Difference between revisions of "1950 AHSME Problems/Problem 22"
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<math> \textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 72\%\qquad\textbf{(D)}\ 28\%\qquad\textbf{(E)}\ \text{None of these} </math> | <math> \textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 72\%\qquad\textbf{(D)}\ 28\%\qquad\textbf{(E)}\ \text{None of these} </math> | ||
| − | ==Solution== | + | ==Solution 1 (Kind of Lame)== |
Without loss of generality, assume something costs <math>100</math> dollars. Then with each successive discount, it would cost <math>90</math> dollars, then <math>72</math> dollars. This amounts to a total of <math>28</math> dollars off, so the single discount would be <math>\boxed{\mathrm{(D)}\ 28\%.}</math> | Without loss of generality, assume something costs <math>100</math> dollars. Then with each successive discount, it would cost <math>90</math> dollars, then <math>72</math> dollars. This amounts to a total of <math>28</math> dollars off, so the single discount would be <math>\boxed{\mathrm{(D)}\ 28\%.}</math> | ||
| + | ==Solution 2 (Technical)== | ||
| + | |||
| + | Let the object cost <math>x</math> dollars. After the <math>10%</math> discount, it's worth <math>(1-10%)x=0.9x</math> dollars. After a <math>20%</math> discount on that, it's worth <math>(1-20%)(0.9x)=0.72x</math>. Say the single discount is of <math>k</math>. Then <math>(1-k)x=0.72x</math>. So <math>k=0.28</math>, or <math>k=28%</math>. So select <math>\boxed{D}</math>. | ||
| + | |||
| + | ~hastapasta | ||
==See Also== | ==See Also== | ||
Revision as of 11:11, 31 March 2022
Problem
Successive discounts of 
and 
are equivalent to a single discount of:
Solution 1 (Kind of Lame)
Without loss of generality, assume something costs 
dollars. Then with each successive discount, it would cost 
dollars, then 
dollars. This amounts to a total of 
dollars off, so the single discount would be 
Solution 2 (Technical)
Let the object cost 
dollars. After the $10%$ (Error compiling LaTeX. Unknown error_msg) discount, it's worth $(1-10%)x=0.9x$ (Error compiling LaTeX. Unknown error_msg) dollars. After a $20%$ (Error compiling LaTeX. Unknown error_msg) discount on that, it's worth $(1-20%)(0.9x)=0.72x$ (Error compiling LaTeX. Unknown error_msg). Say the single discount is of 
. Then 
. So 
, or $k=28%$ (Error compiling LaTeX. Unknown error_msg). So select 
.
~hastapasta
See Also
| 1950 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
