# 8.8: Coin Stumpers

**At Grade**Created by: CK-12

The weight of a pile of pennies and dimes is 142 grams. The total weight of the pennies is 50 grams. How many pennies and dimes are in the pile? In this concept, we will learn how to reason proportionally to answer questions about coins.

### Guidance

In order to answer the questions about coins like the one above, use the problem solving steps.

- Start by
**describing**what you know from the information given. - Next, figure out what
**your job**is in this problem. In all of these problems your job will be to answer a question about the coins. - Then, make a
**plan**for how you will solve. Think about what other information you know about the weight of the coins. See if you can figure out the weight and number of each type of coin in the pile. - Next,
**solve**the problem. - Finally,
**check**to make sure that your answer works with the original information given.

#### Example A

A pile of 100 pennies weighs 250 grams. A nickel weighs twice as much as a penny. What is the weight of a pile of 300 coins, half pennies and half nickels?

**Solution:**

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && 100 \ \text{pennies weigh} \ 250 \ \text{grams.}\\ &&& \text{A nickel weighs twice as much as a penny.}\\ &&& \text{A new pile has} \ 300 \ \text{coins. Half of the} \ 300 \ \text{are pennies and half are nickels.}\\ \\ & \mathbf{My \ job:} && \text{Figure out the weight of the pile of} \ 300 \ \text{coins, half pennies and half nickels.}\\ \\ & \mathbf{Plan:} && \text{Compute the weight of a penny and a nickel. Then, use those weights to figure out}\\ &&& \text{the weight of the pile of} \ 300 \ \text{coins, half pennies and half nickels.}\\ \\ & \mathbf{Solve:} && 100 \ \text{pennies are} \ 250 \ \text{grams.}\\ &&& \text{So, one penny is} \ \frac{250}{100}, \ \text{or} \ 2.5 \ \text{grams.}\\ &&& \text{One nickel is} \ 2 \times 2.5, \ \text{or} \ 5 \ \text{grams.}\\ &&& 150 \times 2.5 = 375 \ \text{grams}\\ &&& 150 \times 5.0 = 750 \ \text{grams}\\ &&& \text{The pile of} \ 300 \ \text{coins is} \ 1125 \ \text{grams.}\\ \\ & \mathbf{Check:} && 100 \ \text{pennies are} \ 250 \ \text{grams. Using a proportion, the weight of} \ 150 \ \text{pennies is} \ 1.5\\ &&& \text{times} \ 250, \ \text{or} \ 375 \ \text{grams.}\\ &&& \text{Since} \ 100 \ \text{pennies are} \ 250 \ \text{grams}, \ 100 \ \text{nickels are} \ 2 \times 250 \ \text{or} \ 500 \ \text{grams. So}, \ 150\\ &&& \text{nickels are} \ 1.5 \times 500, \ \text{or} \ 750 \ \text{grams.}\\ &&& 375 + 750 = 1125 \ \text{grams}\end{align*}

We have already figured out the numbers of grams for a penny and a nickel. Next we will find the number of grams for a dime in Example B. Then we will use those weights to help with all of the other problems in this concept.

#### Example B

The weight of a pile of nickels and dimes is 273 grams. There are 60 coins in the pile. Fifty of the coins are nickels. What is the weight of a dime? Record that number in the list below.

**Solution:**

We can use problem solving steps to help.

**Describe:** 60 coins weigh 273 grams. 50 of the coins are nickels. The rest of the coins are dimes.

**My Job:** Figure out the weight of a dime and record it in the list.

**Plan:** I know the weight of a nickel is 5 grams. Figure out the weight of 50 nickels. Subtract that from the weight of 273 grams to find the weight of 10 dimes. Then, find the weight of one dime.

**Solve:** One nickel is 5 grams, so 50 nickels is \begin{align*}50 \times 5 \end{align*} grams or 250 grams.

- Since the pile of 60 coins weighs 273 grams. \begin{align*}273-250=23 \end{align*} grams is how much the 10 dimes weigh.
- One dime is \begin{align*}\frac{23 \ \text{grams}}{10}=2.3 \ \text{grams}\end{align*}.

**Check:** 50 nickels: \begin{align*}5 \ \text{grams} \times 50=250 \ \text{grams}\end{align*}

- 10 dimes: \begin{align*}2.3 \ \text{grams} \times 10=23 \ \text{grams}\end{align*}
- \begin{align*}250+23=273\end{align*}

#### Example C

The weight of a pile of pennies and nickels is 300 grams. There are 20 nickels. How many pennies are there?

**Solution:**

We can use problem solving steps to help.

**Describe:** There is a pile of pennies and nickels that weighs 300 grams. There are 20 nickels.

**My Job:** Figure out how many pennies there are.

**Plan:** I know that a penny weighs 2.5 grams and a nickel weighs 5 grams. Find the weight of 20 nickels. Subtract that from 300 grams to find the weight of the pennies. Divide the weight of the pennies by 2.5 grams to find out how many pennies there are.

**Solve:** One nickel weighs 5 grams so 20 nickels weighs \begin{align*} 20\times 5 \ \text{grams}=100 \ \text{grams}\end{align*}.

