12. In a survey 96 people liked strawberries, 98 liked raspberries, and 120 liked
blueberries; 18 people liked only strawberries, 20 liked only raspberries, and 24 liked
only blueberries; 38 people liked all three. How many people liked raspberries and
blueberries but not strawberries?
a. 29
b. 30
c. 31
d. 32
Venn Diagram Problem: Raspberries and Blueberries but Not Strawberries
In this survey problem, 29 people liked raspberries and blueberries but not strawberries. The calculation uses standard Venn diagram regions to derive this value precisely.
Problem Breakdown
A total of 96 liked strawberries (S), 98 raspberries (R), and 120 blueberries (B). Exactly 18 liked only S, 20 only R, 24 only B, and 38 all three. The regions are:
- Only S = 18
- Only R = 20
- Only B = 24
- All three (S∩R∩B) = 38
Let AB = S∩R not B, AC = S∩B not R, BC = R∩B not S (the target).
Step-by-Step Solution
Total S equation:
18 + AB + AC + 38 = 96
AB + AC = 40
Total R equation:
20 + AB + BC + 38 = 98
AB + BC = 40
Total B equation:
24 + AC + BC + 38 = 120
AC + BC = 58
Solve the system:
From AB + AC = 40 and AB + BC = 40, subtract to get AC = BC.
Then AC + BC = 58 becomes 2AC = 58 → AC = 29.
BC = 40 – AB and AC = 40 – AB, so (40 – AB) + (40 – AB) = 58 → 80 – 2AB = 58 → 2AB = 22 → AB = 11.
AC = 40 – 11 = 29, BC = 40 – 11 = 29.
Option Analysis
| Option | Value | Explanation |
|---|---|---|
| a. | 29 | Correct – Matches BC = 29 from equations |
| b. | 30 | Incorrect – Off by 1; perhaps from rounding AC + BC ≈58 without solving. |
| c. | 31 | Incorrect – Too high; ignores AB constraint. |
| d. | 32 | Incorrect – Furthest; maybe total B – only B – all three =58, misassigning. |
Target region BC (Raspberries and Blueberries but not Strawberries) = 29 people.
