m81289 Add and Subtract Fractions with Different Denominators By the end of this section, you will be able to: Find the least common denominator (LCD) Convert fractions to equivalent fractions with the LCD Add and subtract fractions with different denominators Identify and use fraction operations Use the order of operations to simplify complex fractions Evaluate variable expressions with fractions 585a56d9-e90a-42c5-a00b-8d275fddd81a Before you get started, take this readiness quiz. Find two fractions equivalent to 56. If you missed this problem, review . 1012,1518 Simplify: 1+5·322+4. If you missed this problem, review . 2

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators? Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals 25 cents and one dime equals 10 cents, so the sum is 35 cents. See .

Together, a quarter and a dime are worth 35 cents, or 35100 of a dollar.

Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is 100. Since there are 100 cents in one dollar, 25 cents is 25100 and 10 cents is 10100. So we add 25100+10100 to get 35100, which is 35 cents. You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators. First, we will use fraction tiles to model finding the common denominator of 12 and 13. We’ll start with one 12 tile and 13 tile. We want to find a common fraction tile that we can use to match both 12 and 13 exactly. If we try the 14 pieces, 2 of them exactly match the 12 piece, but they do not exactly match the 13 piece. If we try the 15 pieces, they do not exactly cover the 12 piece or the 13 piece. If we try the 16 pieces, we see that exactly 3 of them cover the 12 piece, and exactly 2 of them cover the 13 piece. If we were to try the 112 pieces, they would also work. Even smaller tiles, such as 124 and 148, would also exactly cover the 12 piece and the 13 piece. The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of 12 and 13 is 6.Notice that all of the tiles that cover 12 and 13 have something in common: Their denominators are common multiples of 2 and 3, the denominators of 12 and 13. The least common multiple (LCM) of the denominators is 6, and so we say that 6 is the least common denominator (LCD) of the fractions 12 and 13.Doing the Manipulative Mathematics activity "Finding the Least Common Denominator" will help you develop a better understanding of the LCD.The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD. Find the LCD for the fractions 712 and 518. Factor each denominator into its primes. List the primes of 12 and the primes of 18 lining them up in columns when possible. Bring down the columns. Multiply the factors. The product is the LCM. LCM=36 The LCM of 12 and 18 is 36, so the LCD of 712 and 518 is 36. LCD of 712 and 518 is 36. Find the least common denominator for the fractions: 712 and 1115. 60 Find the least common denominator for the fractions: 1315 and 175. 15 To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM. Factor each denominator into its primes. List the primes, matching primes in columns when possible. Bring down the columns. Multiply the factors. The product is the LCM of the denominators. The LCM of the denominators is the LCD of the fractions. Find the least common denominator for the fractions 815 and 1124. To find the LCD, we find the LCM of the denominators. Find the LCM of 15 and 24. The LCM of 15 and 24 is 120. So, the LCD of 815 and 1124 is 120. Find the least common denominator for the fractions: 1324 and 1732. 96 Find the least common denominator for the fractions: 928 and 2132. 224

Earlier, we used fraction tiles to see that the LCD of 14 when 16 is 12. We saw that three 112 pieces exactly covered 14 and two 112 pieces exactly covered 16, so 14=312 and 16=212. We say that 14 and 312 are equivalent fractions and also that 16 and 212 are equivalent fractions. We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference. If a,b,c are whole numbers where b≠0,c≠0, then ab=a·cb·canda·cb·c=ab To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let’s see how to change 14 and 16 to equivalent fractions with denominator 12 without using models. Convert 14 and 16 to equivalent fractions with denominator 12, their LCD. Find the LCD. The LCD of 14 and 16 is 12. Find the number to multiply 4 to get 12. Find the number to multiply 6 to get 12. Use the Equivalent Fractions Property to convert each fraction to an equivalent fraction with the LCD, multiplying both the numerator and denominator of each fraction by the same number. Simplify the numerators and denominators. We do not reduce the resulting fractions. If we did, we would get back to our original fractions and lose the common denominator. Change to equivalent fractions with the LCD: 34 and 56, LCD =12 912,1012 Change to equivalent fractions with the LCD: −712 and 1115, LCD =60 −3560,4460 Find the LCD. For each fraction, determine the number needed to multiply the denominator to get the LCD. Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2. Simplify the numerator and denominator. Convert 815 and 1124 to equivalent fractions with denominator 120, their LCD. The LCD is 120. We will start at Step 2. Find the number that must multiply 15 to get 120. Find the number that must multiply 24 to get 120. Use the Equivalent Fractions Property. Simplify the numerators and denominators. Change to equivalent fractions with the LCD: 1324 and 1732, LCD 96 5296,5196 Change to equivalent fractions with the LCD: 928 and 2732, LCD 224 72224,189224

