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# Expressions with Whole Numbers

## Includes product of a number and the sum of whole numbers

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Expressions for the Product of a Number and a Sum

Terry is going to a friend’s house. On his way there, his friends ask him to pick up some food at the burger place nearby. They order 4 hamburger meals and Terry just wants 2 hamburgers. A hamburger costs $2.10 and the meal costs$3.25 cents extra. What is the total cost of the meal?

In this concept, you will learn how to write a numerical expression for the product of a number and a sum.

### Numerical Expressions

A numerical expression is a statement that has more than one operation in it. A numerical expression should illustrate mathematical information in a correct way. This lesson will focus on a numerical expression that involves the product of a number and a sum.

Let’s break down the phrase “the product of a number and a sum” to understand the meaning of this expression.

The word product means the results of multiplying two or more numbers.

a×b\begin{align*}a \times b\end{align*}

The word sum means the results of adding two or more numbers.

c+d\begin{align*}c + d\end{align*}

The product of a number and a sum is a combination of these operations. The first multiplicand is a number, a\begin{align*}a\end{align*}. The second multiplicand, b\begin{align*}b\end{align*}, will be the entire sum of two or more number. Substitute b\begin{align*}b\end{align*} with the sum of two numbers c+d\begin{align*}c + d\end{align*}. Use parentheses to indicate grouping and also multiplication.

a(c+d)\begin{align*}a (c + d)\end{align*}

Here is an example of an expression using numbers.

the product of 5 and the sum of 4 plus 3\begin{align*}\text{the product of 5 and the sum of 4 plus 3}\end{align*}

Identify the first multiplicand. It is 5. Then, identify the second multiplicand. The second will be the sum of 4 plus 3. Write is as a numerical expression.

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

This is the numerical expression for the information.

### Examples

#### Example 1

Earlier, you were given a problem about Terry at the hamburger shop.

Terry is picking up 2 hamburgers that cost $2.10 each and 4 hamburger meals that cost$3.25 extra per meal. Multiply and add to find the total cost of the meal.

First, think of an expression to find the total cost of the meal. You need to find the sum of the hamburgers and the hamburger meals.

total cost of hamburgers+total cost of hamburger meals\begin{align*}\text{total cost of hamburgers} + \text{total cost of hamburger meals}\end{align*}

Let’s work on the first part, the total cost of the hamburger. The total cost of the hamburgers will be the product of the number of hamburgers times the cost of the burger.

2(2.10)+total cost of the hamburger meals\begin{align*}2(\2.10) + \text{total cost of the hamburger meals}\end{align*} Then, write an expression to find the second part of the expression. The total cost of the meals is the product of the number of meals times the sum of the hamburger plus the added cost. 2(2.10)+4($2.10+$3.25)\begin{align*}2(\2.10) + 4(\2.10 + \3.25)\end{align*}

Next, find the total cost of the order. Order of operations tells you to do the addition in the parentheses first; find each product; and then find the sum.

2($2.10)+4($5.35)$4.20+$21.40  25.60\begin{align*}\begin{array}{rcl} & & 2(\2.10) + 4(\5.35)\\ & &\quad \4.20 + \21.40\\ & & \qquad \ \ \25.60 \end{array}\end{align*} The total cost of the order will be25.60.

#### Example 2

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

First, identify the first multiplicand. It is 2.

Then identify the second multiplicand. It is the sum of 3 and 4 or 3 + 4.

Next, write the product as a numerical expression. Use the parentheses to group the sum.

\begin{align*}2(3 + 4)\end{align*}

#### Example 3

Write the expression as a numerical expression.

\begin{align*}\text{the product of three and the sum of four plus five}\end{align*}

First, identify the first multiplicand. It is 3.

Then identify the second multiplicand. It is the sum of 4 + 5.

Next, write the product as a numerical expression. Use the parentheses to group the sum.

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

#### Example 4

Write the expression as a numerical expression.

\begin{align*}\text{the product of four and the sum of six plus seven}\end{align*}

First, identify the first multiplicand. It is 4.

Then identify the second multiplicand. It is the sum of 6 + 7.

Next, write the product as a numerical expression. Use the parentheses to group the sum.

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

#### Example 5

Write the expression as a numerical expression.

\begin{align*}\text{the product of nine and the sum of one plus eight}\end{align*}

First, identify the first multiplicand. It is 9.

Then identify the second multiplicand. It is the sum of 1 + 8.

Next, write the product as a numerical expression. Use the parentheses to group the sum.

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

### Review

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.

To see the Review answers, open this PDF file and look for section 4.4.

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Color Highlighted Text Notes

### Vocabulary Language: English

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.