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Estimation of Sums of Mixed Numbers and Fractions

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

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Estimation of Sums of Mixed Numbers and Fractions
License: CC BY-NC 3.0

Megan and her mom planted a vegetable garden this summer. They noticed animals getting into the garden so they decided to put fencing around the perimeter of the garden. The garden is a rectangular shape. Megan's mom told her that the length of the garden is \begin{align*}5\frac{1}{6}\end{align*} feet and the width of the garden is \begin{align*}7 \frac{7}{8}\end{align*} feet. The fencing is $6 a foot. How can Megan approximate the perimeter of the garden in order to get a quick estimate of how much the fencing will cost?

In this concept, you will learn to estimate the sums of fractions and mixed numbers.

Estimating Sums of Fractions and Mixed Numbers

Estimation is a method for finding an approximate solution to a problem. You can estimate a solution when you don't need an exact answer or when you want to check if your exact answer is reasonable.

When working with fractions and mixed numbers, you can use the fraction benchmarks of \begin{align*}0, \frac{1}{2},\end{align*} and \begin{align*}1\end{align*} to help you to estimate sums.

Here are the steps for estimating the sums of fractions and mixed numbers using benchmarks.

  1.  Approximate the value of each fraction or mixed number using the benchmarks of \begin{align*}0, \frac{1}{2},\end{align*} and \begin{align*}1\end{align*}.
  2.  Add the approximated values to get an estimated sum.

Here is an example.

Estimate the sum of \begin{align*}\frac{5}{9} + \frac{1}{77}\end{align*}.

First, approximate the value of each individual fraction.

  • \begin{align*}\frac{5}{9}\end{align*} is approximately \begin{align*}\frac{1}{2}\end{align*}
  • \begin{align*}\frac{1}{77}\end{align*} is approximately \begin{align*}0\end{align*}

Next, add the approximated values.

\begin{align*}\frac{1}{2}+0=\frac{1}{2}\end{align*}

The answer is \begin{align*}\frac{5}{9}+\frac{1}{77}\end{align*} is approximately \begin{align*}\frac{1}{2}\end{align*}.

Here is another example.

Estimate the sum of \begin{align*}3 \frac{6}{7}+1 \frac{4}{9}\end{align*}.

First, approximate the value of each individual mixed number.

  • \begin{align*}3 \frac{6}{7}\end{align*} is approximately \begin{align*}4\end{align*}
  • \begin{align*}1 \frac{4}{9}\end{align*} is approximately \begin{align*}1\frac{1}{2}\end{align*}

Next, add the approximated values.

\begin{align*}4+1 \frac{1}{2}=5 \frac{1}{2}\end{align*} The answer is \begin{align*}3 \frac{6}{7}+1 \frac{4}{9}\end{align*} is approximately \begin{align*}5 \frac{1}{2}\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Megan and her vegetable garden.

Her garden is \begin{align*}5 \frac{1}{6}\end{align*} feet by \begin{align*}7 \frac{7}{8}\end{align*} feet. Megan wants to estimate the perimeter of the garden so she can estimate how much fencing she will need to buy to fence it in. The fencing is $6 a foot so Megan wants to use her perimeter estimate to estimate the cost of the fencing.

First, Megan should write an expression for the perimeter of her garden. The garden is a rectangle so it has four sides. The perimeter is the sum of the lengths of the four sides.

\begin{align*}\text{Perimeter} = 5 \frac{1}{6}+7 \frac{7}{8}+5 \frac{1}{6}+7 \frac{7}{8}\end{align*}

Next, Megan should approximate the value of each individual mixed number.

  • \begin{align*}5 \frac{1}{6}\end{align*} is approximately \begin{align*}5\end{align*}
  • \begin{align*}7 \frac{7}{8}\end{align*} is approximately \begin{align*}8\end{align*}

Next, add the approximated values.

\begin{align*}\text{Perimeter} \approx 5+8+5+8 = 26 \ \text{feet}\end{align*}

The perimeter of the garden is approximately 26 feet. If the fencing costs $6 per foot, it will cost approximately \begin{align*}26 \times 6 = \$ 156\end{align*} to fence in the garden.

The answer is the perimeter of Megan's garden is approximately 26 feet. It will cost approximately $156 to put a fence around it.

