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# Product Estimation with Mixed Numbers/Fractions

## Use benchmarks of 0, 1/2 and 1 whole to estimate products of mixed numbers and fractions.

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Estimate Products and Quotients of Fractions and Mixed Numbers

### [Figure1] License: CC BY-NC 3.0

Nisi and Jamie ordered pizza for lunch. They ate \begin{align*}\frac{2}{3}\end{align*} of the pizza which was cut into 24 slices. How many slices did they eat altogether? If Ethan woke up in the morning and ate half of the remaining pizza, what fraction of the 24 slices did he eat?

In this concept, you will learn to estimate products and quotients of fractions and mixed numbers.

### Estimating Products and Quotients of Fractions

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*} round down and if it is more than \begin{align*}\frac{1}{2}\end{align*} round up.

Let's look at an example.

Estimate the product:

\begin{align*}8\frac{4}{11} \times 7 \frac{11}{12}\end{align*}

First, since \begin{align*}\frac{4}{11}\end{align*} is less than \begin{align*}\frac{1}{2}, 8\frac{4}{11}\end{align*} rounds down to 8.

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

Then, multiply to find the estimated product.

\begin{align*}8 \times 8 = 64\end{align*}

Here is another example.

Estimate the quotient:

\begin{align*}22\frac{3}{10} \div 6\frac{9}{13}\end{align*}

First, since \begin{align*}\frac{3}{10}\end{align*} is less than \begin{align*}\frac{1}{2}\end{align*}, \begin{align*}22\frac{3}{10}\end{align*} rounds down to 22.

Next, since \begin{align*}\frac{9}{13}\end{align*} is greater than \begin{align*}\frac{1}{2}\end{align*}, \begin{align*}6\frac{9}{13}\end{align*} rounds up to 7.

Then, divide to find the estimated product.

\begin{align*}22 \div 7=3.14\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Nisi and Jamie and their pizza.

Of the 24 slices, they had eaten \begin{align*}\frac{2}{3}\end{align*} of the pizza.

Therefore, multiply \begin{align*}\frac{2}{3} \times 24\end{align*} to find the number of slices they had eaten.

First, set up the expression to multiply: \begin{align*}\frac{2}{3} \times \frac{24}{1}\end{align*}

Next, multiply the numerator and the denominators.

\begin{align*}\frac{2}{3} \times \frac{24}{1} = \frac{48}{3}\end{align*}

Then, simplify.

\begin{align*}\frac{48}{3}=16\end{align*}

Therefore, Nisi and Jamie ate 16 of the 24 slices.

Now, since they ate 16 of the 24 slices, there were 8 of the 24 slices left. How many did Ethan eat when he got up?

First, remember that Ethan ate \begin{align*}\frac{1}{2}\end{align*} of the remaining pizza.

Next, set up the expression to solve.

\begin{align*}\frac{8}{24} \times \frac{1}{2}\end{align*}

Then solve and simplify.

\begin{align*}\begin{array}{rcl} \frac{8}{24} \times \frac{1}{2} & = & \frac{8}{48}\\ \frac{8}{48} & = & \frac{1}{6} \end{array}\end{align*}

So, Ethan ate \begin{align*}\frac{1}{6}\end{align*} of the original pizza, or 4 slices.

#### Example 2

Estimate the following quotient.

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

First, since \begin{align*}\frac{11}{12}\end{align*} is greater than \begin{align*}\frac{1}{2}\end{align*}, \begin{align*}14\frac{11}{12}\end{align*} rounds up to 15.

Next, since \begin{align*}\frac{7}{8}\end{align*} is greater than \begin{align*}\frac{1}{2}\end{align*}, \begin{align*}2\frac{7}{8}\end{align*} rounds up to 3.

Then, divide to find the estimated product.

\begin{align*}15 \div 3=5\end{align*}

#### Example 3

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

First, since the fraction is \begin{align*}\frac{1}{2}\end{align*}, \begin{align*}6\frac{1}{2}\end{align*} rounds up to 7.

Next, since \begin{align*}\frac{1}{8}\end{align*} is less than \begin{align*}\frac{1}{2}\end{align*}, \begin{align*}4\frac{1}{8}\end{align*} rounds down to 4.

Then, multiply to find the estimated product.

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

#### Example 4

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

First, since \begin{align*}\frac{3}{4}\end{align*} is greater than \begin{align*}\frac{1}{2}, 11\frac{3}{4}\end{align*} rounds up to 12.

Next, since \begin{align*}\frac{1}{10}\end{align*} is less than \begin{align*}\frac{1}{2}, 2\frac{1}{10}\end{align*} rounds down to 2.

Then, divide to find the estimated product.

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

#### Example 5

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

First, since \begin{align*}\frac{3}{4}\end{align*} is greater than \begin{align*}\frac{1}{2}, \frac{3}{4}\end{align*} rounds up to 1.

Next, since \begin{align*}\frac{8}{9}\end{align*} is greater than \begin{align*}\frac{1}{2}, \frac{8}{9}\end{align*} rounds up to 1.

Then, multiply to find the estimated product.

\begin{align*}1 \times 1 = 1\end{align*}

### Review

Estimate each product or quotient.

1. \begin{align*}\frac{7}{8} \times \frac{7}{8} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}3\frac{1}{2} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}6\frac{2}{3} \times \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}8\frac{1}{12} \times 3\frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}9\frac{4}{5} \times 6\frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}12\frac{1}{3} \times 4\frac{5}{6} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}6\frac{4}{7} \times 3\frac{3}{8} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}12\frac{1}{8} \div 3\frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}24\frac{2}{10} \div 3\frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}28\frac{1}{9} \div 7\frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}9\frac{2}{3} \div 1\frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}14\frac{1}{2} \div 3\frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}9\frac{3}{10} \div 3\frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}16\frac{4}{15} \div 2\frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}30\frac{4}{12} \div 3\frac{1}{18} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

To see the Review answers, open this PDF file and look for section 2.9.

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Color Highlighted Text Notes

### Vocabulary Language: English

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.