Track / Stage 3 / 208 of 260 #208 of 1964

Problem 208

AMC 10/12, early questions

Number theory Difficulty 3.8 Find the answer South African Mathematics Olympiad · South Africa

I buy a number of chocolates which cost R25 each and cool-drinks which cost R9 each. I buy more chocolates than cool-drinks. How many cool-drinks do I buy if I pay R839?

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

If I buy $x$ chocolate bars and $y$ cool-drinks, then $25x + 9y = 839$ with $x > y$ . If we put $x = y$ as a first guess, then $x = y = 839/34 \approx 24$ , so we can write $x = 24 + a$ and $y = 24 - b$ . The equation becomes $25a - 9b = 23$ , which by easy trial and error (guess and check) has a solution $a = 2$ and $b = 3$ . Thus $x = 24 + 2 = 26$ and $y = 24 - 3 = 21$ , which is the required answer.

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