Question

Shawna found coins worth $4.32. One-fourth of the found coins are pennies and one-sixth are quarters. The number of nickels found is 1.5 times the number of quarters. How many of each coin did Shawna find?

Answer 1

There are 48 coins in total.

Shawna found:

12 Pennies.

12 Nickels.

16 Dimes.

And 8 Quarters.

Explanation:

We will let
`x` denote the total amount of coins.

We know that the total worth of the coins were $4.32.

One-fourth of the found coins are pennies. Hence:

`\displaystyle (1)/(4)x\text{ are pennies.}`

Since pennies are worth $0.01 each, the worth will be:

`\displaystyle (1)/(4)x(0.01)`

Similarly, we know that one-sixth of the found coins are quarters. Quarter are worth $0.25. So, the total worth in quarters is:

`\displaystyle (1)/(6)x(0.25)`

We have 1.5 times as many nickels as quarters. Therefore, the amount of nickels we have is:

`\displaystyle 1.5((1)/(6)x)=(3)/(2)((1)/(6)x)=(1)/(4)x`

Or one-fourth of the total amount.

Since each nickel is worth $0.05, the total worth in nickels is:

`\displaystyle (1)/(4)x(0.05)`

We will now need to determine the amount to dimes. Notice that 1/4 of the total amount of coins were pennies, 1/6 were quarters, and another 1/4 were nickels. Therefore, the amount of coins that were dimes n must be the remaining fraction of the total amount of coins. In other words, the fraction that were dimes is:

`\displaystyle n=100\%-(1)/(4)-(1)/(6)-(1)/(4)`

Evaluate for n. Let 100% equal 1. Hence:

`\displaystyle \begin{aligned} n&=1-(1)/(4)-(1)/(6)-(1)/(4) \\ &=(12)/(12)-(3)/(12)-(2)/(12)-(3)/(12)\\&=(12-3-2-3)/(12)\\&=(4)/(12)=(1)/(3) \end{aligned}`

Therefore, 1/3 of the coins were dimes.

Since dimes are worth $0.10, the total worth in dimes are:

`\displaystyle (1)/(3)x(0.1)`

The total worth of all the coins found was worth $4.32. Therefore:

`\displaystyle (1)/(4)x(0.01)+(1)/(6)x(0.25)+(1)/(4)x(0.05)+(1)/(3)x(0.1)=4.32`

Solve for
`x`, the total amount of coins. First, let’s multiply everything by 100 to remove the decimals. Hence:

`\displaystyle 100((1)/(4)x(0.01)+(1)/(6)x(0.25)+(1)/(4)x(0.05)+(1)/(3)x(0.1))=100(4.32)`

Distribute:

`\displaystyle (1)/(4)(1)x+(1)/(6)(25)x+(1)/(4)(5)x+(1)/(3)(10)x=432`

Now, let’s multiply everything by 12 to remove the fractions. 12 is the LCM of 4, 6, and 3. Hence:

`\displaystyle 12((1)/(4)(1)x+(1)/(6)(25)x+(1)/(4)(5)x+(1)/(3)(10)x)=12(432)`

Distribute:

`3(1)x+2(25)x+3(5)x+4(10)x=5184`

Multiply:

`3x+50x+15x+40x=5184`

Combine like terms:

`108x=5184`

Divide both sides by 108. Hence, the total amount of coins are:

`x=48`

1/4 of the total coins are pennies. Hence, there are 1/4(48) or 12 pennies.

1/6 of the total coins are quarters. Hence, there are 1/6(48) or 8 quarters.

As we determined, 1/4 of the total coins are also nickels. Hence, there are 1/4(48) or 12 nickels.

Finally, as we determined, 1/3 of the total coins are dimes. Hence, there are 1/3(48) or 16 times.