question 3\nyou have 64 coins, consisting of pennies, nickels, and quarters. the value of the coins is…

Explanation:

Step1: Define variables

Let $p$ be the number of pennies, $n$ be the number of nickels, and $q$ be the number of quarters. We know that:

1. The total number of coins: $p + n+q=64$.
2. The total value of the coins in cents: $p + 5n+25q = 460$ (since $$4.60 = 460$ cents).
3. The relationship between the number of nickels and quarters: $n = 3q$.

Step2: Substitute $n = 3q$ into the first two - equations

Substitute $n = 3q$ into $p + n+q=64$, we get $p+3q + q=64$, which simplifies to $p + 4q=64$, so $p=64 - 4q$. Substitute $n = 3q$ and $p=64 - 4q$ into $p + 5n+25q = 460$. We have $(64 - 4q)+5\times(3q)+25q = 460$.

Step3: Solve the resulting equation for $q$

Expand the left - hand side: $64-4q + 15q+25q=460$. Combine like terms: $64+( - 4q+15q + 25q)=460$, $64 + 36q=460$. Subtract 64 from both sides: $36q=460 - 64=396$. Divide both sides by 36: $q=\frac{396}{36}=11$.

Step4: Find $n$ and $p$

Since $n = 3q$, then $n = 3\times11 = 33$. Since $p=64 - 4q$, then $p=64-4\times11=64 - 44 = 20$.

Answer:

Number of pennies = 20 Number of nickels = 33 Number of quarters = 11