# 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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Estimate Sums and Differences of Fractions and Mixed Numbers

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

Harriet and Matt have two identical jars. Harriet’s jar is \begin{align*}\frac{3}{10}\end{align*} full of pennies. Matt’s jar is \begin{align*}\frac{1}{4}\end{align*} full of pennies. If Matt and Harriet were to combine their pennies into one jar, how full would the jar be?

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

### Sums and Differences of Fractions

When you estimate fraction sums and differences you will need to work with rounding fractions. You can work with rounding fractions in a couple of different ways. One way is to look at the relationship between the fraction and a whole.

Here are some guiding questions:

1. Is this fraction close to one-half or one-whole?
2. If I simplify the fractions that I am adding or subtracting, will they have a common denominator?
3. Is this fraction so close to one-whole that it would make sense to round up to a whole?

Now apply this information and estimate the following sum.

\begin{align*}\frac{4}{25} + \frac{11}{20}\end{align*}

First, notice that the first fraction is close to \begin{align*}\frac{5}{25}\end{align*} or \begin{align*}\frac{1}{5}\end{align*}.

Next, notice that the second fraction is close to \begin{align*}\frac{12}{20}\end{align*} or \begin{align*}\frac{3}{5}\end{align*}.

Then, add the two rounded fractions to get an estimate of the answer.

\begin{align*}\frac{1}{5} + \frac{3}{5} = \frac{4}{5}\end{align*}

The estimate of the answer is \begin{align*}\frac{4}{5}\end{align*}.

With this problem, it made sense to round the fractions so that you could simplify them. The simplified fractions had a common denominator. This makes your work easier.

Here is another problem.

Estimate the difference: \begin{align*}\frac{24}{49} - \frac{7}{31}\end{align*}

First, notice that the first fraction is close to \begin{align*}\frac{25}{50}\end{align*} or \begin{align*}\frac{1}{2}\end{align*}.

Next, notice that the second fraction is close to \begin{align*}\frac{8}{32}\end{align*} or \begin{align*}\frac{1}{4}\end{align*}.

Then, subtract the two rounded fractions to get an estimate of the answer. Remember you need to have a common denominator to add and subtract fractions. Since 4 is divisible by 2, the common denominator is 4. So you need to multiply your first fraction in order to get the common denominator and then subtract the fractions.

\begin{align*}\frac{1}{2} \times \frac{2}{2} &= \frac{2}{4}\\ \frac{2}{4} - \frac{1}{4} &= \frac{1}{4}\end{align*}

The estimate of the answer is \begin{align*}\frac{1}{4}\end{align*}.

### Examples

#### Example 1

Remember Harriett and Matt were going to combine their jars of pennies into one? They wanted to know how full the jar would be.

Let’s look at the addition problem Harriett and Matt have to solve.

\begin{align*}\frac{3}{10} + \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

First, you need to get a common denominator. The first few multiples of 10 are 10, and 20. Since 20 is divisible by 4, it is the lowest common denominator.

Next, convert the fractions so that the denominators are 20.

\begin{align*}\frac{3}{10} \times \frac{2}{2} &= \frac{6}{20}\\ \frac{1}{4} \times \frac{5}{5} &= \frac{5}{20}\end{align*}

\begin{align*}\frac{6}{20} + \frac{5}{20} = \frac{11}{20}\end{align*}

The answer is \begin{align*}\frac{11}{20}\end{align*}.

Therefore, the jar will be \begin{align*}{\frac{11}{20}}^{th}\end{align*} full. Harriett and Matt can estimate that their combined jar would be \begin{align*}\frac{1}{2}\end{align*} full.

#### Example 2

Estimate the following difference.

\begin{align*}5 \frac{11}{12}-2\end{align*}

First, look at the mixed number. \begin{align*}5 \frac{11}{12}\end{align*} is close to \begin{align*}5 \frac{12}{12}\end{align*} or 6.

Next, subtract the two numbers. \begin{align*}6-2=4\end{align*}

The answer for the estimate is 4.

#### Example 3

Estimate the sum for the following.

\begin{align*}\frac{1}{4} + \frac{6}{7}\end{align*}

First, notice that the second fraction is close to \begin{align*}\frac{7}{7}\end{align*} or 1.

Next, add the two rounded fractions to get an estimate of the answer.

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

The estimate of the answer is \begin{align*}1 \frac{1}{4}\end{align*}.

#### Example 4

Estimate the difference for the following.

\begin{align*}\frac{8}{9} - \frac{1}{2}\end{align*}

First, notice that the first fraction is close to \begin{align*}\frac{9}{9}\end{align*} or 1 or \begin{align*}\frac{2}{2}\end{align*}.

Next, subtract the two rounded fractions to get an estimate of the answer.

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

The estimate of the answer is \begin{align*}\frac{1}{2}\end{align*}.

#### Example 5

Estimate the sum for the following.

\begin{align*}\frac{4}{5} + \frac{9}{10}\end{align*}

First, notice that the first fraction is close to \begin{align*}\frac{5}{5}\end{align*} or 1.

Next, notice that the second fraction is close to \begin{align*}\frac{10}{10}\end{align*} or 1.

Then, add the two rounded fractions to get an estimate of the answer.

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

The estimate of the answer is 2.

### Review

Estimate each sum or difference.

1. \begin{align*}\frac{6}{7} + \frac{1}{22}\end{align*}
2. \begin{align*}\frac{1}{2} + \frac{9}{10}\end{align*}
3. \begin{align*}2 \frac{6}{7} + 1 \frac{1}{12}\end{align*}
4. \begin{align*}\frac{8}{9} + \frac{21}{21}\end{align*}
5. \begin{align*}\frac{16}{17} - \frac{1}{22}\end{align*}
6. \begin{align*}\frac{1}{2} + \frac{1}{9}\end{align*}
7. \begin{align*}\frac{11}{12} - \frac{21}{22}\end{align*}
8. \begin{align*}7 \frac{8}{10} - 3 \frac{1}{22}\end{align*}
9. \begin{align*}\frac{4}{6} + \frac{2}{3}\end{align*}
10. \begin{align*}11 \frac{6}{7} + 14 \frac{1}{22}\end{align*}
11. \begin{align*}9 \frac{1}{7} + 14 \frac{1}{22}\end{align*}
12. \begin{align*}\frac{18}{20} - \frac{1}{2}\end{align*}
13. \begin{align*}\frac{16}{32} - \frac{1}{2}\end{align*}
14. \begin{align*}5 \frac{1}{2} - 2 \frac{1}{2}\end{align*}
15. \begin{align*}12 \frac{6}{12} + 15 \frac{1}{22}\end{align*}

To see the Review answers, open this PDF file and look for section 2.7.

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### Vocabulary Language: English

TermDefinition
Denominator The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
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.
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.
Numerator The numerator is the number above the fraction bar in a fraction.