# Sal has quarters, dimes and nickels. She has 52 coins total. She has 3 more quarters than dimes and 5 fewer nickels than nickels. How many dimes does she have?

Jun 1, 2016

Depending upon a correction to the question:
the intended answer was probably $18$ dimes

#### Explanation:

Let
$\textcolor{w h i t e}{\text{XXX}} Q$ represent the number of quarters;
$\textcolor{w h i t e}{\text{XXX}} D$ represent the number of dimes; and
$\textcolor{w h i t e}{\text{XXX}} N$ represent the number of nickels.
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Option 1: line should have read: 5 fewer dimes than nickels.

We are told
$\textcolor{w h i t e}{\text{XXX}} Q + D + N = 52$
$\textcolor{w h i t e}{\text{XXX}} Q = D + 3$
$\textcolor{w h i t e}{\text{XXX}} D = N - 5$

$D = N - 5 \textcolor{w h i t e}{\text{XX")rarrcolor(white)("XX}} N = D + 5$

So we can substitute $D + 3$ for $Q$ and $D + 5$ for $N$ in 

$\textcolor{w h i t e}{\text{XXX}} \left(D + 3\right) + D + \left(D + 5\right) = 52$

$\textcolor{w h i t e}{\text{XXX}} 3 D + 8 = 52$

$\textcolor{w h i t e}{\text{XXX}} 3 D = 44$

$\textcolor{w h i t e}{\text{XXX}} D = 14 \frac{2}{3}$
(since this isn't likely, let's reject Option 1)
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Option 2: line should have read: 5 fewer nickels than dimes.

We are told
$\textcolor{w h i t e}{\text{XXX}} Q + D + N = 52$
$\textcolor{w h i t e}{\text{XXX}} Q = D + 3$
$\textcolor{w h i t e}{\text{XXX}} N = D - 5$

So we can substitute $D + 3$ for $Q$ and $D - 5$ for $N$ in 

$\textcolor{w h i t e}{\text{XXX}} \left(D + 3\right) + D + \left(D - 5\right) = 52$

$\textcolor{w h i t e}{\text{XXX}} 3 D - 2 = 52$

$\textcolor{w h i t e}{\text{XXX}} 3 D = 54$

$\textcolor{w h i t e}{\text{XXX}} D = 18$

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Of course it is possible that one of the "nickels" in
$\textcolor{w h i t e}{\text{XXX}}$...5 fewer nickels than nickels
should have been "quarters"... but let's quit while we have a reasonable answer.