Remember the seventh grade bake sale from the Equivalent Fractions Concept? Well, look at this situation.

Sam baked \begin{align*}8 \frac{9}{12}\end{align*} batches of cookies for the bake sale. When he brought them to school, Tracy asked how many batches he had made.

When Sam told her, she wrote 9 batches of cookies on the check sheet.

How did Tracy know that Sam's quantity was close to 9 batches? Do you know?

**This Concept will teach you how to approximate mixed numbers and fractions using benchmarks. By the end of it, you will know how Tracy came to this conclusion.**

### Guidance

Because a whole can be divided into an infinite number of parts, it is sometimes difficult to get a good sense of the value of a fraction or mixed number when the denominator of the fraction is large. **In order to get an approximate sense of the value of a fraction, we compare the complicated fraction with several simpler fractions, or benchmarks. The three basic fraction benchmarks are: 0, \begin{align*}\frac{1}{2}\end{align*} and 1.**

**When approximating the value of a fraction or mixed number, ask yourself which of these benchmarks is the number closest to?**

Let’s look at how to apply benchmarks.

What is the approximate size of \begin{align*}\frac{17}{18}\end{align*}?

To begin with, we need to determine whether the fraction is closest to 0, one-half or 1 whole. The denominator is 18 and the numerator is 17. The numerator is close in value to the denominator. **The value of \begin{align*}\frac{17}{18}\end{align*} is closest to 1 because \begin{align*}\frac{18}{18}\end{align*} would be equal to one.**

**Our answer is 1.**

**That’s right. When you are looking for a benchmark, you want to choose the one that makes the most sense.**

What is the benchmark for \begin{align*}\frac{24}{49}\end{align*}?

**First, we can look at the relationship between the numerator and the denominator. The numerator in this case is almost half the denominator. Therefore the correct benchmark is one-half.**

**The answer is one-half.**

**What about mixed numbers?**

**We can identify benchmarks for mixed numbers too. The difference is that rather than zero, we look to the whole number of the mixed number, the half and the whole number next in consecutive order.**

What is the benchmark for \begin{align*}7 \frac{1}{8}\end{align*}?

**Here we have 7 and one-eighth. Is this closer to 7, \begin{align*}7 \frac{1}{2}\end{align*} or 8? If you think about it logically, one-eighth is a very small fraction. There is only one part out of eight. Therefore, it makes sense for our benchmark to be 7.**

**The answer is 7.**

Choose the correct benchmark for each example.

#### Example A

\begin{align*}\frac{1}{12}\end{align*}

**Solution: 0**

#### Example B

\begin{align*}\frac{5}{6}\end{align*}

**Solution: 1**

#### Example C

\begin{align*}9 \frac{3}{9}\end{align*}

**Solution: 9**

Here is the original problem once again.

Sam baked \begin{align*}8 \frac{9}{12}\end{align*} batches of cookies for the bake sale. When he brought them to school, Tracy asked how many batches he had made.

When Sam told her, she wrote 9 batches of cookies on the check sheet.

How did Tracy know that Sam's quantity was close to 9 batches? Do you know?

To figure out Tracy's decision, let's look at the fraction part of the mixed number of batches.

\begin{align*}8 \frac{9}{12}\end{align*}

9 is more than half of 12, so rounded up to 9 batches. If the fraction part of 12 would have been less than half, then Tracy would have rounded down to 8 batches.

Tracy thought about this and rounded up to 9.

**This is our answer.**

### Vocabulary

- Whole Number
- a number that is a counting number like 5, 7, 10, or 22.

- Fraction
- a part of a whole.

- Numerator
- the top number in a fraction.

- Denominator
- the bottom number in a fraction. It tells you how many parts the whole is divided into.

- Equivalent Fractions
- equal fractions

- Equivalent
- equal

- Simplifying
- making a fraction smaller

- Greatest Common Factor
- the largest number that will divide into both a numerator and denominator.

- Mixed Number
- a whole number with a fraction

- Improper Fraction
- when the numerator is greater than the denominator in a fraction

### Guided Practice

Here is one for you to try on your own.

Name the common benchmark for this fraction.

\begin{align*}\frac{4}{7}\end{align*}

**Answer**

To begin, we have to look at the relationship between 4 and 7. 4 is a little more than half of seven. Because of this, we can say that this fraction is closest to one - half.

**\begin{align*}\frac{1}{2}\end{align*} is the appropriate benchmark.**

### Practice

Directions: Approximate the value of the following fractions using the benchmarks 0, \begin{align*}\frac{1}{2}\end{align*} and 1.

1. \begin{align*}\frac{9}{10}\end{align*}

2. \begin{align*}\frac{11}{20}\end{align*}

3. \begin{align*}\frac{2}{32}\end{align*}

4. \begin{align*}\frac{21}{22}\end{align*}

5. \begin{align*}\frac{1}{23}\end{align*}

6. \begin{align*}\frac{11}{100}\end{align*}

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

8. \begin{align*}\frac{14}{28}\end{align*}

9. \begin{align*}\frac{16}{30}\end{align*}

10. \begin{align*}\frac{18}{21}\end{align*}

Directions: Approximate the value of the following mixed numbers.

11. \begin{align*}2 \frac{79}{80}\end{align*}

12. \begin{align*}6 \frac{1}{10}\end{align*}

13. \begin{align*}43 \frac{7}{15}\end{align*}

14. \begin{align*}8 \frac{7}{99}\end{align*}

15. \begin{align*}6 \frac{21}{22}\end{align*}