You have have a jar containing 72 coins, all of which are either quarters or nickels. The total value of the coins in the jar is $8.40. How many of each type of coin do you have?

Total number of coins: T=72

Number of quartes ($0.25) : q=?

Number of nickels ($0.10) : n=?

Number of quarters + Number of nickels = Total number of coins

q+n=72 (Equation 1)

Total Value of the coins in the jar: V=$8.40

Value of the quarter's coins: $0.25q

Value of the nickel's coins: $0.05n

Value of the quarter's coins + Value of the nickel's coin=Total value of the coins in the jar

$0.25q+0.05n=$8.40

0.25q+0.05n=8.40 (Equation 2)

We have a system of 2 equations and 2 unknowns (q and n):

(1) q+n=72

(2) 0.25q+0.05n=8.40

Using the method of substitution:

Let's isolate q in the first equation:

(1) q+n=72

(1) q+n-n=72-n

(1) q=72-n

Let's replace q by 72-n in the second equation:

(2) 0.25q+0.05n=8.40

0.25 (72-n) + 0.05n=8.40

Applying the distributive property:

(0.25) (72) - 0.25n+0.05n=8.40

Multiplying the constant terms and adding similar terms of the variable n:

18-0.20n=8.40

Isolating n:

18-0.20n-18=8.40-18

-0.20n=-9.6

(-0.20n) / (-0.20) = (-9.6) / (-0.20)

n=48

Replacing n by 48 in the first equation:

(1) q=72-n

q=72-48

q=24

We have 24 coins of quarters and 48 coins of nickels