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Difference between revisions of "1985 AJHSME Problems/Problem 14"

(New page: ==Solution== The most straightforward method would be to calculate both prices, and subtract. But there's a better method... Before we start, it's always good to convert the word problem...)
 
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==Problem==
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The difference between a <math>6.5\% </math> sales tax and a <math>6\% </math> sales tax on an item priced at <dollar/><math>20</math> before tax is
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<math>\text{(A)}</math> <dollar/><math>.01</math>
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<math>\text{(B)}</math> <dollar/><math>.10</math>
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<math>\text{(C)}</math> <dollar/><math>.50</math>
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<math>\text{(D)}</math> <dollar/><math>1</math>
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<math>\text{(E)}</math> <dollar/><math>10</math>
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==Solution==
 
==Solution==
  
 
The most straightforward method would be to calculate both prices, and subtract. But there's a better method...
 
The most straightforward method would be to calculate both prices, and subtract. But there's a better method...
  
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Before we start, it's always good to convert the word problems into formulas, or at least expressions, we can solve.
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Before we start, it's always good to convert the word problems into expressions, we can solve.
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So we know that the price of the object after a <math>6.5\% </math> increase will be <math>20 \times 6.5\% </math>, and the price of it after a <math>6\% </math> increase will be <math>20 \times 6\% </math>. And what we're trying to find is <math>6.5\% \times 20 - 6\% \times 20</math>, and if you have at least a little experience in the field of algebra, you'll notice that both of the items have a common factor, <math>20</math>, and we can factor the expression into
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<cmath>\begin{align*}
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(6.5\% - 6\% ) \times 20 &= (.5\% )\times 20 \\
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&= \frac{.5}{100}\times 20 \\
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&= \frac{1}{200}\times 20 \\
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&= .10 \\
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\end{align*}</cmath>
  
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So we know that the price of the object after a 6.5% increase will be <math>20 \times 6.5%</math>, and the price of it after a 6% increase will be <math>20 \times 6</math>. And what we're trying to find is <math>6.5% \times 20 - 6 \times 20</math>, and if you have experience in the field of algebra, you'll notice that both of the items have a common factor, 20, and we can factor the expression into...
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<math>.10</math> is choice <math>\boxed{\text{B}}</math>
  
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<math>(6.5% - 6%) \times 20</math>, which simplifies to <math>.5%</math>. We know that <math>.5 = \frac{1}{20}</math>, so <math>.5% = \frac{1}{2000}</math>. <math>20 \times \frac{1}{2000} = \frac{1}{100} = .01</math>, so the answer is 1 cent.
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==See Also==
  
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1 cent = <math>\</math>.01$ = (D)
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[[1985 AJHSME Problems]]

Revision as of 18:45, 13 January 2009

Problem

The difference between a $6.5\%$ sales tax and a $6\%$ sales tax on an item priced at <dollar/>$20$ before tax is

$\text{(A)}$ <dollar/>$.01$

$\text{(B)}$ <dollar/>$.10$

$\text{(C)}$ <dollar/>$.50$

$\text{(D)}$ <dollar/>$1$

$\text{(E)}$ <dollar/>$10$

Solution

The most straightforward method would be to calculate both prices, and subtract. But there's a better method...

Before we start, it's always good to convert the word problems into expressions, we can solve.

So we know that the price of the object after a $6.5\%$ increase will be $20 \times 6.5\%$, and the price of it after a $6\%$ increase will be $20 \times 6\%$. And what we're trying to find is $6.5\% \times 20 - 6\% \times 20$, and if you have at least a little experience in the field of algebra, you'll notice that both of the items have a common factor, $20$, and we can factor the expression into \begin{align*} (6.5\% - 6\% ) \times 20 &= (.5\% )\times 20 \\ &= \frac{.5}{100}\times 20 \\ &= \frac{1}{200}\times 20 \\ &= .10 \\ \end{align*}

$.10$ is choice $\boxed{\text{B}}$

See Also

1985 AJHSME Problems

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