2012/13 AHSMC Part 1 Problem 15

Problem 15 of 16

At the beginning of the year, there were more robots than androids. On the first day of each month, each robot made $7$ androids and each android made $7$ robots. The next day, each old android would pick a fight with a new android, and they would destroy each other. At the end of the year, there were $46875$ million robots and $15625$ million androids. What was the difference between the numbers of robots and androids at the beginning of the year?

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Solution

Let the numbers of robots and androids be $r$ and $a$ respectively. After one month, these numbers became $r + 7a$ and $7r - a$ . After another month, they became $(r + 7a) + 7(7r - a) = 50r$ and $7(r + 7a) - (7r - a) = 50a$ . Hence after a two-month period, the number of robots became $50$ times the original number, and the same goes for the number of androids. There being $6$ two-month periods in a year, the initial number of robots was $46875000000 \div 50^6 = 3$ and the initial number of androids was $15625000000 \div 50^6 = 1$ . The answer is (A).

University of Alberta

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