Problem 13 · 2016 Math Kangaroo
Medium
Algebra & Patterns
substitution
There are 30 girls and boys in a class. Two students always share a desk. Every boy shares a desk with a girl. Exactly half the girls share a desk with a boy. How many boys are in the class?
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Answer: D — 10
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Hint 1 of 3
A boy-girl desk has one boy and one girl, so counting those desks counts the boys.
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Hint 2 of 3
Those same boy-girl desks use up exactly half the girls, so there are as many boys as half-the-girls.
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Hint 3 of 3
That means there are twice as many girls as boys.
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Approach: match boys to half the girls
- Every boy sits at a boy-girl desk, so the number of boy-girl desks equals the number of boys.
- Those desks contain exactly half the girls, so the number of boys equals half the girls; in other words there are twice as many girls as boys.
- Split 30 into 1 part boys and 2 parts girls: 3 equal parts make 30, so each part is 10, and the boys are 1 part, giving 10 boys, choice (D).
With algebra
If \(b\) boys and \(g\) girls, then \(b+g=30\) and \(g=2b\), so \(3b=30\) and \(b=10\).
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