m81290 Add and Subtract Mixed Numbers By the end of this section, you will be able to: Model addition of mixed numbers with a common denominator Add mixed numbers with a common denominator Model subtraction of mixed numbers Subtract mixed numbers with a common denominator Add and subtract mixed numbers with different denominators a5956e79-3c5d-45b3-a564-4de0a2ded1ae Before you get started, take this readiness quiz. Draw a model of the fraction 73. If you missed this problem, review . Change 114 to a mixed number. If you missed this problem, review . 234 Change 312 to an improper fraction. If you missed this problem, review . 72

So far, we’ve added and subtracted proper and improper fractions, but not mixed numbers. Let’s begin by thinking about addition of mixed numbers using money. If Ron has 1 dollar and 1 quarter, he has 114 dollars. If Don has 2 dollars and 1 quarter, he has 214 dollars. What if Ron and Don put their money together? They would have 3 dollars and 2 quarters. They add the dollars and add the quarters. This makes 324 dollars. Because two quarters is half a dollar, they would have 3 and a half dollars, or 312 dollars. 114+214________324=312 When you added the dollars and then added the quarters, you were adding the whole numbers and then adding the fractions. 114+214We can use fraction circles to model this same example: 114+214 Start with 114. one whole and one 14 pieces Add 214 more. two wholes and one 14 pieces The sum is: three wholes and two 14's Doing the Manipulative Mathematics activity "Model Mixed Number Addition/Subtraction" will help you develop a better understanding of adding and subtracting mixed numbers. Model 213+123 and give the sum. We will use fraction circles, whole circles for the whole numbers and 13 pieces for the fractions. two wholes and one 13 plus one whole and two 13s sum is three wholes and three 13s This is the same as 4 wholes. So, 213+123=4. Use a model to add the following. Draw a picture to illustrate your model. 125+335 5 Use a model to add the following. Draw a picture to illustrate your model. 216+256 5 Model 135+235 and give the sum as a mixed number. We will use fraction circles, whole circles for the whole numbers and 15 pieces for the fractions. one whole and three 15s plus two wholes and three 15s. sum is three wholes and six 15s Adding the whole circles and fifth pieces, we got a sum of 365. We can see that 65 is equivalent to 115, so we add that to the 3 to get 415. Model, and give the sum as a mixed number. Draw a picture to illustrate your model. 256+156 423 Model, and give the sum as a mixed number. Draw a picture to illustrate your model. 158+178 312

Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible. Step 1. Add the whole numbers. Step 2. Add the fractions. Step 3. Simplify, if possible. Add: 349+229. 349+229 Add the whole numbers. Add the fractions. Simplify the fraction. Find the sum: 447+127. 567 Find the sum: 2311+5611. 7911 In , the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction. Find the sum: 959+579. 959+579 Add the whole numbers and then add the fractions. 959 +579_____ 14129 Rewrite 129 as a mixed number. 14+139 Add. 1539 Simplify. 1513 Find the sum: 878+758. 1612 Find the sum: 679+859. 1513 An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally. Add by converting the mixed numbers to improper fractions: 378+438. 378+438 Convert to improper fractions. 318+358 Add the fractions. 31+358 Simplify the numerator. 668 Rewrite as a mixed number. 828 Simplify the fraction. 814 Since the problem was given in mixed number form, we will write the sum as a mixed number. Find the sum by converting the mixed numbers to improper fractions: 559+379. 913 Find the sum by converting the mixed numbers to improper fractions: 3710+2910. 635 compares the two methods of addition, using the expression 325+645 as an example. Which way do you prefer? Mixed Numbers Improper Fractions 325+6459659+659+1151015 325+645175+3455151015

Let’s think of pizzas again to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half of the pizza. What do you have to do to the pizza to give him half? You have to cut it into at least two pieces. Then you can give him half. We will use fraction circles (pizzas!) to help us visualize the process. Start with one whole. Algebraically, you would write: Use a model to subtract: 1−13. Use a model to subtract: 1−14. 34 Use a model to subtract: 1−15. 45 What if we start with more than one whole? Let’s find out. Use a model to subtract: 2−34. Use a model to subtract: 2−15. 95 Use a model to subtract: 2−13. 53 In the next example, we’ll subtract more than one whole. Use a model to subtract: 2−125. Use a model to subtract: 2−113. 23 Use a model to subtract: 2−114. 34 What if you start with a mixed number and need to subtract a fraction? Think about this situation: You need to put three quarters in a parking meter, but you have only a $1 bill and one quarter. What could you do? You could change the dollar bill into 4 quarters. The value of 4 quarters is the same as one dollar bill, but the 4 quarters are more useful for the parking meter. Now, instead of having a $1 bill and one quarter, you have 5 quarters and can put 3 quarters in the meter. This models what happens when we subtract a fraction from a mixed number. We subtracted three quarters from one dollar and one quarter. We can also model this using fraction circles, much like we did for addition of mixed numbers. Use a model to subtract: 114−34 Rewrite vertically. Start with one whole and one fourth. Since the fractions have denominator 4, cut the whole into 4 pieces. You now have 44 and 14 which is 54. Take away 34. There is 12 left. Use a model to subtract. Draw a picture to illustrate your model. 113−23 23 Use a model to subtract. Draw a picture to illustrate your model. 115−45 25

