Difference between revisions of "1967 AHSME Problems/Problem 30"
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== Solution == | == Solution == | ||
− | <math>\fbox{D}</math> | + | |
+ | The first <math>2</math> radios are sold for <math>\frac{d}{2}</math> dollars each, for income of <math>2 \cdot \frac{d}{2}</math>, or simply <math>d</math>. | ||
+ | |||
+ | The remaining <math>n-2</math> radios are sold for <math>d+8</math> dollars each, for income of <math>(n-2)(d+8)</math>, which expands to <math>nd +8n -2d - 16</math>. | ||
+ | |||
+ | The amount invested in buying <math>n</math> radios for <math>d</math> dollars is <math>nd</math>. | ||
+ | |||
+ | Thus, the total profit is <math>(d) + (nd - 2d + 8n - 16) - nd</math>, which simplifies to <math>8n - d - 16</math>. | ||
+ | |||
+ | Setting this equal to <math>72</math> gives <math>8n - d - 16 = 72</math>, or <math>n = 11 + \frac{d}{8}</math>. | ||
+ | |||
+ | Since <math>n</math> must be an integer, <math>d</math> must be a multiple of <math>8</math>. If we want to minimize <math>n</math>, we want to minimize <math>d</math>. Since <math>d>0</math>, setting <math>d=8</math> yields <math>n=12</math>, giving the answer of <math>\fbox{D}</math> | ||
== See also == | == See also == |
Latest revision as of 10:25, 12 July 2019
Problem
A dealer bought radios for dollars, a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $8 on each radio sold. If the overall profit was $72, then the least possible value of for the given information is:
Solution
The first radios are sold for dollars each, for income of , or simply .
The remaining radios are sold for dollars each, for income of , which expands to .
The amount invested in buying radios for dollars is .
Thus, the total profit is , which simplifies to .
Setting this equal to gives , or .
Since must be an integer, must be a multiple of . If we want to minimize , we want to minimize . Since , setting yields , giving the answer of
See also
1967 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 31 | |
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 |
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