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# Mental Math to Evaluate Products

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Mental Math to Evaluate Products

Mr. Thompson is preparing supplies for an art lesson. Each student will be given a bag with 15 color pencils and markers. There are total of 25 students in the class. Mr. Thompson doesn’t have a calculator or a pencil with him. How can Mr. Thompson find the total number of drawing utensils he will need for the class?

In this concept, you will use mental math to evaluate products using the distributive property.

### Mental Math

Some multiplication problems can be solved by doing the calculations in your head. That is called mental math. Some multiplication problems may seem too large to do in your head. But remember, a large number can be broken down into the sum of two smaller numbers.

\begin{align*}109 = 100 + 9\end{align*}

The distributive property can help you evaluate the product of larger number. Remember that the distributive property is a property that allows you to multiply a number and a sum by distributing the multiplier outside the parentheses with each numbers inside the parentheses.

\begin{align*}a(b+c) = ab+ac\end{align*}

Here is a multiplication problem.

\begin{align*}11 \times 109 = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Think of 109 as the sum of 100 plus 9.

\begin{align*}11 \times 109 = 11(100 + 9)\end{align*}

Instead of multiplying 11 times 109, use the distributive property to distribute the multiplication to equal the sum of 11 times 100 and 11 times 9.

\begin{align*}11(100 + 9) = 11(100) + 11(9)\end{align*}

Try to find the products using mental math. Multiply 11 times 100 in your head. Remember than when you multiply a number by a multiple of 10, ignore the zeros and place it into the product at the end.

\begin{align*}11 \times 100 \rightarrow 11 \times 100 = 11{\color{red}00}\end{align*}

Think 11 times 9 is 99. Now find the sum of the products.

\begin{align*}\begin{array}{rcl} && \quad 11(100) + 11(9)\\ && 1,100 + 99 = 1,199 \end{array}\end{align*}

You have found the product of 11 times 109.

\begin{align*}11 \times 109 = 1,199\end{align*}

Here is another multiplication problem.

\begin{align*}12 \times 53 = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

This problem would be difficult to do in your head. However, think of 53 as the sum of 50 plus 3.

\begin{align*}12 \times 53 = 12(50 + 3)\end{align*}

Use the distributive property to distribute the multiplication of 12 between 50 and 3.

\begin{align*}12(50 + 3) = 12(50) + 12(3)\end{align*}

Multiply 12 times 50 in your head. 12 times 5 is 60, so 12 times 50 is 600. Multiply 12 times 3 in your head. 12 times 3 is 36. Now, add the sum of the products.

\begin{align*}\begin{array}{rcl} && 12(50) + 12(3)\\ && 600 + 36 = 636 \end{array}\end{align*}

The product of 12 times 53 is 636.

\begin{align*}12 \times 53 = 636\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Mr. Thompson’s art lesson.

He needs 15 color pencils and markers for 25 students. Multiply the numbers using mental math to find out how many drawing utensils Mr. Thompson needs.

First, change one of the numbers to the sum of two smaller numbers. 15 is 10 plus 5. Note that the larger number does not always have to be the number broken down into two smaller numbers. Finding multiplies of 25 is similar to finding multiples of a quarter.

Then, use the distributive property to find the product of a number and a sum.

\begin{align*}25(10 + 5) = 25(10) + 25(5)\end{align*}

Next, calculate the products using mental math and add the sum. 25 times 10 is 250. 25 times 5 is 125. (5 quarters is equal to \$1.25 or 125 cents.

\begin{align*}\begin{array}{rcl} && \ \ 25(10) + 25(5)\\ && 250 + 125 = 375 \end{array}\end{align*}

Mr. Thompson will need a total of 375 drawing utensils.

#### Example 2

Use mental math to solve this problem.

\begin{align*}9 \times 81 = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

First, change one of the numbers to the sum of two smaller numbers. 81 is \begin{align*}80 + 1\end{align*}.

Then, use the distributive property to find the product of a number and a sum.

\begin{align*}9(80 + 1) = 9(80) + 9(1)\end{align*}

Next, calculate the products using mental math. Multiply 9 times 80. Think 9 times 80 is 9 times 8, then add a zero, 720. 9 times 1 is 9.

\begin{align*}9(80) + 9(1) = 720 + 9\end{align*}

Finally, add the sum of the products.

\begin{align*}720 + 9 = 729\end{align*}

The product of 9 times 81 is 729.

#### Example 3

Use mental math to evaluate the product.

\begin{align*}5(99)\end{align*}

First, change one of the numbers to the sum of two smaller numbers. 99 is 90 plus 9.

Then, use the distributive property to find the product of a number and a sum.

\begin{align*}5(90 + 9) = 5(90) + 5(9)\end{align*}

Next, calculate the products using mental math and add the sum. 5 times 90 is 450. 5 times 9 is 45.

\begin{align*}\begin{array}{rcl} && \ \ 5(90) + 5(9)\\ && 450 + 45 = 495 \end{array}\end{align*}

The product of 5 times 99 is 495.

#### Example 4

Use mental math to evaluate the product.

\begin{align*}4(65)\end{align*}

First, change one of the numbers to the sum of two smaller numbers. 65 is 60 plus 5.

Then, use the distributive property to find the product of a number and a sum.

\begin{align*}4(60 + 5) = 4(60) + 4(5)\end{align*}

Next, calculate the products using mental math and add the sum. 4 times 60 is 240. 4 times 5 is 20.

\begin{align*}\begin{array}{rcl} && \ \ 4(60) + 4(5)\\ && 240 + 20 = 260 \end{array}\end{align*}

The product of 4 times 65 is 260.

#### Example 5

Use mental math to evaluate the product.

\begin{align*}3(140)\end{align*}

First, change one of the numbers to the sum of two smaller numbers. 120 is 100 plus 40.

Then, use the distributive property to find the product of a number and a sum.

\begin{align*}3(100 + 40) = 3(100) + 3(40)\end{align*}

Next, calculate the products using mental math and add the sum. 3 times 100 is 300. 3 times 40 is 120.

\begin{align*}\begin{array}{rcl} && \ \ 3(100) + 3(20)\\ && 300 + 120 = 420 \end{array}\end{align*}

The product of 3 times 140 is 420.

### Review

Use mental math to evaluate the following expressions.

1. \begin{align*}4(68)﻿\end{align*}
2. \begin{align*}8(45)\end{align*}
3. \begin{align*}3(61)\end{align*}
4. \begin{align*}2(53)\end{align*}
5. \begin{align*}9(22)\end{align*}
6. \begin{align*}6(44)\end{align*}
7. \begin{align*}5(120)\end{align*}
8. \begin{align*}12(12)\end{align*}
9. \begin{align*}11(18)\end{align*}
10. \begin{align*}13(12)\end{align*}
11. \begin{align*}15(22)\end{align*}
12. \begin{align*}20(106)\end{align*}
13. \begin{align*}12(310)\end{align*}
14. \begin{align*}25(16)\end{align*}
15. \begin{align*}50(720)\end{align*}
16. \begin{align*}100(210)\end{align*}
17. \begin{align*}110(180)\end{align*}
18. \begin{align*}1,200(14)\end{align*}

To see the Review answers, open this PDF file and look for section 4.6.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

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.

Property

A property is a rule that works for a given set of numbers.

Sum

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