A person has a total of 65 coins in a jar consisting of quarters and nickles. The total value of the coins is $10.25. How many quarters and nickles does the person have?

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Answer 1 · 2017-06-14T20:41:11

See a solution process below:

First:

Let's call the number of quarters: $q$ $q$

Let's call the number of nickles: $n$ $n$

We know:

$q + n = 65$ $q + n = 65$ because there are 65 coins in the jar.

We also know multiplying the value of the coins by the number of coins gives: $0.25 q + 0.05 n = $ 10.25$ $0.25q + 0.05n = $10.25$

Step 1) We can now solve the first equation for $q$ $q$ :

$q + n = 65$ $q + n = 65$

$q + n - n = 65 - n$ $q + n - color(red)(n) = 65 - color(red)(n)$

$q + 0 = 65 - n$ $q + 0 = 65 - n$

$q = 65 - n$ $q = 65 - n$

Step 2) Substitute $(65 - n)$ $(65 - n)$ for $q$ $q$ in the second equation and solve for $n$ $n$ :

$0.25 q + 0.05 n = 10.25$ $0.25q + 0.05n = 10.25$ becomes:

$0.25 (65 - n) + 0.05 n = 10.25$ $0.25(65 - n) + 0.05n = 10.25$

$(0.25 \cdot 65) - (0.25 \cdot n) + 0.05 n = 10.25$ $(0.25 * 65) - (0.25 * n) + 0.05n = 10.25$

$16.25 - 0.25 n + 0.05 n = 10.25$ $16.25 - 0.25n + 0.05n = 10.25$

$16.25 + (- 0.25 + 0.05) n = 10.25$ $16.25 + (-0.25 + 0.05)n = 10.25$

$16.25 + (- 0.2) n = 10.25$ $16.25 + (-0.2)n = 10.25$

$16.25 - 0.2 n = 10.25$ $16.25 - 0.2n = 10.25$

$- 16.25 + 16.25 - 0.2 n = - 16.25 + 10.25$ $-color(red)(16.25) + 16.25 - 0.2n = -color(red)(16.25) + 10.25$

$0 - 0.2 n = - 6$ $0 - 0.2n = -6$

$- 0.2 n = - 6$ $-0.2n = -6$

$\frac{- 0.2 n}{- 0.2} = - \frac{6}{- 0.2}$ $(-0.2n)/color(red)(-0.2) = -6/color(red)(-0.2)$

$(color(red)(cancel(color(black)(-0.2)))n)/cancel(color(red)(-0.2)) = 30$

$n = 30$

Step 3) Substitute $30$ for $n$ in the solution to the first equation at the end of Step 1 and calculate $q$ :

$q = 65 - n$ becomes:

$q = 65 - 30$

$q = 35$

The solution is: $n = 30$ and $q = 35$

So there are $30$ nickles and $35$ quarters.