Have you ever tried to estimate when working with fractions and mixed numbers? Well, Travis is working on measuring pieces of wood. Take a look.

Travis needs to piece two different pieces of wood together to form a whole piece. This wood will form a trim around one of the rooms in the house.

The first piece of wood is \begin{align*}2 \frac{1}{2}\end{align*} feet long.

The second piece of wood is \begin{align*}5 \frac{7}{8}\end{align*} feet long.

Travis wants to estimate the total length of the two pieces combined. Do you know how to do this? Estimation is necessary for Travis to be successful.

**In this Concept, you will learn how to estimate the sums and differences of fractions and mixed numbers. By the end of the Concept, you will know how to help Travis with this problem.**

### Guidance

Previously we worked on how to round fractions to the nearest half and mixed numbers to the nearest whole, we can apply this information to estimating ** sums** and

**?**

*differences*
**How do we estimate a sum?**

Remember back that a **sum is the answer in addition problem**. You have already learned how to estimate a whole number sum and a decimal sum. Now we are going to learn how to estimate fraction sums.

**To estimate a sum of two fractions, you must use what you have learned about rounding to the nearest half. That is the first thing that you do when estimating sums of fractions.**

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

**To estimate this sum, we must first round each fraction to the nearest half.** Let’s start with three-eighths.

**Is** \begin{align*}\frac{3}{8}\end{align*} **closest to zero, one – half or one whole?**

We know that \begin{align*} \frac{4}{8} = \frac{1}{2}\end{align*}, so we can say that \begin{align*}\frac{3}{8}\end{align*} is closest to one-half.

**Is** \begin{align*}\frac{1}{7}\end{align*} **closest to zero, one-half or one whole?**

We can say that \begin{align*}\frac{1}{7}\end{align*} is closest to zero because it such a small part of a whole.

**Now we rewrite the problem.**

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

**Our answer is** \begin{align*}\frac{1}{2}\end{align*}.

**How do we estimate a sum for two mixed numbers?**

**When working with mixed numbers, we round to the nearest whole number, not the nearest half. We round each mixed number and then add to find our estimate.**

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

**First, let’s round** \begin{align*}3 \frac{4}{5}\end{align*} **to the nearest whole.** We know that we are either going to round down to 3 or up to 4 because this mixed number is between those two whole numbers. Four-fifths is almost one whole. We can see this because if we had five-fifths, we would have another whole. This mixed number is closest to 4. We need to round up to 4.

**Next, let’s round** \begin{align*}2 \frac{1}{9}\end{align*} **to the nearest whole.** We know that we are either going to round down to 2 or up to 3 because this mixed number is between those two whole numbers. One-ninth is a very small fraction. Think about it, we would need eight-ninths more to make one whole. Therefore, we round down to 2. **Now we can rewrite the problem.**

4 + 2 = 6

**Our estimate is 6.**

**What about estimating the differences of fractions?**

Remember that when we talk about a ** difference**, that we are talking about subtraction. We can approach estimating the differences of fractions and mixed numbers in the same way that we approached sums.

**We need to round each fraction or mixed number**and then subtract to find the estimate.

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

\begin{align*}\frac{4}{5}\end{align*} rounds up to 1.

\begin{align*}\frac{3}{7}\end{align*} rounds to \begin{align*}\frac{1}{2}\end{align*}.

**Next, we rewrite the problem.**

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

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

Now it is time for you to practice. Estimate each sum.

#### Example A

\begin{align*}\frac{4}{9} + \frac{7}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

**Solution:\begin{align*}1 \frac{1}{2}\end{align*}**

#### Example B

\begin{align*}\frac{6}{7} + \frac{1}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

**Solution:\begin{align*}1\end{align*}**

#### Example C

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

**Solution:\begin{align*}2\end{align*}**

Remember Travis? Here is the original problem once again.

Travis needs to piece two different pieces of wood together to form a whole piece. This wood will form a trim around one of the rooms in the house.

The first piece of wood is \begin{align*}2 \frac{1}{2}\end{align*} feet long.

The second piece of wood is \begin{align*}5 \frac{7}{8}\end{align*} feet long.

Travis wants to estimate the total length of the two pieces combined. Do you know how to do this? Estimation is necessary for Travis to be successful.

First, Travis needs to use rounding. The first piece of wood can stay the same since we are working with halves. The second piece of wood can be rounded up to 6 feet.

Now we can add.

\begin{align*}2 \frac{1}{2} + 6 = 8 \frac{1}{2}\end{align*}

**This is our answer.**

### Vocabulary

- Fraction
- a part of a whole written with a fraction bar, a numerator and a denominator.

- Estimate
- to find an approximate answer that is reasonable and makes sense given the problem.

- Mixed number
- a number made up of a whole number and a fraction.

- Sum
- the answer to an addition problem.

- Difference
- the answer to a subtraction problem.

### Guided Practice

Here is one for you to try on your own.

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

**Answer**

\begin{align*}6 \frac{3}{4}\end{align*} rounds to 7.

\begin{align*}2 \frac{1}{8}\end{align*} rounds to 2.

**Now we rewrite the problem.**

7 - 2 = 5

**Our estimate is 5.**

### Practice

Directions: 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*}

Directions: Estimate each difference.

16. \begin{align*}\frac{4}{5} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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

18. \begin{align*}\frac{9}{10} - \frac{3}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}\frac{11}{12} - \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}\frac{10}{13} - \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}