Example 2

Estimate the sum of \begin{align*}\frac{18}{20}+5 \frac{9}{10}\end{align*}.

First, approximate the value of each individual fraction and mixed number.

  • \begin{align*}\frac{18}{20}\end{align*} is approximately \begin{align*}1\end{align*}
  • \begin{align*}5 \frac{9}{10}\end{align*} is approximately \begin{align*}6\end{align*}

Next, add the approximated values.

\begin{align*}1+6=7\end{align*}The answer is \begin{align*}\frac{18}{20}+5 \frac{9}{10}\end{align*} is approximately \begin{align*}7\end{align*}.

Example 3

Estimate the sum of \begin{align*}\frac{6}{7}+\frac{1}{2}\end{align*}.

First, approximate the value of each individual fraction. \begin{align*}\frac{1}{2}\end{align*} is already equal to a benchmark value so it does not need to be approximated.

  • \begin{align*}\frac{6}{7}\end{align*} is approximately \begin{align*}1\end{align*}
  • \begin{align*}\frac{1}{2}\end{align*} is equal to \begin{align*}\frac{1}{2}\end{align*}

Next, add the approximated values.

\begin{align*}1+\frac{1}{2}=1 \frac{1}{2}\end{align*}The answer is \begin{align*}\frac{6}{7} + \frac{1}{2}\end{align*} is approximately \begin{align*}1 \frac{1}{2}\end{align*}.

Example 4

Estimate the sum of \begin{align*}\frac{29}{30}+7 \frac{8}{10}\end{align*}.

First, approximate the value of each individual fraction and mixed number.

  • \begin{align*}\frac{29}{30}\end{align*} is approximately \begin{align*}1\end{align*}
  • \begin{align*}7 \frac{8}{10}\end{align*} is approximately \begin{align*}8\end{align*}

Next, add the approximated values.

\begin{align*}1+8=9\end{align*}The answer is \begin{align*}\frac{29}{30}+7 \frac{8}{10}\end{align*} is approximately \begin{align*}9\end{align*}.

Example 5

Estimate the sum of \begin{align*}1\frac{1}{2}+3 \frac{5}{6}\end{align*}.

First, approximate the value of each individual mixed number. \begin{align*}1 \frac{1}{2}\end{align*} is already equal to a benchmark value so it does not need to be approximated.

  • \begin{align*}1 \frac{1}{2}\end{align*} is equal to \begin{align*}1 \frac{1}{2}\end{align*}
  • \begin{align*}3 \frac{5}{6}\end{align*} is approximately \begin{align*}4\end{align*}

Next, add the approximated values.

\begin{align*}1 \frac{1}{2}+ 4 = 5 \frac{1}{2}\end{align*}

The answer is \begin{align*}1\frac{1}{2}+3 \frac{5}{6}\end{align*} is approximately \begin{align*}5 \frac{1}{2}\end{align*}.

Review

Estimate the sums.

  1. \begin{align*}\frac{1}{29}+\frac{4}{5}\end{align*}
  2. \begin{align*}\frac{9}{11}+\frac{4}{10}\end{align*}
  3. \begin{align*}\frac{2}{5}+\frac{12}{13}\end{align*}
  4. \begin{align*}\frac{2}{71}+\frac{1}{29}\end{align*}
  5. \begin{align*}\frac{1}{29}+\frac{4}{5}\end{align*}
  6. \begin{align*}\frac{3}{20}+\frac{14}{15}\end{align*}
  7. \begin{align*}\frac{6}{7}+\frac{1}{5}\end{align*}
  8. \begin{align*}\frac{9}{18}+\frac{5}{6}\end{align*}
  9. \begin{align*}\frac{12}{13}+\frac{1}{25}\end{align*}
  10. \begin{align*}\frac{7}{9}+\frac{1}{30}\end{align*}
  11. \begin{align*}3 \frac{6}{7}+2 \frac{10}{11}\end{align*}
  12.  \begin{align*}8 \frac{1}{12}+6 \frac{3}{7}\end{align*}
  13.  \begin{align*}2 \frac{9}{10} + 3 \frac{1}{17}\end{align*}
  14. \begin{align*}1 \frac{2}{12}+\frac{44}{46}\end{align*}
  15. \begin{align*}8 \frac{1}{29}+10 \frac{4}{5}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.5.

Vocabulary

Mixed Number

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

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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