SOLUTION: Jack has dimes, nickels, and quarters worth $10.00. If he has 50 coins in total, how many coins will Jack have of each?

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Question 848318: Jack has dimes, nickels, and quarters worth $10.00. If he has 50 coins in total, how many coins will Jack have of each?
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
d count of dimes
n count of nickels
q count of quarters

d%2Bn%2Bq=50 and 0.10d%2B0.05n%2B0.25q=10.
Simplify the money count equation for whole number coefficients.
10d%2B5n%2B25q=1000
2d%2Bn%2B5q=200--------

SYSTEM:
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2d+n+5q=200
'
d+n+q=50
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That is two equations and three unknown variables. Something must be initially assumed constant. Try taking n as a constant, k=n.

SYSTEM with k=n:
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2d+5q=200-k
'
d+q=50-k
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Try elimination. Multiply the second equation by 2 and subtract from the first equation.
2d%2B5q-%282d%2B2q%29=200-k-%282%2A50-2k%29
3q=200-k-100%2B2k
highlight_green%283q=100%2Bk%29
highlight_green%28q=%28k%2B100%29%2F3%29
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Try elimination again for the other variable. Multiply the second equation by 5 and ...
2d%2B5q-%285d%2B5q%29=200-k-%285%2A50-5k%29
-3d=200-k-250%2B5k
-3d=-50%2B4k
highlight_green%28d=%2850-4k%29%2F%283%29%29
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A choice must be found of some whole number for k so that d and q each are also whole numbers. MAYBE k=2 worth a test.
k_____________q____________d
2____________34____________14----------Combination appears to work, since k+q+d=50.

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ANSWER: A good result then is 2 nickels, 14 dimes, 34 quarters.
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