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CEMC Gauss (Grade 8), 2013

Algebra Difficulty 1.4 Junior Find the answer Canada

The ratio of the value of four nickels (5¢ coins) to the value of six dimes (10¢ coins) to the value of two quarters (25¢ coins) can be written as

Pick one

Solution

To make the difference $UVW-XYZ$ as large as possible, we make $UVW$ as large as possible and $XYZ$ as small as possible.

The hundreds digit of a number contributes more to its value than its tens digit, and its tens digit contributes more to its value than its units digit.

Thus, we construct the largest possible number $UVW$ by choosing 9 (the largest digit) to be its hundreds digit, $U$ , and by choosing 8 (the second largest digit) to be its tens digit, $V$ , and by choosing 7 (the third largest digit) to be the units digit, $W$ .

Similarly, we construct the smallest possible number $XYZ$ by choosing 1 (the smallest allowable digit) to be its hundreds digit, $X$ , and 2 (the second smallest allowable digit) to be its tens digit, $Y$ , and by choosing 3 (the third smallest allowable digit) to be its units digit, $Z$ .
The largest possible difference is $UVW-XYZ$ or $987-123$ or 864.

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