Question

How many separate dimes and how many separate nickels do you have?

If you have 10 dimes and nickles in your pocket that total to $0.70, how many separate dimes and how many separate nickels do you have?

Answer 1

There are `4` dimes and `6` nickels.

Explanation:

We can create a system of equations based on the two true statements we know:

`{ ("The value of all coins" = $0.70),("Number of nickels + number of dimes" = 10) :}`

Now, let `x` equal the number of dimes in your pocket and let `y` equal the number of nickels. We can create a new system of equations:

`{ ($0.10*x + $0.05*y=$0.70, qquad(1)), (x+y=10, qquad(2)):}`

Using equation `(2)`, we can solve for `x`:

`x+y=10=>x=10-y`

Now, use this new value of `x` in equation `(1)`:

`$0.10*x + $0.05*y=$0.70`

`=> $0.10*(10-y) + $0.05*y=$0.70`

`$1.00-$0.10*y+$0.05*y=$0.70`

`-$0.10*y+$0.05*y=-$0.30`

`-$0.05*y=-$0.30`

`y=(-$0.30)/(-$0.05)`

`y=6`

Now we know that there are `6` nickels. Using equation `(2)` again, we can find out how many dimes there are:

`x+y=10`

`=>x+6=10`

`x=4`

Finally, we know that there are `4` dimes and `6` nickels. We can check our work by plugging in these two values into one of the equations. I'll use equation `(2)` just because it is a simpler expression:

`x+y=10`

`=>4+6=10`

`10=10`

Since this expression is true, we know that our answer is correct.