, try the extension problem Ratios and Dilutions.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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