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# Quotient Estimation with Mixed Numbers/Fractions

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# Quotient Estimation with Mixed Numbers/Fractions

Have you ever made something out of cloth? Take a look at this dilemma.

Casey is going to make a new sleeping bag. She has $5 \frac{3}{4} \ yards$ of material. She needs to divide this in half.

About how many yards of material will be in each section after she divides the material?

This Concept is about estimating quotients of fractions and mixed numbers. You will be able to answer this question by the end of the Concept.

### Guidance

In real-world situations, we use estimation every day. In every real-world problem that involves math, the solution is usually estimated before an exact answer is found. “I think we’ll need about $3 \frac{1}{2}$ of the long pieces of wood.” “Stephen estimates that the project will take about $36 \frac{1}{4}$ hours.” Observe your language as you engage in everyday activities. You are probably using estimation all of the time. Now that you already know how to estimate sums, differences and products of fractions, we are going to see how easy it is to use estimation with division of fractions, too.

Estimating quotients of fractions is pretty similar to estimating products of fractions, but there is one difference. As you already know, when finding exact quotients in dividing fractions, the first step is to invert the divisor and rewrite the problem as a multiplication problem. You have to complete this same first step when you estimate quotients of fractions. Once you invert the divisor and rewrite as a multiplication problem, you find approximate values for the fractions using the three benchmarks, 0, $\frac{1}{2}$ and 1. Is the fraction closer to 0, $\frac{1}{2}$ or 1? If it’s closest to $\frac{1}{2}$ , we say that the value of the fraction is “about $\frac{1}{2}$ .” Once we identify the approximate value of both fractions, we simply multiply and we will have the estimate quotient.

Estimate $\frac{7}{8} \div \frac{1}{3}$

First, we have to rewrite this as a multiplication problem. Then we can use benchmarks to estimate the product.

$\frac{7}{8} \cdot \frac{3}{1}$

Now we can use benchmarks. Seven-eighths is close to 1. Three over one is close to three.

$1 \times 3 = 3$

$\frac{7}{8} \div \frac{1}{3}$ is an estimate of 3.

Three does make sense as an answer if you think about what a division problem is asking. A division problem is asking how many groups or how many in each groups. In other words, this problem is asking how many groups can you divide seven-eighths into if you divide that quantity into thirds. You can divide it into three groups.

What about estimating with dividing mixed numbers?

Working with mixed numbers is a little bit different, but we are still simply answering the question, “what is a reasonable answer for this division problem?” When estimating quotients where the dividend is a mixed number, we first estimate the value of the dividend and the divisor before we convert to an improper fraction or divide. Consider a divisor that is $6 \frac{1}{29}$ . $6 \frac{1}{29}$ is really just 6. It is a lot easier to divide by 6 than to convert to an improper fraction and invert. Making a lot of work defeats the purpose of estimating. If you do have a fraction in your estimated divisor, you will go ahead and convert to an improper fraction and multiply. Consider a divisor that is $6 \frac{15}{29}$ . We approximate this at $6 \frac{1}{2}$ . Now we can convert to an improper fraction of $\frac{13}{2}$ and multiply. We always approximate the value of mixed numbers before we convert to improper fractions or invert.

$2 \frac{2}{3} \div 1 \frac{7}{8}$

Since we are dividing with mixed numbers, let’s approximate the values of the mixed numbers before we divide. $2 \frac{2}{3}$ is about 3 and $1 \frac{7}{8}$ is about 2. The problem rewritten with the approximate values looks like this: $3 \div 2$ . We can simply invert the divisor and rewrite as a multiplication problem, $3 \cdot \frac{1}{2}$ .

Our answer is that $2 \frac{2}{3} \div 1 \frac{7}{8}$ is about $1 \frac{1}{2}$ .

Now it's time for you to try a few. Estimate each quotient.

#### Example A

$\frac{9}{10} \div \frac{1}{13}$

Solution: $1$

#### Example B

$5 \frac{5}{6} \div 2 \frac{3}{4}$

Solution: $2$

#### Example C

$\frac{8}{9} \div \frac{1}{2}$

Solution: $\frac{1}{2}$

Here is the original problem once again.

Casey is going to make a new sleeping bag. She has $5 \frac{3}{4} \ yards$ of material. She needs to divide this in half.

About how many yards of material will be in each section after she divides the material?

To figure this out, first we can write the problem out using a division sign.

$5 \frac{3}{4} \div \frac{1}{2}$

Now to estimate, we don't need an exact quotient, so we can round the length of the material to help us with the estimate.

$5 \frac{3}{4} = 6$

Next, divide.

$6 \div \frac{1}{2} = 3$

There will be approximately 3 yards in each section of the material.

### Vocabulary

Reciprocal
the flip or inverted form of a fraction.
Estimation
Finding an approximate answer by using benchmarks.

### Guided Practice

Here is one for you to try on your own.

Estimate the following quotient.

$7 \frac{7}{8} \div 4$

First, let's round $7 \frac{7}{8}$ . It rounds up to 8.

$8 \div 4 = 2$

Our estimate is $2$ .

### Video Review

http://www.youtube.com/watch?v=3ahgPUBdanE - This is a James Sousa video on dividing fractions.

### Practice

Directions: Estimate the quotient.

1. $\frac{5}{6} \div \frac{1}{36}$

2. $\frac{1}{12} \div \frac{6}{7}$

3. $\frac{18}{37} \div \frac{10}{11}$

4. $\frac{13}{15} \div \frac{4}{9}$

5. $6 \frac{2}{3} \div 2 \frac{6}{11}$

6. $5 \frac{27}{29} \div 3 \frac{1}{18}$

7. $12 \div \frac{6}{7}$

8. $1 \frac{1}{9} \div 2 \frac{4}{5}$

9. $6 \frac{2}{3} \div 2 \frac{6}{11}$

10. $8 \frac{2}{3} \div 2 \frac{1}{11}$

11. $9 \frac{2}{3} \div 2 \frac{10}{11}$

12. $16 \frac{1}{13} \div 2 \frac{1}{9}$

13. $26 \frac{1}{4} \div 13 \frac{2}{30}$

14. $44 \frac{3}{4} \div 15 \frac{1}{20}$

15. $26 \frac{1}{13} \div 2 \frac{1}{18}$