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# Inverse Property of Addition in Fraction Equations

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Have you ever made a pie crust?

Scott made a crust for a peach pie. He couldn't find the recipe, so he decided to work off of his memory of the last time he'd made a pie with his mom. He added $\frac{3}{4}$ of a cup of flour, but the dough was too sticky to work with. After he'd added more flour, he realized that he had used $1 \frac{3}{4}$ cups of flour.

Given this information, can you write an equation to show how Scott added the flour?

Can you solve it using an inverse operation?

This Concept will teach you exactly how to do this.

### Guidance

Do you remember the problem in the Estimation with Mixed Number/Fraction Subtraction Concept where Teri was making the blueberry pie?

Imagine that the recipe calls for $3 \frac{3}{4}$ cups of blueberries. You look in the refrigerator and you find $1 \frac{1}{2}$ cups of blueberries. You need to go to the store to buy the rest of the blueberries you need for the recipe. This is the problem that you are now going to tackle.

Let’s consider the information that you know and the information that you don’t know. You know: the amount of blueberries required by the recipe $= 3 \frac{3}{4}$ cups; the amount of blueberries in your refrigerator $= 1 \frac{1}{2}$ cups. You don’t know the amount of blueberries you need to buy at the store. Let’s call that $x$ , since it’s an unknown. If we write an equation to model the relationship between these amounts, it looks like this.

blueberries in refrigerator + blueberries you buy at the store = blueberries required for the pie recipe

When you insert the values (numerical and unknown) into the equation, it looks like this.

$1 \frac{1}{2} + x = 3 \frac{3}{4}$

This is how we can write an equation to solve a real-world problem.

How do we solve this equation and figure out the value of the variable?

We can use the Inverse Property of Addition to help us in solving equations.

The Inverse Property of Addition states that a number and its opposite is equal to 0. Let’s test it out.

$5 + (-5) = 0$

Even if you can’t remember how to work with negative numbers, you can still think this through and have it make sense. We have five and we add taking away five. Well, that answer is 0.

What does this have to do with solving equations?

Well, to solve an equation, we need to get the variable by itself on one side of the equals. In the example above, the $x$ is on one side with one and one-half. We need to move one and one-half to the other side of the equals. We can do this by using the Inverse Property of Addition. To get the $x$ alone, we need to have zero on the same side of the equation with it. Here is how we use the Inverse Property of Addition.

$1 \ \frac{1}{2} + x &= 3 \ \frac{3}{4}\\ \left(-1 \ \frac{1}{2}\right)+1 \ \frac{1}{2} + x &= 3 \ \frac{3}{4}-1 \frac{1}{2}\\0+x &= 3 \ \frac{3}{4}-1 \ \frac{1}{2}\\x &= 3 \ \frac{3}{4}-1 \ \frac{1}{2}$

Now we can simply subtract the mixed numbers. First, we need to rename each fraction in terms of fourths. Three and three-fourths is all set. We convert one and one-half to one and two-fourths.

Our answer is $2 \frac{1}{4}$ .

Solve each equation using the Inverse Property of Addition. Be sure that your answer is in simplest form.

#### Example A

$\frac{1}{7}+x=\frac{5}{7}$

Solution: $\frac{4}{7}$

#### Example B

$\frac{4}{6}+ y= \frac{5}{6}$

Solution: $\frac{1}{6}$

#### Example C

$y + \frac{3}{6} = \frac{5}{6}$

Solution: $\frac{2}{6} = \frac{1}{3}$

Here is the original problem once again.

Scott made a crust for a peach pie. He couldn't find the recipe, so he decided to work off of his memory of the last time he'd made a pie with his mom. He added $\frac{3}{4}$ of a cup of flour, but the dough was too sticky to work with. After he'd added more flour, he realized that he had used $1 \frac{3}{4}$ cups of flour.

Given this information, can you write an equation to show how Scott added the flour?

Can you solve it using an inverse operation?

To write this equation, we use the fractions from the dilemma.

$\frac{3}{4} + x = 1 \frac{3}{4}$

Notice that we used a variable for the unknown amount of flour that Scott added. We have the total amount of flour at the end of the equation.

Now we can use an inverse operation and solve.

$x = 1 \frac{3}{4} - \frac{3}{4} = 1$

$x = 1$

Scott added 1 cup of flour to the three - fourths he started with.

### 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
when you add the inverse of a number, the answer is zero.

### Guided Practice

Here is one for you to try on your own.

$x + \frac{2}{3} = \frac{5}{6}$

First, we have to write an equation showing how we can solve this equation using an inverse operation.

$x = \frac{5}{6} - \frac{2}{3}$

Now we need to rename these fractions with a common denominator.

$x = \frac{5}{6} - \frac{4}{6}$

$x = \frac{1}{6}$

### Practice

Directions: Solve for $x$ .

1. $x + \frac{2}{5} = \frac{5}{5}$

2. $x + \frac{2}{8} = \frac{6}{8}$

3. $x + \frac{3}{9}= \frac{4}{9}$

4. $x + \frac{5}{7} = \frac{6}{7}$

5. $x + \frac{10}{12} = \frac{11}{12}$

6. $x + \frac{3}{15} = \frac{10}{15}$

7. $x + \frac{9}{13} = \frac{12}{13}$

8. $x + \frac{6}{14} = \frac{12}{14}$

9. $x + \frac{1}{5} = \frac{6}{10}$

10. $x + \frac{1}{2} = \frac{11}{12}$

11. $x - 1 \frac{1}{2} = 4$

12. $2 \frac{1}{4} + x = 3 \frac{3}{4}$

13. $x - \frac{7}{8} = 2 \frac{3}{4}$

14. Ludmilla, Brent and Rudy have $8 \frac{5}{6}$ feet of taffy that they have to sell to raise money for the school drama club. Brent has already sold $3 \frac{2}{3}$ feet of taffy and Rudy plans to sell exactly $2 \frac{3}{4}$ feet. How much taffy does Ludmilla have to sell, if they sell all of the taffy?

15. Ron, Jung-Ho and Sarah have a lawn mowing business. Today they are cutting an enormous lawn. Sarah agrees to start and will mow $\frac{3}{8}$ of the lawn. Ron will only mow $\frac{1}{7}$ of the lawn, but he’s willing to work during the hottest time of the day. How much of the lawn is Jung-Ho responsible for completing?