SOLUTION: A box contains 140 dimes and nickles. The total value is $11.5.How many dimes and nickels are there?

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Question 103692: A box contains 140 dimes and nickles. The total value is $11.5.How many dimes and nickels are there?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=# of dimes, y=# of nickels

So the word problem translates into this system:
x%2By=140 "A box contains 140 dimes and nickles"
0.1x%2B0.05y=11.5 "The total value is $11.5"

10x%2B5y=1150 Multiply the second equation by 100 to make every value a whole number


So now we get this system:

x%2By=140
10x%2B5y=1150



Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=140
10%2Ax%2B5%2Ay=1150

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=140-1%2AxSubtract 1%2Ax from both sides

y=%28140-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=140-1%2Ax Now we've fully isolated y

Since y equals 140-1%2Ax we can substitute the expression 140-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


10%2Ax%2B5%2Ahighlight%28%28140-1%2Ax%29%29=1150 Replace y with 140-1%2Ax. Since this eliminates y, we can now solve for x.

10%2Ax%2B5%2A%28140%29%2B5%28-1%29x=1150 Distribute 5 to 140-1%2Ax

10%2Ax%2B700-5%2Ax=1150 Multiply



10%2Ax%2B700-5%2Ax=1150 Reduce any fractions

10%2Ax-5%2Ax=1150-700 Subtract 700 from both sides


10%2Ax-5%2Ax=450 Combine the terms on the right side



5%2Ax=450 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%28450%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply 450%2F1 and 1%2F5 (and simplify) we get



x=90 <---------------------------------One answer

Now that we know that x=90, lets substitute that in for x to solve for y

10%2890%29%2B5%2Ay=1150 Plug in x=90 into the 2nd equation

900%2B5%2Ay=1150 Multiply

5%2Ay=1150-900Subtract 900 from both sides

5%2Ay=250 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ay=%28250%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.

y=250%2F5 Multiply the terms on the right side


y=50 Reduce


So this is the other answer


y=50<---------------------------------Other answer


So our solution is

x=90 and y=50

which can also look like

(90,50)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=140
10%2Ax%2B5%2Ay=1150

we get


graph of 1%2Ax%2B1%2Ay=140 (red) and 10%2Ax%2B5%2Ay=1150 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (90,50). This verifies our answer.


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Check:

Plug in (90,50) into the system of equations


Let x=90 and y=50. Now plug those values into the equation 1%2Ax%2B1%2Ay=140

1%2A%2890%29%2B1%2A%2850%29=140 Plug in x=90 and y=50


90%2B50=140 Multiply


140=140 Add


140=140 Reduce. Since this equation is true the solution works.


So the solution (90,50) satisfies 1%2Ax%2B1%2Ay=140



Let x=90 and y=50. Now plug those values into the equation 10%2Ax%2B5%2Ay=1150

10%2A%2890%29%2B5%2A%2850%29=1150 Plug in x=90 and y=50


900%2B250=1150 Multiply


1150=1150 Add


1150=1150 Reduce. Since this equation is true the solution works.


So the solution (90,50) satisfies 10%2Ax%2B5%2Ay=1150


Since the solution (90,50) satisfies the system of equations


1%2Ax%2B1%2Ay=140
10%2Ax%2B5%2Ay=1150


this verifies our answer.






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Answer:

Since x=90 and y=50, this means that we have 90 dimes and 50 nickels