Problem 23 · 2019 Math Kangaroo
Stretch
Arithmetic & Operations
order-of-operationssubstitution
To find the value of \(\dfrac{a+b}{c}\) (where \(a\), \(b\) and \(c\) are positive integers), Sara types \(a + b \div c =\) into a calculator and gets 11. Then she types \(b + a \div c =\) and is surprised to get 14. She realises the calculator follows the order of operations, doing division before addition. What is the actual value of \(\dfrac{a+b}{c}\)?
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Answer: E — 5
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Hint 1 of 2
The calculator computes \(a + \dfrac{b}{c} = 11\) and \(b + \dfrac{a}{c} = 14\).
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Hint 2 of 2
Subtracting and adding the two equations lets you pin down \(c\) and then \(a + b\).
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Approach: set up the two order-of-operations equations
- The two calculator results give \(a + \dfrac{b}{c} = 11\) and \(b + \dfrac{a}{c} = 14\).
- Subtracting: \((a - b)\left(1 - \dfrac{1}{c}\right) = -3\), so \((a-b)(c-1) = -3c\); the only positive-integer fit is \(c = 4\) with \(a - b = -4\).
- With \(c = 4\), the first equation gives \(a + \dfrac{b}{4} = 11\), and together with \(a - b = -4\) we get \(a = 8\), \(b = 12\), so \(a + b = 20\).
- The actual value is \(\dfrac{a+b}{c} = \dfrac{20}{4} = 5\) — answer (E).
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