Part-to-part / part-to-whole swap sprint

Treat a part-to-part ratio as part-to-whole (or vice versa).

An error-pattern drill: practice spotting and avoiding this specific SHSAT trap.

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How to avoid mixing up part-to-part and part-to-whole

A class has boys and girls in a ratio of 3:5, and 40 students in all. How many boys are there? The trap is to take \(\tfrac{3}{5}\) of 40, which is 24. But 3:5 compares boys to girls, two parts. The whole is \(3 + 5 = 8\) parts. So the number of boys is \(\tfrac{3}{8} \times 40 = 15\).

This trap works because a ratio like 3:5 looks like the fraction \(\tfrac{3}{5}\). It only means "3 out of 5" when the 5 is the total.

Check habits:

Worked examples

Example 1.
A movie list has 15 movies, split between comedies and dramas in the ratio 2:3. How many comedies are there?
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Answer:
C

Total parts \(= 2 + 3 = 5\), so each part is worth \(15 \div 5 = 3\) movies. Comedies \(= 2 \times 3 = 6\). (Using \(2/3\) of the total instead of \(2/5\) gives 10. Finding the dramas instead gives 9. Stopping at the size of one part gives 3.)

Example 2.
A school orchestra has violins and cellos in the ratio 4:7. What fraction of the instruments are violins?
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Answer:
B

Total instruments \(= 4 + 7 = 11\). Fraction that are violins \(= 4/11\), which is already in lowest terms. (Writing the part-to-part ratio 4:7 as a fraction gives 4/7. Finding the fraction that are cellos gives 7/11. Putting the total on top gives 11/4.)

Example 3.
A team's season has 8 wins and 13 losses. What is the ratio of wins to the total number of games?
Show answer and explanation
Answer:
B

Total games \(= 8 + 13 = 21\). Ratio of wins to total \(= 8:21\), which is already in lowest terms. (Using losses instead of wins against the total gives the wrong answer 13:21. Writing the total first gives 21:8. Leaving it as a part-to-part ratio gives 8:13.)