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2023/24 AHSMC Part 1 Problem 6

Problem 6 of 16

Oranges cost 9999 cents each, and mangoes cost $1.76\$1.76 each. You have a $13\$13 gift certificate from the fruit store, and want to use all of it to buy some oranges and mangoes. You want the total cost of the fruit to be as small as possible, provided it is at least $13\$13. What would be the total cost?

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Solution

Both 9999 and 176176 are divisible by 1111, so the total (in cents) is a multiple of 1111. Thus the answer must be one of $13.09\$13.09, $13.20\$13.20 or $13.31\$13.31. For $13.09\$13.09, the equation 1309=99x+176y1309 = 99x + 176y becomes 119=9x+16y119 = 9x + 16y. Starting at 119119 and repeatedly subtracting 1616 gives 119,103,87,71,55,39,23,7119,103,87,71,55,39,23,7, none of which is a multiple of 99. For 1320=11⋅1201320 = 11 \cdot 120, one has 120−3⋅16=72=9⋅8120 - 3\cdot 16 = 72 = 9\cdot 8, so $13.20=8($0.99)+3($1.76)\$13.20 = 8(\$0.99) + 3(\$1.76). The answer is (C).

University of Alberta