Difference between revisions of "1956 AHSME Problems/Problem 16"
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| + | == Problem 16== | ||
| + | |||
| + | The sum of three numbers is <math>98</math>. The ratio of the first to the second is <math>\frac {2}{3}</math>, | ||
| + | and the ratio of the second to the third is <math>\frac {5}{8}</math>. The second number is: | ||
| + | |||
| + | <math>\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 33 </math> | ||
| + | |||
==Solution== | ==Solution== | ||
| − | Let the <math>3</math> numbers be <math>a</math> <math>b</math> and <math>c</math>. We see that | + | Let the <math>3</math> numbers be <math>a,</math> <math>b,</math> and <math>c</math>. We see that |
<cmath>a+b+c = 98</cmath> | <cmath>a+b+c = 98</cmath> | ||
and | and | ||
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~JustinLee2017 | ~JustinLee2017 | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{AHSME box|year=1956|num-b=15|num-a=17}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 21:34, 12 February 2021
Problem 16
The sum of three numbers is 
. The ratio of the first to the second is 
,
and the ratio of the second to the third is 
. The second number is:
Solution
Let the 
numbers be 
and 
. We see that
![\[a+b+c = 98\]](http://latex.artofproblemsolving.com/a/1/9/a19e174a2a977a30668dae420893fae94ac3ca77.png)
and
![\[\frac{a}{b} = \frac{2}{3} \Rrightarrow 3a = 2b\]](http://latex.artofproblemsolving.com/8/6/f/86ff42e081fc9cb125c59ae0f7e7b97a611a7372.png)
![\[\frac{b}{c} = \frac{5}{8} \Rrightarrow 8b = 5c\]](http://latex.artofproblemsolving.com/3/1/3/31371e78ea1eabe0bcae23ba5361493f9eb76102.png)
Writing 
and in terms of 
we have
and 
.
Substituting in the sum, we have
![\[\frac{2}{3} b + b + \frac{8}{5} b = 98\]](http://latex.artofproblemsolving.com/2/e/a/2ea3357eda2600e331064430f70094432acdf7a0.png)
![\[\frac{49}{15} b = 98\]](http://latex.artofproblemsolving.com/f/4/1/f4108342884665c2b91b4ba224902ed450bec928.png)
![\[b = 98 \cdot \frac{15}{49} \Rrightarrow b = 30\]](http://latex.artofproblemsolving.com/d/6/c/d6c05997150a8b5c6aad1e28afa15274b743c759.png)
~JustinLee2017
See Also
| 1956 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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. 
