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

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==Problem==
 
==Problem==
  
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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The [[subtraction|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
  
 
<math>\text{(A)}</math> <dollar/><math>.01</math>  
 
<math>\text{(A)}</math> <dollar/><math>.01</math>  
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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...
  
Before we start, it's always good to convert the word problems into expressions, we can solve.
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Before we start, it's always good to convert the word problems into [[Expression|expressions]], we can solve.
  
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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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 [[divisor|factor]], <math>20</math>, and we can [[factoring|factor]] the expression into  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
(6.5\% - 6\% ) \times 20 &= (.5\% )\times 20 \\
 
(6.5\% - 6\% ) \times 20 &= (.5\% )\times 20 \\
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==See Also==
 
==See Also==
  
[[1985 AJHSME Problems]]
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{{AJHSME box|year=1985|num-b=13|num-a=15}}
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[[Category:Introductory Algebra Problems]]

Revision as of 21:21, 15 May 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 (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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All AJHSME/AMC 8 Problems and Solutions
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