Create an equation with one variable

Question

You have exactly 20 coins, consisting of dimes, nickels and quarters. There are 10 nickels. If the total value of all 20 coins is $1.95, how many of each coin do you have?

Answer 1

 

We have 20 coins: dimes ($0.10 apiece), nickels ($0.05 apiece) and quarters ($0.25 apiece).

Ten of these 20 coins are nickels, accounting for 50¢.

The value of all 20 coins is $1.95

 

I know the number of nickels is 10 and there are 20 coins, so I can say

D+10+Q=20

which I can simplify to

D+Q=10

 

The problem with that equation is that it has two variables, but I think we can fix that. We should start by solving for one of the variables. I'm going to do that by subtracting Q from both sides

 

Q+D=10

-Q         -Q

     D=10-Q

 

Ok, let's move on to the currency values. The number of each type of coin times its respective value should equal $1.95. I can write that in an equation too:

 

0.10D+0.50+0.25Q=1.95

 

Notice that I put in 0.50 for the nickels, because I already know how much my nickels are worth. In this equation also, there are two variables, so we're not there yet. Except we solved for D earlier, so now we can use the substitution method to substitute every D we see with (10-Q)

 

0.10D+0.50+0.25Q=1.95

becomes

0.10(10-Q)+0.50+0.25Q=1.95

 

If your assignment was just to create an equation with one variable, then you're done! That's it!

Of course, we can use some distribution, association, and subtraction to simplify and solve for Q.

 

.1(10-Q)+.5+.25Q=1.95

1-.1Q+.5+.25Q=1.95

1.5+.15Q=1.95

.15Q=.45

Q=3

 

Given that there are 20 coins, we should be able to say that 3+10+D=20