Now we will subtract mixed numbers without using a model. But it may help to picture the model in your mind as you read the steps. Rewrite the problem in vertical form. Compare the two fractions.If the top fraction is larger than the bottom fraction, go to Step 3.If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction. Subtract the fractions. Subtract the whole numbers. Simplify, if possible. Find the difference: 535−245. Rewrite the problem in vertical form. Since 35 is less than 45, take 1 from the 5 and add it to the 35:(55+35=85) Subtract the fractions. Subtract the whole parts. The result is in simplest form. Since the problem was given with mixed numbers, we leave the result as mixed numbers. Find the difference: 649−379. 223 Find the difference: 447−267. 157 Just as we did with addition, we could subtract mixed numbers by converting them first to improper fractions. We should write the answer in the form it was given, so if we are given mixed numbers to subtract we will write the answer as a mixed number. Step 1. Rewrite the mixed numbers as improper fractions. Step 2. Subtract the numerators. Step 3. Write the answer as a mixed number, simplifying the fraction part, if possible. Find the difference by converting to improper fractions: 9611−71011. 9611−71011 Rewrite as improper fractions. 10511−8711 Subtract the numerators. 1811 Rewrite as a mixed number. 1711 Find the difference by converting the mixed numbers to improper fractions: 649−379. 223 Find the difference by converting the mixed numbers to improper fractions: 447−267. 157

To add or subtract mixed numbers with different denominators, we first convert the fractions to equivalent fractions with the LCD. Then we can follow all the steps we used above for adding or subtracting fractions with like denominators. Add: 212+523. Since the denominators are different, we rewrite the fractions as equivalent fractions with the LCD, 6. Then we will add and simplify. We write the answer as a mixed number because we were given mixed numbers in the problem. Add: 156+434. 6712 Add: 345+812. 12310 Subtract: 434−278. Since the denominators of the fractions are different, we will rewrite them as equivalent fractions with the LCD 8. Once in that form, we will subtract. But we will need to borrow 1 first. We were given mixed numbers, so we leave the answer as a mixed number. Find the difference: 812−345. 4710 Find the difference: 434−156. 21112 Subtract: 3511−434. We can see the answer will be negative since we are subtracting 4 from 3. Generally, when we know the answer will be negative it is easier to subtract with improper fractions rather than mixed numbers. 3511−434 Change to equivalent fractions with the LCD. 35·411·4−43·114·1132044−43344 Rewrite as improper fractions. 15244−20944 Subtract. −5744 Rewrite as a mixed number. −11344 Subtract: 134−678. −418 Subtract: 1037−2249. −75763 Adding Mixed Numbers Subtracting Mixed Numbers

Add mixed numbers with a common denominator.Add the whole numbers. Add the fractions. Simplify, if possible. Subtract mixed numbers with common denominators.Rewrite the problem in vertical form. Compare the two fractions. If the top fraction is larger than the bottom fraction, go to Step 3. If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction. Subtract the fractions. Subtract the whole numbers. Simplify, if possible. Subtract mixed numbers with common denominators as improper fractions.Rewrite the mixed numbers as improper fractions. Subtract the numerators. Write the answer as a mixed number, simplifying the fraction part, if possible.

Model Addition of Mixed Numbers In the following exercises, use a model to find the sum. Draw a picture to illustrate your model. 115+315 213+113 323 138+178 156+156 323 Add Mixed Numbers with a Common Denominator In the following exercises, add. 513+613 249+519 759 458+938 7910+3110 11 345+645 923+123 1113 6910+8310 849+289 1113 Model Subtraction of Mixed Numbers In the following exercises, use a model to find the difference. Draw a picture to illustrate your model. 116−56 118−58 12 Subtract Mixed Numbers with a Common Denominator In the following exercises, find the difference. 278−138 2712−1512 116 81720−4920 191315−13715 625 837−447 529−349 179 258−178 2512−1712 56 Add and Subtract Mixed Numbers with Different Denominators In the following exercises, write the sum or difference as a mixed number in simplified form. 314+613 216+534 71112 158+412 723+812 1616 9710−213 645−114 51120 223−312 278−413 −11124 Mixed Practice In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form. 258·134 123·416 61718 27+47 29+59 79 1512÷112 2310÷110 23 13512−9712 1558−678 834 59−49 1115−715 415 4−34 6−25 535 920÷34 724÷143 116 9611+71011 8513+4913 13113 325+534 256+415 7130 815·1019 512·89 1027 678−213 659−425 2745 529−445 438−323 1724

Sewing Renata is sewing matching shirts for her husband and son. According to the patterns she will use, she needs 238 yards of fabric for her husband’s shirt and 118 yards of fabric for her son’s shirt. How much fabric does she need to make both shirts? Sewing Pauline has 314 yards of fabric to make a jacket. The jacket uses 223 yards. How much fabric will she have left after making the jacket? 712 yards Printing Nishant is printing invitations on his computer. The paper is 812 inches wide, and he sets the print area to have a 112-inch border on each side. How wide is the print area on the sheet of paper? Framing a picture Tessa bought a picture frame for her son’s graduation picture. The picture is 8 inches wide. The picture frame is 258 inches wide on each side. How wide will the framed picture be? 1314 inches

Draw a diagram and use it to explain how to add 158+278. Edgar will have to pay $3.75 in tolls to drive to the city. ⓐ Explain how he can make change from a $10 bill before he leaves so that he has the exact amount he needs.ⓑ How is Edgar’s situation similar to how you subtract 10−334? Answers will vary. Add 4512+378 twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why? Subtract 378−4512 twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why? Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?