# 4.4: Expressions for the Product of a Number and a Sum

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

Have you ever been to an Omni - Theater? Well, Kyle and his class hope to visit one at the science museum. But of course, there are new problems to tackle with this added adventure. Take a look.

Three days before the trip, Mrs. Andersen comes running up to Kyle. She has discovered that there is an Omni Theater at the Science Museum and they are showing a film on the Rainforest. Kyle is thrilled. He loves the Omni Theater. However, the problem is that it will cost an additional two dollars for each of the students to attend the showing. The Chaperones can all go for free. “Can you work this out?” Mrs. Andersen asks Kyle. “There are fifty dollars in our class account plus the money that you have already collected from the students. How much money total will we need to go to both the museum and the Omni Theater?”

“I will handle it,” Kyle says. “I think we have enough money for everything. Let me figure it out.”

Mrs. Andersen smiles and goes back to work. Kyle takes out a piece of paper and a pencil. He writes down the following information.

22 students with an admission price of $8.95

22 students with an Omni Theater price of $2.00

Ah! Kyle remembers that he can use parentheses to help him out with this problem. He isn't sure how.

**In this Concept, you will learn how to write numerical expressions just like the one that Kyle will need. Pay close attention and you can help him out at the end of the Concept.**

### Guidance

In an earlier Concept, you learned about numerical expressions. So, you know that a ** numerical expression** is a statement that has more than one operation in it. When we write an expression, we want it to illustrate mathematical information in a correct way. We can write expressions that contain all kinds of combinations of operations.

Today, we are going to learn about how to write an expression that involves the product of a number and a sum.

**How do we write an expression that involves the product of a number and a sum?**

The first thing that we need to do is to decipher these words so that we can understand what we are actually talking about.

**The product of a number** – we know that product means multiplication. We are going to be multiplying this number.

**And a sum** – the word sum means addition. We are going to have a sum here. That means that we will have two numbers that are being added together.

The tricky thing about this wording is that it talks about the product of a number AND a sum. That means that we are going to be multiplying a number by an ENTIRE sum.

**We can figure out what this looks like by first taking a number.**

**Let’s use 5.**

**Then we take a sum.**

**Let’s use 4 + 3.**

**Now because we want to multiply the number times the sum, we need to put the sum into parentheses.**

**Here is our answer.**

**5(4 + 3)**

**This is a numerical expression for the information.**

Try writing a few of these on your own.

#### Example A

The product of three and the sum of four plus five.

**Solution: \begin{align*}3(4 + 5)\end{align*}**

#### Example B

The product of four and the sum of six plus seven.

**Solution:\begin{align*}4(6 + 7)\end{align*}**

#### Example C

The product of nine and the sum of one plus eight.

**Solution:\begin{align*}9(1 + 8)\end{align*}**

Now you can help Kyle. With the information given to him by his teacher, Kyle wrote down the following information.

22 students with an admission price of $8.95

22 students with an Omni Theater price of $2.00

Ah! Kyle remembers that he can use parentheses to help him out with this problem. Here is what he finally writes.

**\begin{align*}22(8.95 + 2.00)\end{align*}**

**This is a numerical expression that makes sense for Kyle and his dilemma.**

### Vocabulary

Here are the vocabulary words found in this Concept.

- Numerical expression
- a number sentence that has at least two different operations in it.

- Product
- the answer in a multiplication problem

- Sum
- the answer in an addition problem

### Guided Practice

Here is one for you to try on your own.

Write a numerical expression for the product of 2 times the sum of 3 and 4.

**Answer**

Here we know that two is going to be outside the parentheses-“the product of 2.”

The grouping of 3 plus 4 will be inside the parentheses-this is the sum.

Here is our expression.

**Our answer is \begin{align*}2(3 + 4)\end{align*}.**

### Practice

Directions: Write a numerical expression for each sentence.

1. The product of two and the sum of five and six.

2. The product of three and the sum of three and seven.

3. The product of five and the sum of two and three.

4. The product of four and the sum of three and five.

5. The product of seven and the sum of four and five.

6. The product of ten and the sum of five and seven.

7. The product of six and the sum of five and two.

8. The product of five and the sum of four and nine.

9. The product of thirteen and the sum of five and twelve.

10. The sum of six and seven times three.

11. The sum of eight and ten times four.

12. The sum of six and fifteen times eight.

13. The sum of four and nine times twelve.

14. The sum of three and eight times sixteen.

15. The product of eight and the sum of four and fourteen.

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Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.Product

The product is the result after two amounts have been multiplied.Sum

The sum is the result after two or more amounts have been added together.### Image Attributions

Here you'll learn to write a numerical expression for the product of a number and a sum.

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