Difference between revisions of "1967 AHSME Problems/Problem 19"
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== Solution == | == Solution == | ||
− | <math>\fbox{E}</math> | + | We are given <math>xy = (x+\frac{5}{2})(y-\frac{2}{3}) = (x - \frac{5}{2})(y + \frac{4}{3})</math> |
+ | |||
+ | FOILing each side gives: | ||
+ | |||
+ | <math>xy = xy - \frac{2}{3}x + \frac{5}{2}y - \frac{5}{3} = xy + \frac{4}{3}x - \frac{5}{2}y - \frac{10}{3}</math> | ||
+ | |||
+ | Taking the last two parts and moving everything to the left gives: | ||
+ | |||
+ | <math>-2x + 5y + \frac{5}{3} = 0</math> | ||
+ | |||
+ | Taking the first two parts and multiplying by <math>3</math> gives <math>-2x + \frac{15}{2}y - 5 = 0</math> | ||
+ | |||
+ | Solving both equations for <math>-2x</math> and setting them equal to each other gives <math>5y + \frac{5}{3} = \frac{15}{2}y - 5</math>, which leads to <math>y = \frac{8}{3}</math> | ||
+ | |||
+ | Plugging that in to <math>-2x + 5y + \frac{5}{3} = 0</math> gives <math>x = \frac{15}{2}</math>. | ||
+ | |||
+ | The area of the rectangle is <math>xy = 20</math>, or <math>\fbox{E}</math>. | ||
== See also == | == See also == |
Latest revision as of 03:20, 13 July 2019
Problem
The area of a rectangle remains unchanged when it is made inches longer and inch narrower, or when it is made inches shorter and inch wider. Its area, in square inches, is:
Solution
We are given
FOILing each side gives:
Taking the last two parts and moving everything to the left gives:
Taking the first two parts and multiplying by gives
Solving both equations for and setting them equal to each other gives , which leads to
Plugging that in to gives .
The area of the rectangle is , or .
See also
1967 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.