Difference between revisions of "1954 AHSME Problems/Problem 28"
Katzrockso (talk | contribs) (Created page with "== Problem 28== If <math>\frac{m}{n}=\frac{4}{3}</math> and <math>\frac{r}{t}=\frac{9}{14}</math>, the value of <math>\frac{3mr-nt}{4nt-7mr}</math> is: <math> \textbf{(A)}\...") |
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Because the ratio works for any set of integers satisfying <math>\frac{m}{n}=\frac{4}{3}</math> and <math>\frac{r}{t}=\frac{9}{14}</math>, it has to satisfy <math>m=4</math>, <math>n=3</math>, <math>r=9</math>, and <math>t=14</math>. From here it is just simple arithmetic. | Because the ratio works for any set of integers satisfying <math>\frac{m}{n}=\frac{4}{3}</math> and <math>\frac{r}{t}=\frac{9}{14}</math>, it has to satisfy <math>m=4</math>, <math>n=3</math>, <math>r=9</math>, and <math>t=14</math>. From here it is just simple arithmetic. | ||
− | <math>\frac{3\cdot4\cdot9-3\cdot14}{4\cdot3\cdot14-7\cdot4\cdot9}\implies \frac{3(36-14)}{4(42-63)}\implies \frac{3(22)}{4(-21)}\implies \boxed{\frac{-11}{14} (\textbf{B})}</math> | + | <math>\frac{3mr-nt}{4nt-7mr}\implies\frac{3\cdot4\cdot9-3\cdot14}{4\cdot3\cdot14-7\cdot4\cdot9}\implies \frac{3(36-14)}{4(42-63)}\implies \frac{3(22)}{4(-21)}\implies \boxed{\frac{-11}{14} (\textbf{B})}</math> |
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AHSME 50p box|year=1954|num-b=27|num-a=29}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 00:31, 28 February 2020
Contents
[hide]Problem 28
If and , the value of is:
Solution 1
From , we have . From , we have
This simplifies the fraction to
Solution 2
Because the ratio works for any set of integers satisfying and , it has to satisfy , , , and . From here it is just simple arithmetic.
See Also
1954 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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