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Mixed Number and Fraction Estimation

Estimating sums and differences

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Mixed Number and Fraction Estimation
License: CC BY-NC 3.0

Travis was practicing for the long jump. His first jump was \begin{align*}8 \frac{3}{4}\end{align*} feet. After practicing for a month, he was able to add \begin{align*}2 \frac{1}{3}\end{align*} feet to his record. About how many feet is Travis able to jump? Estimate the distance. 

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

Estimating Mixed Numbers and Fractions

A sum is the answer in addition problem. You can also estimate the sum of fractions. To estimate a sum of two fractions, round the fractions to the nearest half. Here is an addition problem.

\begin{align*}\frac{3}{8} + \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, round each fraction to the nearest half.

Let’s start with \begin{align*}\frac{3}{8}\end{align*}. For a whole divided into 8 parts, \begin{align*}\frac{1}{2}=\frac{4}{8}\end{align*}. \begin{align*}\frac{3}{8}\end{align*} is rounded to \begin{align*}\frac{1}{2}\end{align*}.

For a whole divided into 7 parts, one-half is between \begin{align*}\frac{3}{7}\end{align*} and \begin{align*}\frac{4}{7}\end{align*}\begin{align*}\frac{1}{7}\end{align*} is rounded 0. 

\begin{align*}\frac{3}{8}& \approx \frac{1}{2}\\ \frac{1}{2} & \approx 0\end{align*}Then, rewrite the problem and find the sum.

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

The estimate sum is \begin{align*}\frac{1}{2}\end{align*}.

When estimating the sum of mixed numbers, round to the nearest whole number instead of the nearest half. Here is an addition problem involving mixed numbers. 

\begin{align*}3 \frac{4}{5} + 2 \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, round each mixed number to the nearest whole number.

 \begin{align*}3 \frac{4}{5}\end{align*}  will either round down to 3 or up to 4. Look at the fraction part of the mixed number. \begin{align*}\frac{4}{5}\end{align*} is greater than \begin{align*}\frac{1}{2}\end{align*}. Round up to 4.

\begin{align*}2 \frac{1}{9}\end{align*}  will either round down to 2 or up to 3. One-ninth is a very small fraction and less than \begin{align*}\frac{1}{2}\end{align*}. Round down to 2. 

\begin{align*}3\frac{4}{5} & \approx 4 \\ 2\frac{1}{9} & \approx 2\end{align*} 

Then, rewrite the problem and find the sum.

\begin{align*}4+2=6\end{align*}

The estimate sum is 6.

When talking about a difference, you are talking about subtraction. Estimating the difference of fractions and mixed numbers is similar to estimating sums. Round each fraction or mixed number and then subtract to find the estimate.

\begin{align*}\frac{4}{5} - \frac{3}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, round each fraction to the nearest half. 

For a whole divided into 5 parts, one-half is between \begin{align*}\frac{2}{5}\end{align*} and \begin{align*}\frac{3}{5}\end{align*}\begin{align*}\frac{4}{5}\end{align*} is rounded to 1. 

For a whole divided into 7 parts, is one-half is between \begin{align*}\frac{3}{7}\end{align*} and \begin{align*}\frac{4}{7}\end{align*}.  \begin{align*}\frac{3}{7}\end{align*} is rounded to \begin{align*}\frac{1}{2}\end{align*}.

\begin{align*}\frac{4}{5} & \approx 1 \\ \frac{3}{7} & \approx \frac{1}{2}\end{align*}

Then, rewrite the problem and subtract.

\begin{align*} 1 - \frac{1}{2} = \frac{1}{2}\end{align*}

The estimate difference is \begin{align*}\frac{1}{2}\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Travis practicing the long jump.

Travis started at \begin{align*}8\frac{3}{4}\end{align*} feet and was able to add \begin{align*}2\frac{1}{3}\end{align*} feet to his long jump record. Round the numbers and add the numbers to find an estimate of his current long jump record. 

 \begin{align*}8\frac{3}{4} + 2\frac {1}{3} \end{align*}

First, round the mixed numbers to the nearest whole number. 

 \begin{align*}8\frac{3}{4} \approx 9 \\ 2\frac {1}{3} \approx 2\end{align*}

Then, rewrite the problem and add.

 \begin{align*}9+2=11\end{align*}

Travis can now jump approximately 11 feet.

Example 2

Estimate the difference:

\begin{align*}6 \frac{3}{4} - 2 \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}.

First, round each mixed number to the nearest whole number. Look at the fraction part of each mixed number. 

\begin{align*} \frac{3}{4}\end{align*} is greater than \begin{align*}\frac{1}{2}\end{align*}. \begin{align*}6\frac{3}{4}\end{align*} rounds up to 7.

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

Then, rewrite the problem and subtract.

\begin{align*}7-2=5\end{align*}

The estimate sum is 5.

Example 3

Estimate the sum: \begin{align*}\frac{4}{9} + \frac{7}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}.

First, round each fraction to the nearest half.

 \begin{align*}\frac{4}{9}&\approx \frac{1}{2}\\\frac{7}{8}&\approx 1\end{align*}

Then, rewrite the problem and add.

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

The estimate sum is \begin{align*}1 \frac{1}{2}\end{align*}.

Example 4

Estimate the sum: \begin{align*}\frac{6}{7} + \frac{1}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}.

First, round each fraction to the nearest half.

\begin{align*}\frac{6}{7} & \approx 1 \\ \frac{1}{11} & \approx 0\end{align*}

Then, rewrite the problem and add.

 \begin{align*}1+0=1\end{align*}

The estimate sum is \begin{align*}1\end{align*}.

Example 5

Estimate the difference: \begin{align*}5 \frac{1}{3} - 2 \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}.

First, round the mixed numbers to the nearest whole number.

 \begin{align*}5 \frac{1}{3} & \approx 5 \\ 2 \frac{3}{4} & \approx 3\end{align*}

Then, rewrite the problem and find the difference.

 \begin{align*}5-3=2\end{align*}

The estimate difference is \begin{align*}2\end{align*}.

Review

Estimate the following sums.

  1. \begin{align*}\frac{1}{5} + \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  2. \begin{align*}\frac{8}{9} + \frac{4}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  3. \begin{align*}\frac{2}{9} + \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  4. \begin{align*}\frac{3}{6} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  5. \begin{align*}\frac{5}{6} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  6. \begin{align*}\frac{1}{12} + \frac{9}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  7. \begin{align*}\frac{6}{12} + \frac{10}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  8. \begin{align*}1 \frac{1}{10} + 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  9. \begin{align*}4 \frac{2}{3} + 5 \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  10. \begin{align*}7 \frac{1}{9} + 8 \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  11. \begin{align*}14 \frac{5}{9} + 8 \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  12. \begin{align*}4 \frac{2}{3} + 7 \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  13. \begin{align*}18 \frac{1}{13} + 7 \frac{2}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  14. \begin{align*}11 \frac{12}{13} + 4 \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  15. \begin{align*}22 \frac{5}{7} + 11 \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Estimate each difference.

  1. \begin{align*}\frac{4}{5} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  2. \begin{align*}\frac{4}{5} - \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  3. \begin{align*}\frac{9}{10} - \frac{3}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  4. \begin{align*}\frac{11}{12} - \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  5. \begin{align*}\frac{10}{13} - \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.3. 

Resources

Vocabulary

Difference

The result of a subtraction operation is called a difference.

Estimate

To estimate is to find an approximate answer that is reasonable or makes sense given the 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.

Mixed Number

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

Sum

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

Image Attributions

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

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