Kevin and Randy Muise have a jar containing 36 ​coins, all of which are either quarters or nickels. The total value of the coins in the jar is ​$5.80. How many of each type of coin do they​ have?

Answer 1

You can solve this problem using a system of equations. The two equations in the system are (a) the number of coins must total 36, and (b) the value of the coins must total $5.80. Let Q be the number of quarters and N be the number of nickels.

The first equation represents the number of coins:

Q + N = 36

The second equation represents the value of the coins. A quarter is equal to $0.25, so we multiply the number of quarters (Q) by 0.25 to get the total value of quarters. A nickel is equal to $0.05, so we multiply the number of nickels (N) by 0.05 to get the total value of nickels. The total value of the two groups of coins must together equal $5.80, so we get the following for the second equation:

0.25Q + 0.05N = 5.8

This system can be solved by either substitution or elimination. Solving by substitution, I'll pick to rearrange the first equation because the coefficients of the variables are already 1 (so I don't have to multiply or divide by anything at the end of the rearrangement). I'll solve for Q:

Q + N = 36

Subtract N from both sides:

Q + N - N = 36 - N

Simplify:

Q = 36 - N

Now substitute this into the other equation:

0.25Q + 0.05N = 5.8

0.25 * (36 - N) + 0.05N = 5.8

In order of operations (PEMDAS), we don't have any parentheses to simplify, and we don't have any exponents, so we start with multiplication, by distributing the 0.25 into the parentheses:

0.25 * 36 - 0.25N + 0.05N = 5.8

Simplify:

9 - 0.25N + 0.05N = 5.8

Combine like terms:

9 - 0.20N = 5.8

Subtract 9 from both sides:

9 - 0.20N - 9 = 5.8 - 9

Simplify:

-0.20N = -3.2

Divide both sides by -0.20 to solve for N:

-0.20N/-0.20 = -3.2/-.20

The negatives cancel on both sides, and we're left with:

N = 16

So the number of nickels is 16. Plugging this into the first equation (which we already rearranged to solve for the number of quarters, Q, which we're still looking for), we get:

Q = 36 - N

Q = 36 - 16

Q = 20

So you must have N = 16 nickels and Q = 20 quarters to meet the restraints of the problem.

Answer 2

Since a quarter is worth 25 cents and a Nickel is worth 5 cents, I would start by figuring out how many quarters there are. That will make finding how many nickels easier.