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# Estimation with Mixed Number/Fraction Subtraction

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Remember Jesse and the peaches from the Subtraction of Fractions Concept? Well, what if you had estimated the difference? Would this have worked?

Here is the problem once again.

Jesse has decided to make peach pie for the bake sale. His recipe calls for $2 \frac{1}{2}$ pounds of peaches. Jesse’s older brother Jeff drives him to the farmer’s market to pick up his peaches. When Jesse gets there, he is amazed at how fast paced the market is. There are several people working behind the counter and they seem to be adding up all of the figures in their heads. Jesse is amazed. He loves math but he can’t even imagine adding up so many numbers in his head at one time.

Jesse is fascinated. So much so that he loses his focus and puts many, many peaches in his cloth bag. When the girl weighs it, Jesse watches her add up the math in her head. She tells Jesse how much and he pays her. Then she hands Jesse back the bag.

“How many pounds is this?” Jesse asks.

“You bought $6 \frac{1}{4}$ pounds of peaches,” She says focusing on another customer.

Jesse is surprised. He knows that he wasn’t paying very good attention, but he has a lot more peaches than he needs. How much more does he have? After Jesse makes his pie, how many pounds of peaches will be left over?

In this Concept you will learn how to estimate using benchmarks and figure out the difference.

### Guidance

Estimation is a method for finding an approximate solution to a problem. For example, $52 - 21$ is exactly 31. We can estimate the difference by rounding to the tens place and subtracting $50 - 20$ . Then we can say that $52 - 21$ is “about 30.”

To approximate the values of fractions, we compare the fractions in relation to three benchmarks, 0, $\frac{1}{2}$ and 1. Is the fraction closer to 0, $\frac{1}{2}$ or 1? If it’s closer to 1, we say that the value of the fraction is “about 1.”

We can use these approximate values of fractions to estimate the differences of fractions and mixed numbers.

First, we approximate the value of each fraction or mixed number using the benchmarks 0, $\frac{1}{2}$ and 1.

Next, we find the difference between the approximate values.

Even when you are asked to find an exact answer, estimation is a useful way to get an idea of a reasonable solution to a problem. Once you have finished solving for an exact answer of a problem, you can check your answer against the estimate. Refer back to the following steps if necessary.

Estimate the difference between $\frac{7}{8}-\frac{4}{9}$

First, we need to rewrite each fraction in terms of its benchmark. Seven-eighths is almost one, so we can say that the best benchmark is 1. Four - ninths is a little less than one-half, so we can use one-half as the benchmark.

Next, to find the best estimate, we subtract the benchmarks.

$1 - \frac{1}{2}=\frac{1}{2}$

Our estimate is one-half.

We can also estimate differences of mixed numbers.

Estimate the difference between $5 \frac{14}{16}-5 \frac{1}{16}$

First, we need to rewrite each of these in terms of a benchmark. Now we use the whole numbers of 5 and 6 as our starting points. Five and fourteen-sixteenths is close to 6. We use 6 as our benchmark for the first mixed number.

Five and one-sixteenth is closest to five. We use 5 as our second benchmark.

Now we subtract the benchmarks to find an accurate estimate.

$6 - 5 = 1$

Our estimate is 1.

Use benchmarks to estimate the differences.

#### Example A

$6 \frac{5}{6}-\frac{9}{10}$

Solution: $6$

#### Example B

$\frac{8}{9}-\frac{1}{7}$

Solution: $1$

#### Example C

$4 \frac{4}{5}-1 \frac{2}{5}$

Solution: $4$

Now we can solve Jesse's dilemma using estimation. Here is the original problem once again.

Jesse has decided to make peach pie for the bake sale. His recipe calls for $2 \frac{1}{2}$ pounds of peaches. Jesse’s older brother Jeff drives him to the farmer’s market to pick up his peaches. When Jesse gets there, he is amazed at how fast paced the market is. There are several people working behind the counter and they seem to be adding up all of the figures in their heads. Jesse is amazed. He loves math but he can’t even imagine adding up so many numbers in his head at one time.

Jesse is fascinated. So much so that he loses his focus and puts many, many peaches in his cloth bag. When the girl weighs it, Jesse watches her add up the math in her head. She tells Jesse how much and he pays her. Then she hands Jesse back the bag.

“How many pounds is this?” Jesse asks.

“You bought $6 \frac{1}{4}$ pounds of peaches,” She says focusing on another customer.

Jesse is surprised. He knows that he wasn’t paying very good attention, but he has a lot more peaches than he needs. How much more does he have? After Jesse makes his pie, how many pounds of peaches will be left over?

To work through this problem, we must first figure out the benchmark for each mixed number.

Jesse bought $6 \frac{1}{4}$ pounds of peaches.

This can simply be estimated at 6 pounds.

Jesse needed $2 \frac{1}{2}$ pounds of peaches.

We want to figure out how many pounds he will have left over.

$6 - 2 \frac{1}{2}$

Now you would think that this answer is 4 pounds, but we have the extra one - half to consider.

The estimate is that Jesse will have $3 \frac{1}{2}$ pounds of peaches left over.

### Vocabulary

Lowest Common Denominator
when two fractions have different denominators, we use the lowest common denominator to rename each fraction in terms of that common number. The lowest common denominator is also a least common multiple of the denominators.
Equivalent Fractions
equal fractions
Improper Fractions
when the numerator of a fraction is larger than the denominator
Estimation

### Guided Practice

Here is one for you to try on your own.

Estimate the difference.

$4 \frac{1}{15} - 2 \frac{3}{4}$

To begin, we have to figure out the benchmark for each value.

$4 \frac{1}{15} = 4$

$2 \frac{3}{4} = 3$

$4 - 3 = 1$

### Practice

Directions: Estimate each difference by using benchmarks.

1. $\frac{14}{16}-\frac{1}{11}$

2. $\frac{28}{60}-\frac{6}{13}$

3. $\frac{6}{7}-\frac{2}{145}$

4. $\frac{32}{33}-\frac{5}{12}$

5. $\frac{32}{33}-\frac{1}{12}$

6. $14 \frac{20}{21}-3 \frac{18}{19}$

7. $2 \frac{1}{49}-1 \frac{13}{14}$

8. $3 \frac{2}{21}-2 \frac{6}{11}$

9. $4 \frac{8}{17}-\frac{71}{73}$

10. $2 \frac{1}{49}-1 \frac{13}{14}$

11. $12 \frac{3}{4}-11 \frac{1}{14}$

12. $23 \frac{6}{8}- 14 \frac{3}{4}$

13. $18 \frac{1}{9}- 12\frac{3}{10}$

14. $22 \frac{1}{2}- 5 \frac{12}{34}$

15. $27 \frac{1}{5}- 18 \frac{7}{8}$