Two banana bunches: the first has “one-quarter again as many” as the second, and the second has 3 fewer than the first. How many bananas are in the first bunch?
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A9
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B10
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C12
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D15
Answer
Correct Answer: 15
Explanation
Introduction / Context:This is an algebraic word problem involving proportional language. “One-quarter again as many” means the first quantity equals the second plus one-quarter of the second (i.e., 125% of the second). We also have a difference statement linking the two counts.
Given Data / Assumptions:
- Let F = number of bananas in the first bunch.
- Let S = number of bananas in the second bunch.
- “First has one-quarter again as many as second” means F = (5/4) * S.
- “Second has 3 less than first” means S = F - 3.
Concept / Approach:Translate the English phrases into equations, then solve the simultaneous system. Substitute one equation into the other to find the exact integer values that satisfy both conditions.
Step-by-Step Solution:
F = (5/4) * S.S = F - 3.Substitute S into the first: F = (5/4) * (F - 3).Multiply both sides by 4: 4F = 5F - 15 ⇒ 5F - 4F = 15 ⇒ F = 15.Hence, the first bunch has 15 bananas.Verification / Alternative check:Compute S: S = F - 3 = 12. Check “one-quarter again as many”: (5/4) * 12 = 15. Both conditions hold perfectly.
Why Other Options Are Wrong:
- 9, 10, 12 — none satisfy both the proportional and difference constraints simultaneously.
Common Pitfalls:Misreading “one-quarter again as many” as “one-quarter as many.” The word “again” indicates adding the fraction to the whole: 1 + 1/4 = 5/4.
Final Answer:15