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Difference between revisions of "1950 AHSME Problems/Problem 22"

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so the single discount would be <math>\boxed{\mathrm{(D)}\ 28\%.}</math>
 
so the single discount would be <math>\boxed{\mathrm{(D)}\ 28\%.}</math>
  
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==Solution 2 (Technical)==
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==Solution 3 (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> dollars. 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>.
 
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> dollars. 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>.

Revision as of 10:42, 21 September 2025

Problem

Successive discounts of $10\%$ and $20\%$ are equivalent to a single discount of:

$\textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 72\%\qquad\textbf{(D)}\ 28\%\qquad\textbf{(E)}\ \text{None of these}$

Solution 1 (Kind of Lame)

Without loss of generality, assume something costs $100$ dollars. Then with each successive discount, it would cost $90$ dollars, then $72$ dollars. This amounts to a total of $28$ dollars off, so the single discount would be $\boxed{\mathrm{(D)}\ 28\%.}$


Solution 2

Total discount if the Product is $100$ dollars is 10/100 of 100 + 20/100 of (100 - 10/100 of 100) 

                                                 = 10/100  * 100 +  20/100  * (100 - 10/100  * 100) 
                                                 = 10 + 2/10 * 90
                                                 = 10 + 9*2
                                                 = 10+18
                                                 = 28

So discount is $28$ dollars so the single discount would be $\boxed{\mathrm{(D)}\ 28\%.}$

Solution 3 (Technical)

Let the object cost $x$ dollars. After the $10\%$ discount, it's worth $(1-10\%)x=0.9x$ dollars. After a $20\%$ discount on that, it's worth $(1-20\%)(0.9x)=0.72x$ dollars. Say the single discount is of $k$. Then $(1-k)x=0.72x$. So $k=0.28$, or $k=28\%$. So select $\boxed{D}$.

~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

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