find the total coins

Question

Gail has $8 worth of 5-cent and 10-cent coins. If the number of 10-cent coins is 20 less than twice the number of 5-cent coins, how many coins are there altogether

Answer 1

You can write two equations from the problem statement.

First assign variable names for the coins

d = the number of 10-cent coins (dimes)

n = the number of 5-cent coins (nickels)

The easy equation comes from the first statement.  You can calculate the total value of the coins:

0.05n + 0.10d = 8.00

The second statement is more involved.  Start by looking for things you can write expressions for, then look for how to put those expressions together.

"twice the number of 5-cent coins" is 2n.

"20 less than" that number is 2n-20.

The number of dimes is thus

d = 2n-20

 

This is a system of equations.  Since one of our equations is "d =  ...", we can substitute for d in the other equation.

0.05n + 0.10d = 8.00

Substituting 2n-20 for d:

0.05n + 0.1(2n-20) = 8

Solve for n:

0.05n + 0.2n - 2 = 8

0.25n = 8 + 2

0.25n = 10

n = 40

 

Substitute the value for n in either equation and solve for d.

The second equation looks easier to work with.

d = 2n-20
d = 2(40) - 20

d = 60

 

Check:

0.05n + 0.10d = 8.00

0.05(40) + 0.10(60) = 8.00

2.00 + 6.00 = 8.00

8.00 = 8.00

 

The total number of coins is 100.

Answer 2

Vy, this requires a system of equations. Let's let x = the number of 5-cent and y = the number of 10-cent coins.

 

Now, we are going to deal with cents in order to avoid the decimals.

 

We have that 5x + 10y = 800 ($8 is 800 cents)

                     y = 2x - 20

 

Now, we can substitute the "value" for y into the first equation.

 

5x + 10(2x - 20) = 800

5x + 20x - 200 = 800

       25x - 200 = 800

                25x = 1000

                    x = 40

 

Now, we can plug in our value for x into either equation. Let's do the second.

 

y = 2(40) - 20

 y = 80 - 20

 y = 60

 

Let's check.

 

5(40) + 10(60) = 800

200 + 600 = 800. Check.

 

We have 100 coins all together!