Solving Ratio Problems
A ratio compares quantities. The common SAT task is splitting a total according to a ratio — which means working with the sum of the parts.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
For a ratio $a : b$ and a total $T$, each share is the fraction of the total: $\frac{a}{a+b} \cdot T$ and $\frac{b}{a+b} \cdot T$. Keep part-to-part and part-to-whole straight.
How to solve one, step by step
Example: split $30$ in the ratio $3 : 2$.
- Parts total $3 + 2 = 5$.
- Shares: $\frac{3}{5}(30) = 18$ and $\frac{2}{5}(30) = 12$.
The mistakes that cost points
- Inverting the ratio. Keep track of which part goes with which quantity, and whether it's part-to-part or part-to-whole.
- Adding parts instead of distributing. Divide the total by the sum of the parts, then scale each share.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A recipe uses flour and sugar in a ratio of $4:1$. If the total amount of flour and sugar is $k$ cups and there are 8 cups of flour, what is the value of $k$?
- A$k = 9$
- B$k = 2$
- C$k = 8$
- D$k = 10$
Show solution
Answer: D, $k = 10$. Ratio $4:1$ means for every 4 cups of flour, there is 1 cup of sugar. If flour $= 8$, then sugar $= 8 \div 4 = 2$. Total $= 8 + 2 = 10$.
Medium
Three friends split a prize in the ratio $2:3:k$. The total prize is $\$360$ and the smallest share is $\$72$. What is the value of $k$?
- A$k = 5$
- B$k = 2$
- C$k = 10$
- D$k = 20\%$
Show solution
Answer: A, $k = 5$. Total parts $= 2 + 3 + k = 5 + k$. Value per part $= \frac{360}{5+k}$. The smallest share (ratio 2) equals 72: $2 \cdot \frac{360}{5+k} = 72 \Rightarrow \frac{720}{5+k} = 72 \Rightarrow 5+k = 10 \Rightarrow k = 5$.
Hard
The ratio of boys to girls in a class is $3:k$. There are 12 boys and 20 students total. What is the value of $k$?
- A$k = 2$
- B$k = 60\%$
- C$k = 8$
- D$k = 20$
Show solution
Answer: A, $k = 2$. Girls $= 20 - 12 = 8$. Ratio boys to girls $= 12:8$. Simplify: $12:8 = 3:2$. So $k = 2$.
Find your exact gaps
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