Question 224659
Jake has 205 coins consisting of nickels, dimes, and quarters. If the total value of the coins is $22 and there are three times as many dimes as nickels, how many of each coin did he have?


Step 1.  Let x be the number of nickels.


Step 2.  Let 0.05x be the dollar value of nickels.


Step 3.  Let y be the number of dimes


Step 4.  Let 0.10y be the dollar value of dimes.


Step 5.  Let z be the number of quarters.


Step 6.  Let 0.25z be the dollar value of quarters.


Step 7.  Then x+y+z=205 since the total number of coins is 205.


Step 8.  Then  0.05x+0.10y+0.25z=22 since the dollar value of nickels, dimes, and quarters is 22.


Step 9.  We are also given y=3x since there are three times as many dimes as nickels.


Step 10.  Here's our system of linear equations given in Steps 7, 8, and 9.


{{{x+y+z=205}}}  Equation A
{{{0.05x+0.10y+0.25z=22}}}  Equation B
{{{y=3x}}}  Equation C


Substitute Equation C into Equations A and B


{{{x+3x+z=205}}} or {{{4x+z=205}}}  Equation A1


{{{0.05x+0.10(3x)+0.25z=22}}} or {{{0.35x+.25z=22}}} Equation B1


*[invoke linear_substitution "x", "z", 4, 1, 205, 0.35, 0.25, 22 ]


With {{{x=45}}} and {{{z=25}}}  then {{{y=205-45-25=135}}}


Check dollar value...{{{45*0.05+135*0.10+25*0.25= 2.25+13.50+6.25=22}}}...which is a true statement.


Step 11.  ANSWER:  The number of nickels is 45, the number of dimes is 135, and the number of quarters is 25.


I hope the above steps were helpful.


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Good luck in your studies!


Respectfully,
Dr J