IMC 2016 Q12
12. The sum of two numbers \(a\) and \(b\) is 7 and the difference between them is 2.
What is the value of \(a \times b\)?
- A \(8\frac{1}{4}\)
- B \(9\frac{1}{4}\)
- C \(10\frac{1}{4}\)
- D \(11\frac{1}{4}\)
- E \(12\frac{1}{4}\)
Intermediate Mathematical Challenge 2016, Question 12. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(11\frac{1}{4}\)
Assume that \(a > b\). Then \(a + b = 7\) and \(a - b = 2\). Adding these two equations together gives \(2a = 9\). So \(a = \frac{9}{2}\) and hence \(b = 7 - \frac{9}{2} = \frac{14 - 9}{2} = \frac{5}{2}\).
Therefore \(a \times b = \frac{9}{2} \times \frac{5}{2} = \frac{45}{4} = 11\frac{1}{4}\).
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).