Ratios and Proportions on the SAT
Most ratio questions are not hard, and that is exactly why points get lost here. Students rush, use the wrong total, put the units in different places on each side of a proportion, or forget that areas and volumes do not scale the way lengths do.
Start here
A ratio of $3 : 5$ means $3$ parts of one thing for every $5$ parts of the other. To turn it into real amounts, call one part $k$. The amounts are $3k$ and $5k$, and the total is $8k$.
For rates and recipes, set up a proportion with the same units in the same places on both sides, then cross-multiply. For unit conversions, multiply by fractions equal to $1$ until the units you don't want cancel.
Worked examples
A ratio and a total
The ratio of boys to girls in a class is $3 : 5$. There are $40$ students. How many are girls?
- Boys $= 3k$, girls $= 5k$, total $= 8k$.
- $8k = 40$, so $k = 5$.
- Girls $= 5k = 25$. Using $\dfrac{5}{3}$ of something, or dividing $40$ by $5$, are the usual slips.
Combining two ratios
$a : b = 2 : 3$ and $b : c = 4 : 5$. What is $a : c$?
- The shared term $b$ is $3$ in one ratio and $4$ in the other. Make both $12$.
- $a : b = 8 : 12$ and $b : c = 12 : 15$.
- So $a : c = 8 : 15$.
A ratio that changes
A bag has red and blue marbles in the ratio $4 : 7$. After $6$ red marbles are added, the ratio is $2 : 3$. How many blue marbles are there?
- Before: red $= 4k$, blue $= 7k$. After: red $= 4k + 6$, blue still $7k$.
- $\dfrac{4k + 6}{7k} = \dfrac{2}{3}$. Cross-multiply: $12k + 18 = 14k$, so $k = 9$.
- Blue $= 7k = 63$. Check: red becomes $42$, and $42 : 63 = 2 : 3$.
Converting units
Convert $72$ kilometers per hour to meters per second.
- $72 \dfrac{\text{km}}{\text{h}} \cdot \dfrac{1{,}000 \text{ m}}{1 \text{ km}} \cdot \dfrac{1 \text{ h}}{3{,}600 \text{ s}}$.
- The kilometers and hours cancel, leaving $\dfrac{72{,}000}{3{,}600} = 20$ meters per second.
What the questions look like
- A ratio and a total. Write the parts as $3k$ and $5k$, add them, set equal to the total, solve for $k$.
- A rate, find a new amount. Proportion: $\dfrac{12 \text{ gal}}{315 \text{ mi}} = \dfrac{20 \text{ gal}}{x \text{ mi}}$. Same units across from each other.
- A scale on a map or drawing. Multiply the drawing length by the scale factor.
- Two ratios sharing a term ($a:b$ and $b:c$). Rescale both so the shared term is the same number, then read $a : c$.
- A ratio that changes after adding some. Write the before amounts as $4k$ and $7k$, add to one, set the new ratio equal, and solve.
- Similar figures. Lengths scale by $k$, areas by $k^2$, volumes by $k^3$.
Watch out for these
- Using the wrong total. In a $3 : 5$ ratio, the whole is $8$ parts. Girls are $\dfrac{5}{8}$ of the class, not $\dfrac{5}{3}$.
- Mismatched proportions. If gallons are on top on the left, gallons go on top on the right.
- Combining ratios without matching the shared term. From $2 : 3$ and $4 : 5$ you cannot just read $2 : 5$.
- Scaling area like length. If lengths are $2.5$ times bigger, the area is $2.5^2 = 6.25$ times bigger.
- Flipping a conversion factor. Check that the unit you want to get rid of appears once on top and once on the bottom.
Practice
Write the ratio as amounts with $k$ before you calculate anything.
Easy
The ratio of boys to girls in a class is $3 : 5$. There are $40$ students in the class. How many of the students are girls?
- A$15$
- B$24$
- C$25$
- D$8$
Show solution
Answer: C. There are $3 + 5 = 8$ parts, so each part is $\dfrac{40}{8} = 5$ students. Girls are $5$ parts: $25$. Choice A is the number of boys.
