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13 Mar 2026, 11:53
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
67% (02:50) correct
33%
(02:46)
wrong
based on 30
sessions
History
Date
Time
Result
Not Attempted Yet
A total of 120 people visited an open-air market. Of these people, 36 bought lentils, 37 bought chickpeas, and 35 bought peas.
Among the people who bought at least two of these legumes, one-third bought all three. For every one person who bought at least two of these legumes, two people bought none of them.
How many people bought none of these legumes?
A. 24
B. 30
C. 36
D. 48
E. 60
Among the people who bought at least two of these legumes, one-third bought all three. For every one person who bought at least two of these legumes, two people bought none of them.
How many people bought none of these legumes?
A. 24
B. 30
C. 36
D. 48
E. 60
13 Mar 2026, 12:31
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This is an Inclusion-Exclusion (Overlapping Sets) problem, and the key is setting up one variable smartly using the ratio relationship given in the problem. The trap most students fall into is trying to use the full Venn diagram formula (L + C + P - both pairs + all three + none = total) without first noticing that the problem gives you a clean ratio between "at least two" and "none."
Key concept: Inclusion-Exclusion with a bridging ratio.
Let x = number of people who bought at least two legumes.
Then: people who bought all three = x/3 (one-third of "at least two")
And: people who bought none = 2x (two people bought none for every one who bought at least 2)
Step 1: Use the identity for the sum of individual group counts.
L + C + P = (exactly 1) + 2(exactly 2) + 3(all three)
36 + 37 + 35 = 108
Step 2: Express the sub-groups in terms of x.
Exactly 2 = x - x/3 = 2x/3
All three = x/3
So: 108 = (exactly 1) + 2(2x/3) + 3(x/3) = (exactly 1) + 4x/3 + x = (exactly 1) + 7x/3
Step 3: Use the total to express "exactly 1" in terms of x.
Total = (exactly 1) + (at least 2) + none
120 = (exactly 1) + x + 2x
→ exactly 1 = 120 - 3x
Step 4: Substitute into the equation from Step 2.
108 = (120 - 3x) + 7x/3
108 - 120 = -3x + 7x/3 = x(-9/3 + 7/3) = -2x/3
-12 = -2x/3
x = 18
Step 5: Calculate "none."
None = 2x = 2(18) = 36
Quick check: exactly 1 = 120 - 3(18) = 66; exactly 2 = 12; all three = 6
Total = 66 + 12 + 6 + 36 = 120 ✓
L + C + P = 66 + 2(12) + 3(6) = 66 + 24 + 18 = 108 ✓
Answer: C (36)
Takeaway: In three-set Overlapping Sets problems, always look for a ratio relationship connecting your unknowns — it lets you work with one variable instead of three, and the problem unravels very quickly.
Key concept: Inclusion-Exclusion with a bridging ratio.
Let x = number of people who bought at least two legumes.
Then: people who bought all three = x/3 (one-third of "at least two")
And: people who bought none = 2x (two people bought none for every one who bought at least 2)
Step 1: Use the identity for the sum of individual group counts.
L + C + P = (exactly 1) + 2(exactly 2) + 3(all three)
36 + 37 + 35 = 108
Step 2: Express the sub-groups in terms of x.
Exactly 2 = x - x/3 = 2x/3
All three = x/3
So: 108 = (exactly 1) + 2(2x/3) + 3(x/3) = (exactly 1) + 4x/3 + x = (exactly 1) + 7x/3
Step 3: Use the total to express "exactly 1" in terms of x.
Total = (exactly 1) + (at least 2) + none
120 = (exactly 1) + x + 2x
→ exactly 1 = 120 - 3x
Step 4: Substitute into the equation from Step 2.
108 = (120 - 3x) + 7x/3
108 - 120 = -3x + 7x/3 = x(-9/3 + 7/3) = -2x/3
-12 = -2x/3
x = 18
Step 5: Calculate "none."
None = 2x = 2(18) = 36
Quick check: exactly 1 = 120 - 3(18) = 66; exactly 2 = 12; all three = 6
Total = 66 + 12 + 6 + 36 = 120 ✓
L + C + P = 66 + 2(12) + 3(6) = 66 + 24 + 18 = 108 ✓
Answer: C (36)
Takeaway: In three-set Overlapping Sets problems, always look for a ratio relationship connecting your unknowns — it lets you work with one variable instead of three, and the problem unravels very quickly.
14 Mar 2026, 02:11
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This question can be borderline considered as hard because of the framing.
Equation:
108 - 2/3x - 2/3x + 2x = 120
Solving for x you get 18 and None = 2x = 36
Answer: Option C
Equation:
108 - 2/3x - 2/3x + 2x = 120
Solving for x you get 18 and None = 2x = 36
Answer: Option C
kevincan