- The pile is 300 grams, so that means that the pennies weigh \begin{align*}300 \ \text{grams}-100 \ \text{grams}=200 \ \text{grams}\end{align*}.
- One penny weighs 2.5 grams. \begin{align*}\frac{200 \ \text{grams}}{2.5 \ \text{grams}}=80\end{align*} so 80 pennies weigh 200 grams. There are 80 pennies in the pile.

**Check:** 20 nickels: \begin{align*}5 \ \text{grams} \times 20=100 \ \text{grams}\end{align*}

- 80 pennies: \begin{align*}2.5 \ \text{grams} \times 80=200 \ \text{grams}\end{align*}
- \begin{align*}100+200=300\end{align*}

#### Concept Problem Revisited

The weight of a pile of pennies and dimes is 142 grams. The total weight of the pennies is 50 grams. How many pennies and dimes are in the pile?

We can use the information from the examples and problem solving steps to help us solve this problem.

**Describe:** There is a pile of pennies and dimes that weighs 142 grams. The pennies weigh 50 grams.

**My Job:** Figure out how many pennies and dimes are in the pile.

**Plan:** Use the fact that the whole pile is 142 grams and the pennies are 50 grams to find the weight of the dimes. I know that a penny weighs 2.5 grams and a dime weighs 2.3 grams. Use this information to find out how many pennies and dimes there are.

**Solve:** \begin{align*}142 \ \text{grams}-50 \ \text{grams}=92 \ \text{grams}\end{align*} so the dimes weigh 92 grams.

- One penny weighs 2.5 grams. \begin{align*}\frac{50 \ \text{grams}}{2.5 \ \text{grams}}=20 \end{align*} so there are 20 pennies.
- One dime weighs 2.3 grams. \begin{align*}\frac{92 \ \text{grams}}{2.3 \ \text{grams}}=40 \end{align*} so there are 40 dimes.

**Check:** 20 pennies: \begin{align*}2.5 \ \text{grams} \times 20=50 \ \text{grams}\end{align*}

- 40 dimes: \begin{align*}2.3 \ \text{grams} \times 40=92 \ \text{grams}\end{align*}
- \begin{align*}50+92=142\end{align*}

### Vocabulary

In this concept we used ** proportional reasoning** when we used what we knew about a pile of coins to figure out information about one coin. Any time there is a constant ratio between two quantities (such as the number of pennies in a pile and the weight of the pile) we can use

**to solve.**

*proportional reasoning*### Guided Practice

1. The total weight of a pile of pennies, nickels, and dimes is 191.4 grams. There are 20 pennies in the pile. There are 2 fewer dimes than pennies. How many coins are nickels?

2. The total weight of a pile of pennies and dimes is 96 grams. There are 20 dimes in the pile. What is the total value of the coins?

3. The total weight of a pile of nickels and pennies is 3000 grams. The total value of the nickels is $25. How many pennies are in the pile?

**Answers:**

1. 20 nickels

- 20 pennies are \begin{align*}20 \times 2.5\end{align*} grams \begin{align*}= 50\end{align*} grams; 18 dimes are \begin{align*}18 \times 2.3\end{align*} grams \begin{align*}= 41.4\end{align*} grams. \begin{align*}191.4 - 91.4 = 100\end{align*} grams.
- \begin{align*}\frac{100}{5} = 20\end{align*} nickels.

2. $2.20

- 20 dimes is \begin{align*}20 \times 2.3\end{align*} grams, or 46 grams
- \begin{align*}96 - 46 = 50\end{align*} grams
- \begin{align*}\frac{50}{2.5} = 20\end{align*} pennies
- 20 dimes is $2.00
- 20 pennies is $0.20
- \begin{align*}\$2 + \$0.20 = \$2.20\end{align*}

3. 200 pennies

- \begin{align*}\$25 \times 20\end{align*} nickels/dollar \begin{align*}= 500\end{align*} nickels
- \begin{align*}500 \times 5\end{align*} grams \begin{align*}= 2500\end{align*} grams of nickels
- \begin{align*}3000 - 2500 = 500\end{align*} grams of pennies
- \begin{align*}\frac{500}{2.5} = 200\end{align*} pennies

### Practice

- A pile of pennies, nickels and dimes weighs 146 grams. There are 30 pennies in the pile. There are 10 less dimes than pennies. What is the total value of the nickels in the pile?
- A pile of pennies, nickels and dimes weighs 195.5 grams. There are 15 pennies in the pile. There are 12 more nickels than pennies. What is the total value of the coins in the pile?
- The total weight of a pile of pennies, nickels, and dimes is 84.6 grams. There are 22 pennies in the pile. There are 20 fewer dimes than pennies. How many coins are nickels?
- The total weight of a pile of pennies and nickels is 142.5 grams. There are 15 nickels in the pile. What is the total value of the coins?
- The total weight of a pile of nickels and dimes is 1273.1 grams. The total value of the nickels is $10.50. How many dimes are in the pile?

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### Image Attributions

Students use proportional reasoning to answer questions about coins. Students use problem solving steps to help.