These are the first sentences in the short story, "The Gift of the Magl," by O. Henry: "One dollar and elghty-seven cents. That was all. And sixty cents of it was in pennies." A. Simplify the equation that represents the value of the coins. Make it an equation that expresses the value of quarters, dimes, and nickels. Show your work B. Is it possible to have exactly 60 pennies in $$ \$ 1.87 $$ ? Explain.

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UpStudy AI · Accepted Answer

Certainly! Let's tackle both parts of your question step by step.

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## **A. Simplifying the Equation for Quarters, Dimes, and Nickels**

**Given:**
- **Total Amount:** \$1.87
- **Amount in Pennies:** \$0.60

Since \$0.60 is in pennies, the remaining amount to be made up with quarters, dimes, and nickels is:
$$
\$1.87 - \$0.60 = \$1.27
$$
We'll express \$1.27 using quarters (25¢), dimes (10¢), and nickels (5¢).

**Let:**
- $$ q $$ = Number of quarters
- $$ d $$ = Number of dimes
- $$ n $$ = Number of nickels

**Convert Dollars to Cents for Simplicity:**
$$
\$1.27 = 127 \text{ cents}
$$
$$
\$0.25 = 25 \text{ cents (quarter)}
$$
$$
\$0.10 = 10 \text{ cents (dime)}
$$
$$
\$0.05 = 5 \text{ cents (nickel)}
$$

**Formulating the Equation:**
$$
25q + 10d + 5n = 127
$$

**Simplifying the Equation:**
To make it simpler, we can divide the entire equation by 5:
$$
\frac{25q}{5} + \frac{10d}{5} + \frac{5n}{5} = \frac{127}{5}
$$
$$
5q + 2d + n = 25.4
$$

However, since the number of coins must be whole numbers, having a decimal on the right side indicates that the equation might not have a valid solution with whole numbers for $$ q $$, $$ d $$, and $$ n $$.

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## **B. Is It Possible to Have Exactly 60 Pennies in \$1.87?**

**Analysis:**
- **Pennies:** 60 pennies = 60 cents = \$0.60
- **Remaining Amount:** \$1.87 - \$0.60 = \$1.27

We need to determine if \$1.27 can be composed using only quarters (25¢), dimes (10¢), and nickels (5¢).

**Key Point:**
- All coins other than pennies are multiples of 5 cents.
- Therefore, any amount made with quarters, dimes, and nickels must be a multiple of 5 cents.

**Checking \$1.27:**
$$
127 \text{ cents} \div 5 = 25.4
$$
Since 127 is **not** divisible by 5 (it leaves a remainder), it's **impossible** to make \$1.27 using only quarters, dimes, and nickels.

**Conclusion:**
No, it is **not possible** to have exactly 60 pennies in \$1.87 when using only quarters, dimes, and nickels for the remaining amount.

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**Summary:**
- **Part A:** The equation $$ 25q + 10d + 5n = 127 $$ represents the value of quarters, dimes, and nickels that sum up to \$1.27.
- **Part B:** It's impossible to have exactly 60 pennies in \$1.87 because the remaining \$1.27 cannot be composed using only quarters, dimes, and nickels.
It's impossible to have exactly 60 pennies in \$1.87 because the remaining \$1.27 cannot be formed using only quarters, dimes, and nickels.

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