Respuesta :
Answer:
There are 48 coins in total.
Shawna found:
12 Pennies.
12 Nickels.
16 Dimes.
And 8 Quarters.
Step-by-step explanation:
We will let [tex]x[/tex] 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:
[tex]\displaystyle \frac{1}{4}x\text{ are pennies.}[/tex]
Since pennies are worth $0.01 each, the worth will be:
[tex]\displaystyle \frac{1}{4}x(0.01)[/tex]
Similarly, we know that one-sixth of the found coins are quarters. Quarter are worth $0.25. So, the total worth in quarters is:
[tex]\displaystyle \frac{1}{6}x(0.25)[/tex]
We have 1.5 times as many nickels as quarters. Therefore, the amount of nickels we have is:
[tex]\displaystyle 1.5(\frac{1}{6}x)=\frac{3}{2}(\frac{1}{6}x)=\frac{1}{4}x[/tex]
Or one-fourth of the total amount.
Since each nickel is worth $0.05, the total worth in nickels is:
[tex]\displaystyle \frac{1}{4}x(0.05)[/tex]
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:
[tex]\displaystyle n=100\%-\frac{1}{4}-\frac{1}{6}-\frac{1}{4}[/tex]
Evaluate for n. Let 100% equal 1. Hence:
[tex]\displaystyle \begin{aligned} n&=1-\frac{1}{4}-\frac{1}{6}-\frac{1}{4} \\ &=\frac{12}{12}-\frac{3}{12}-\frac{2}{12}-\frac{3}{12}\\&=\frac{12-3-2-3}{12}\\&=\frac{4}{12}=\frac{1}{3} \end{aligned}[/tex]
Therefore, 1/3 of the coins were dimes.
Since dimes are worth $0.10, the total worth in dimes are:
[tex]\displaystyle \frac{1}{3}x(0.1)[/tex]
The total worth of all the coins found was worth $4.32. Therefore:
[tex]\displaystyle \frac{1}{4}x(0.01)+\frac{1}{6}x(0.25)+\frac{1}{4}x(0.05)+\frac{1}{3}x(0.1)=4.32[/tex]
Solve for [tex]x[/tex], the total amount of coins. First, let’s multiply everything by 100 to remove the decimals. Hence:
[tex]\displaystyle 100(\frac{1}{4}x(0.01)+\frac{1}{6}x(0.25)+\frac{1}{4}x(0.05)+\frac{1}{3}x(0.1))=100(4.32)[/tex]
Distribute:
[tex]\displaystyle \frac{1}{4}(1)x+\frac{1}{6}(25)x+\frac{1}{4}(5)x+\frac{1}{3}(10)x=432[/tex]
Now, let’s multiply everything by 12 to remove the fractions. 12 is the LCM of 4, 6, and 3. Hence:
[tex]\displaystyle 12(\frac{1}{4}(1)x+\frac{1}{6}(25)x+\frac{1}{4}(5)x+\frac{1}{3}(10)x)=12(432)[/tex]
Distribute:
[tex]3(1)x+2(25)x+3(5)x+4(10)x=5184[/tex]
Multiply:
[tex]3x+50x+15x+40x=5184[/tex]
Combine like terms:
[tex]108x=5184[/tex]
Divide both sides by 108. Hence, the total amount of coins are:
[tex]x=48[/tex]
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.
