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2.9: Estimate Products and Quotients of Fractions and Mixed Numbers

Difficulty Level: At Grade Created by: CK-12
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Have you ever tried to estimate a product or a quotient? Take a look at this dilemma with fractions and mixed numbers.

\begin{align*}4 \frac{1}{12} \times 12 \frac{3}{2}\end{align*}4112×1232

To figure this out, you need to know how to estimate a product. You will know how to do this by the end of the Concept.

Guidance

Estimation is a useful strategy to use to check that your computation is reasonable. It is also a way to find an approximate answer to a solution.

To estimate products and quotients of mixed numbers, round the fractions to the nearest whole number. If the fraction is less than \begin{align*}\frac{1}{2}\end{align*}12, round down. If the fraction is greater than \begin{align*}\frac{1}{2}\end{align*}12, round up.

Estimate the product: \begin{align*}8 \frac{4}{11} \times 7\frac{11}{12}\end{align*}8411×71112

Round the first number.

Since \begin{align*}\frac{4}{11}\end{align*}411 is less than \begin{align*}\frac{1}{2},8\frac{4}{11}\end{align*}12,8411 rounds down to 8.

Round the second number.

Since \begin{align*}\frac{11}{12}\end{align*}1112 is greater than \begin{align*}\frac{1}{2},7\frac{11}{12}\end{align*}12,71112 rounds up to 8.

Now multiply to find the estimated product.

\begin{align*}8\times8=64\end{align*}8×8=64

A reasonable estimate is 64.

Here is another one.

Estimate the quotient: \begin{align*}22\frac{3}{10}\div 6\frac{9}{13}\end{align*}22310÷6913

Round the first number.

Since \begin{align*}\frac{3}{10}\end{align*}310 is less than \begin{align*}\frac{1}{2},22\frac{3}{10}\end{align*}12,22310 rounds down to 22.

Round the second number.

Since \begin{align*}\frac{9}{13}\end{align*}913 is greater than \begin{align*}\frac{1}{2},6\frac{9}{13}\end{align*}12,6913rounds up to 7.

Now divide to find the estimated product. Since 22 is not divisible by 7, round it down to 21 to make compatible numbers that are easier to divide.

\begin{align*}21\div 7=3\end{align*}21÷7=3

A good estimate for the quotient is 3.

Notice that you have to use some common sense and thinking to figure out that you had to round 22 down to 21 to find a good estimate.

Estimate each product or quotient.

Example A

\begin{align*}6 \frac{1}{2} \times 4 \frac{1}{8}\end{align*}612×418

Solution:  \begin{align*}7 \times 4 = 28\end{align*}7×4=28

Example B

\begin{align*}11 \frac{3}{4} \div 2 \frac{1}{10}\end{align*}1134÷2110

Solution:  \begin{align*}12 \div 2 = 6\end{align*}12÷2=6

Example C

\begin{align*}\frac{3}{4} \times \frac{8}{9}\end{align*}34×89

Solution:  \begin{align*}1 \times 1 = 1\end{align*}1×1=1

Now let's go back to the dilemma from the beginning of the Concept.

\begin{align*}4 \frac{1}{12} \times 12 \frac{3}{12}\end{align*}4112×12312

First, round each value.

\begin{align*}4 \frac{1}{12}\end{align*}4112 becomes \begin{align*}4\end{align*}4

\begin{align*}12 \frac{3}{12}\end{align*}12312 becomes \begin{align*}12\end{align*}12

Now we can write a new problem.

\begin{align*}4 \times 12 = 48\end{align*}4×12=48

Our reasonable estimate is \begin{align*}48\end{align*}48.

Vocabulary

Greatest Common Factor
a number that will divide evenly into both the numerator and the denominator of a fraction.
Product
the answer in a multiplication problem.
Quotient
the answer in a division problem.
Fraction
a part of a whole
Mixed Number
a number with a whole number and a fraction.
Improper Fraction
a number that is greater than a whole with a larger top number and a smaller bottom number.
Estimation
a reasonable answer

Guided Practice

Here is one for you to try on your own.

Estimate the following quotient.

\begin{align*}14 \frac{11}{12} \div 2 \frac{7}{8}\end{align*}141112÷278

Solution

Begin by rounding each value. Here is the new equation.

\begin{align*}15 \div 3 = x\end{align*}15÷3=x

Now solve for \begin{align*}x\end{align*}x.

\begin{align*}x = 5\end{align*}x=5

This is our estimate.

Video Review

Practice

Directions: Estimate each product or quotient.

1. \begin{align*}\frac{7}{8} \times \frac{7}{8}\end{align*}78×78

2. \begin{align*}3 \frac{1}{2} \times \frac{3}{4}\end{align*}312×34

3. \begin{align*}6 \frac{2}{3} \times \frac{4}{5}\end{align*}623×45

4. \begin{align*}8 \frac{1}{12} \times 3 \frac{1}{8}\end{align*}8112×318

5. \begin{align*}9 \frac{4}{5} \times 6 \frac{1}{9}\end{align*}945×619

6. \begin{align*}12 \frac{1}{3} \times 4 \frac{5}{6}\end{align*}1213×456

7. \begin{align*}6 \frac{4}{7} \times 3 \frac{3}{8}\end{align*}647×338

8. \begin{align*}12 \frac{1}{8} \div 3 \frac{1}{3}\end{align*}1218÷313

9. \begin{align*}24 \frac{2}{10} \div 3 \frac{1}{3}\end{align*}24210÷313

10. \begin{align*}28 \frac{1}{9} \div 7 \frac{1}{10}\end{align*}2819÷7110

11. \begin{align*}9 \frac{2}{3} \div 1 \frac{4}{5}\end{align*}923÷145

12. \begin{align*}14 \frac{1}{2} \div 3 \frac{1}{10}\end{align*}1412÷3110

13. \begin{align*}9 \frac{3}{10} \div 3 \frac{1}{9}\end{align*}9310÷319

14. \begin{align*}16 \frac{4}{15} \div 2 \frac{1}{5}\end{align*}16415÷215

15. \begin{align*}30 \frac{4}{12} \div 3 \frac{6}{18}\end{align*}30412÷3618

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Vocabulary

Estimation

Estimation is the process of finding an approximate answer to a problem.

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as 4\frac{3}{5}.

Product

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

Quotient

The quotient is the result after two amounts have been divided.

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