Add and Subtract Fractions with Common Denominators

Model Fraction AdditionHow many quarters are pictured? One quarter plus 2 quarters equals 3 quarters.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that 142434one quarter+two quarters=three quarters Let’s use fraction circles to model the same example, 14+24. Start with one 14 piece.

Add two more 14pieces.

The result is 34.

So again, we see that 14+24=34 Doing the Manipulative Mathematics activity "Model Fraction Addition" will help you develop a better understanding of adding fractions Use a model to find the sum 38+28. Solution Start with three 18 pieces.

Add two 18pieces.

How many 18pieces are there?

There are five 18 pieces, or five-eighths. The model shows that 38+28=58. Use a model to find each sum. Show a diagram to illustrate your model. 18+48 58

Use a model to find each sum. Show a diagram to illustrate your model. 16+46 56

Add Fractions with a Common Denominator shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces. Fraction Addition If a,b, and c are numbers where c≠0, then ac+bc=a+bc To add fractions with a common denominator, add the numerators and place the sum over the common denominator. Find the sum: 35+15. Solution 35+15 Add the numerators and place the sum over the common denominator. 3+15 Simplify. 45

Find each sum: 36+26. 56 Find each sum: 310+710. 1 Find the sum: x3+23. Solution x3+23 Add the numerators and place the sum over the common denominator. x+23

Note that we cannot simplify this fraction any more. Since x and 2 are not like terms, we cannot combine them. Find the sum: x4+34. x+34 Find the sum: y8+58. y+58 Find the sum: −9d+3d. Solution We will begin by rewriting the first fraction with the negative sign in the numerator. −ab=−ab −9d+3d Rewrite the first fraction with the negative in the numerator. −9d+3d Add the numerators and place the sum over the common denominator. −9+3d Simplify the numerator. −6d Rewrite with negative sign in front of the fraction. −6d

Find the sum: −7d+8d. 1d Find the sum: −6m+9m. 3m Find the sum: 2n11+5n11. Solution 2n11+5n11 Add the numerators and place the sum over the common denominator. 2n+5n11 Combine like terms. 7n11

Find the sum: 3p8+6p8. 9p8 Find the sum: 2q5+7q5. 9q5 Find the sum: −312+(−512). Solution −312+(−512) Add the numerators and place the sum over the common denominator. −3+(−5)12 Add. −812 Simplify the fraction. −23

Find each sum: −415+(−615). −23 Find each sum: −521+(−921). −23

Model Fraction SubtractionSubtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 12 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 712 of the pizza) left in the box. If Leonardo eats 2 of these remaining pieces (or 212 of the pizza), how much is left? There would be 5 pieces left (or 512 of the pizza). 712−212=512 Let’s use fraction circles to model the same example, 712−212. Start with seven 112 pieces. Take away two 112 pieces. How many twelfths are left?

Again, we have five twelfths, 512. Doing the Manipulative Mathematics activity "Model Fraction Subtraction" will help you develop a better understanding of subtracting fractions. Use fraction circles to find the difference: 45−15. Solution Start with four 15 pieces. Take away one 15 piece. Count how many fifths are left. There are three 15 pieces left.

Use a model to find each difference. Show a diagram to illustrate your model. 78−48 38, models may differ. Use a model to find each difference. Show a diagram to illustrate your model. 56−46 16, models may differ

Subtract Fractions with a Common Denominator We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator. Fraction Subtraction If a,b, and c are numbers where c≠0, then ac−bc=a−bc To subtract fractions with a common denominator, we subtract the numerators and place the difference over the common denominator. Find the difference: 2324−1424. Solution 2324−1424 Subtract the numerators and place the difference over the common denominator. 23−1424 Simplify the numerator. 924 Simplify the fraction by removing common factors. 38

Find the difference: 1928−728. 37 Find the difference: 2732−1132. 12 Find the difference: y6−16. Solution y6−16 Subtract the numerators and place the difference over the common denominator. y−16

The fraction is simplified because we cannot combine the terms in the numerator. Find the difference: x7−27. x−27 Find the difference: y14−1314. y−1314 Find the difference: −10x−4x. Solution Remember, the fraction −10x can be written as −10x. −10x−4x Subtract the numerators. −10−4x Simplify. −14x Rewrite with the negative sign in front of the fraction. −14x

Find the difference: −9x−7x. −16x Find the difference: −17a−5a. −22a Now lets do an example that involves both addition and subtraction. Simplify: 38+(−58)−18. Solution 38+(−58)−18 Combine the numerators over the common denominator. 3+(−5)−18 Simplify the numerator, working left to right. −2−18 Subtract the terms in the numerator. −38 Rewrite with the negative sign in front of the fraction. −38

Simplify: 25+(−45)−35. −1 Simplify: 59+(−49)−79. −23 ACCESS ADDITIONAL ONLINE RESOURCES Adding Fractions With Pattern Blocks Adding Fractions With Like Denominators Subtracting Fractions With Like Denominators

Practice Makes Perfect Model Fraction Addition In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model. 25+15 310+410

710 16+36 38+38

34 Add Fractions with a Common Denominator In the following exercises, find each sum. 49+19 29+59 79 613+713 915+715 1615 x4+34 y3+23 y+23 7p+9p 8q+6q 14q 8b9+3b9 5a7+4a7 9a7 −12y8+3y8 −11x5+7x5 −4x5 −18+(−38) −18+(−58) −34 −316+(−716) −516+(−916) −78 −817+1517 −919+1719 819 613+(−1013)+(−1213) 512+(−712)+(−1112) −1312 Model Fraction Subtraction In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model. 58−28 56−26

12 Subtract Fractions with a Common Denominator In the following exercises, find the difference. 45−15 45−35 15 1115−715 913−413 513 1112−512 712−512 16 421−1921 −89−169 −83 y17−917 x19−819 x−819 5y8−78 11z13−813 11z−813 −8d−3d −7c−7c −14c −23u−15u −29v−26v −55v 6c7−5c7 12d11−9d11 3d11 −4r13−5r13 −7s3−7s3 −14s3 −35−(−45) −37−(−57) 27 −79−(−59) −811−(−511) −311 Mixed Practice In the following exercises, perform the indicated operation and write your answers in simplified form. −518·910 −314·712 −18 n5−45 611−s11 6−s11 −724+224 −518+118 −29 815÷125 712÷928 4927

Everyday Math Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has 610 of a pound of nuts and 310 of a pound of raisins. How much trail mix can he make? Baking Janet needs 58 of a cup of flour for a recipe she is making. She only has 38 of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow? 14 cup

Writing Exercises Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know. Bits in case: 116, 18, ___, ___, 516, 38, ___, 12, 916, 58. Bits that fell out: 716, 316, 14. After a party, Lupe has 512 of a cheese pizza, 412 of a pepperoni pizza, and 412 of a veggie pizza left. Will all the slices fit into 1 pizza box? Explain your reasoning. No, adding up the number of pieces gives 1312, which is greater than 1. (Answers may vary.)

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

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?