EasyStudent-produced response
A recipe uses $2$ cups of flour for every $3$ eggs. How many cups of flour are needed for $9$ eggs?
Enter your answer.
Show solution
Answer: $6$. $9$ eggs is $3$ times as many as $3$ eggs, so the flour triples too: $2 \cdot 3 = 6$ cups.
Medium
If $a : b = 2 : 3$ and $b : c = 4 : 5$, what is $a : c$?
- A$2 : 5$
- B$10 : 12$
- C$8 : 15$
- D$15 : 8$
Show solution
Answer: C. Make the shared term match: $a : b = 8 : 12$ and $b : c = 12 : 15$. So $a : c = 8 : 15$. Choice A reads the outer numbers without matching $b$.
MediumStudent-produced response
On a map, $1$ inch represents $25$ miles. Two towns are $3.5$ inches apart on the map. How many miles apart are the towns?
Enter your answer.
Show solution
Answer: $87.5$. $3.5 \cdot 25 = 87.5$ miles.
Medium
A car travels $315$ miles on $12$ gallons of gas. At this rate, how many miles can it travel on $20$ gallons?
- A$525$
- B$500$
- C$540$
- D$323$
Show solution
Answer: A. $\dfrac{315}{12} = \dfrac{x}{20}$, so $x = \dfrac{315 \cdot 20}{12} = 525$. Choice D adds the $8$ extra gallons as if they were miles.
MediumStudent-produced response
A train travels at $72$ kilometers per hour. What is this speed in meters per second? ($1$ kilometer $= 1{,}000$ meters)
Enter your answer.
Show solution
Answer: $20$. $72 \cdot \dfrac{1{,}000}{3{,}600} = \dfrac{72{,}000}{3{,}600} = 20$ meters per second.
HardStudent-produced response
A bag contains red and blue marbles in the ratio $4 : 7$. After $6$ red marbles are added, the ratio of red to blue marbles is $2 : 3$. How many blue marbles are in the bag?
Enter your answer.
Show solution
Answer: $63$. Write red $= 4k$ and blue $= 7k$. Then $\dfrac{4k + 6}{7k} = \dfrac{2}{3}$, so $12k + 18 = 14k$ and $k = 9$. Blue $= 7 \cdot 9 = 63$.
Hard
If $\dfrac{x}{y} = \dfrac{3}{4}$, what is the value of $\dfrac{x + y}{y}$?
- A$\dfrac{3}{7}$
- B$\dfrac{4}{7}$
- C$\dfrac{7}{3}$
- D$\dfrac{7}{4}$
Show solution
Answer: D. $\dfrac{x + y}{y} = \dfrac{x}{y} + 1 = \dfrac{3}{4} + 1 = \dfrac{7}{4}$. Or let $x = 3$ and $y = 4$: $\dfrac{7}{4}$. Choice B is $\dfrac{y}{x + y}$.
Hard
A mixture of nuts contains peanuts and cashews in the ratio $5 : 3$ by weight. The mixture weighs $24$ pounds. How many more pounds of peanuts than cashews are in the mixture?
- A$2$
- B$9$
- C$15$
- D$6$
Show solution
Answer: D. $8$ parts weigh $24$ pounds, so each part is $3$ pounds. Peanuts $= 15$, cashews $= 9$, and the difference is $6$ pounds. Choice A is the difference in parts, not pounds.
Hard
Two rectangles are similar. The ratio of a side of the smaller rectangle to the matching side of the larger rectangle is $2 : 5$. The area of the smaller rectangle is $12$ square inches. What is the area of the larger rectangle, in square inches?
- A$30$
- B$75$
- C$60$
- D$187.5$
Show solution
Answer: B. Lengths scale by $\dfrac{5}{2}$, so areas scale by $\left(\dfrac{5}{2}\right)^2 = \dfrac{25}{4}$. $12 \cdot \dfrac{25}{4} = 75$. Choice A scales the area like a length.
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