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators. Find the LCD. Convert each fraction to an equivalent form with the LCD as the denominator. Add or subtract the fractions. Write the result in simplified form. Add: 12+13. 12+13 Find the LCD of 2, 3. Change into equivalent fractions with the LCD 6. Simplify the numerators and denominators. 36+26 Add. 56 Remember, always check to see if the answer can be simplified. Since 5 and 6 have no common factors, the fraction 56 cannot be reduced. Add: 14+13. 712 Add: 12+15. 710 Subtract: 12−(−14). 12−(−14) Find the LCD of 2 and 4. Rewrite as equivalent fractions using the LCD 4. Simplify the first fraction. 24−(−14) Subtract. 2−(−1)4 Simplify. 34 One of the fractions already had the least common denominator, so we only had to convert the other fraction. Simplify: 12−(−18). 58 Simplify: 13−(−16). 12 Add: 712+518. 712+518 Find the LCD of 12 and 18. Rewrite as equivalent fractions with the LCD. Simplify the numerators and denominators. 2136+1036 Add. 3136 Because 31 is a prime number, it has no factors in common with 36. The answer is simplified. Add: 712+1115. 7960 Add: 1315+1720. 10360 When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need. The LCD, 36, has 2 factors of 2 and 2 factors of 3. Twelve has two factors of 2, but only one of 3—so it is ‘missing‘ one 3. We multiplied the numerator and denominator of 712 by 3 to get an equivalent fraction with denominator 36. Eighteen is missing one factor of 2—so you multiply the numerator and denominator 518 by 2 to get an equivalent fraction with denominator 36. We will apply this method as we subtract the fractions in the next example. Subtract: 715−1924. 715−1924 Find the LCD.15 is 'missing' three factors of 224 is 'missing' a factor of 5 Rewrite as equivalent fractions with the LCD. Simplify each numerator and denominator. 56120−95120 Subtract. −39120 Rewrite showing the common factor of 3. −13·340·3 Remove the common factor to simplify. −1340 Subtract: 1324−1732. 196 Subtract: 2132−928. 75224 Add: −1130+2342. −1130+2342 Find the LCD. Rewrite as equivalent fractions with the LCD. Simplify each numerator and denominator. −77210+115210 Add. 38210 Rewrite showing the common factor of 2. 19·2105·2 Remove the common factor to simplify. 19105 Add: −1342+1735. 37210 Add: −1924+1732. −2596 In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers. Add: 35+x8. The fractions have different denominators. 35+x8 Find the LCD. Rewrite as equivalent fractions with the LCD. Simplify the numerators and denominators. 2440+5x40 Add. 24+5x40 We cannot add 24 and 5x since they are not like terms, so we cannot simplify the expression any further. Add: y6+79. 3y+1418 Add: x6+715. 5x+1430

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions Fraction multiplication: Multiply the numerators and multiply the denominators.ab·cd=acbd Fraction division: Multiply the first fraction by the reciprocal of the second.ab÷cd=ab·dc Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.ac+bc=a+bc Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.ac−bc=a−bc Simplify: ⓐ −14+16 ⓑ −14÷16 First we ask ourselves, “What is the operation?” ⓐ The operation is addition.Do the fractions have a common denominator? No. −14+16 Find the LCD. Rewrite each fraction as an equivalent fraction with the LCD. Simplify the numerators and denominators. −312+212 Add the numerators and place the sum over the common denominator. −112 Check to see if the answer can be simplified. It cannot. ⓑ The operation is division. We do not need a common denominator. −14÷16 To divide fractions, multiply the first fraction by the reciprocal of the second. −14·61 Multiply. −64 Simplify. −32 Simplify each expression: ⓐ −34−16 ⓑ −34·16 ⓐ −1112 ⓑ −18 Simplify each expression: ⓐ 56÷(−14) ⓑ 56−(−14) ⓐ −103 ⓑ 1312 Simplify: ⓐ 5x6−310 ⓑ 5x6·310 ⓐ The operation is subtraction. The fractions do not have a common denominator. 5x6−310 Rewrite each fraction as an equivalent fraction with the LCD, 30. 5x·56·5−3·310·3 25x30−930 Subtract the numerators and place the difference over the common denominator. 25x−930 ⓑ The operation is multiplication; no need for a common denominator. 5x6·310 To multiply fractions, multiply the numerators and multiply the denominators. 5x·36·10 Rewrite, showing common factors. 5·x·32·3·2·5 Remove common factors to simplify. x4 Simplify: ⓐ (27a−32)36 ⓑ 2a3 ⓐ (27a−32)36 ⓑ 2a3 Simplify: ⓐ (24k+25)30 ⓑ 24k5 ⓐ (24k+25)30 ⓑ 24k5

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,3458=34÷58 Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator. Simplify the numerator. Simplify the denominator. Divide the numerator by the denominator. Simplify if possible. Simplify: (12)24+32. (12)24+32 Simplify the numerator. 144+32 Simplify the term with the exponent in the denominator. 144+9 Add the terms in the denominator. 1413 Divide the numerator by the denominator. 14÷13 Rewrite as multiplication by the reciprocal. 14·113 Multiply. 152 Simplify: (13)223+2. 190 Simplify: 1+42(14)2. 272 Simplify: 12+2334−16. 12+2334−16 Rewrite numerator with the LCD of 6 and denominator with LCD of 12. 36+46912−212 Add in the numerator. Subtract in the denominator. 76712 Divide the numerator by the denominator. 76÷712 Rewrite as multiplication by the reciprocal. 76·127 Rewrite, showing common factors. 7·6·26·7·1 Simplify. 2 Simplify: 13+1234−13. 2 Simplify: 23−1214+13. 27

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify. Evaluate x+13 when ⓐ x=−13 ⓑ x=−34. ⓐ To evaluate x+13 when x=−13, substitute −13 for x in the expression. x+13 Simplify. 0 ⓑ To evaluate x+13 when x=−34, we substitute −34 for x in the expression. x+13 Rewrite as equivalent fractions with the LCD, 12. −3·34·3+1·43·4 Simplify the numerators and denominators. −912+412 Add. −512 Evaluate: x+34 when ⓐ x=−74 ⓑ x=−54 ⓐ −1 ⓑ −12 Evaluate: y+12 when ⓐ y=23 ⓑ y=−34 ⓐ 76 ⓑ −14 Evaluate y−56 when y=−23. We substitute −23 for y in the expression. y−56 Rewrite as equivalent fractions with the LCD, 6. −46−56 Subtract. −96 Simplify. −32 Evaluate: y−12 when y=−14. −34 Evaluate: x−38 when x=−52. −238 Evaluate 2x2y when x=14 and y=−23. Substitute the values into the expression. In 2x2y, the exponent applies only to x. Simplify exponents first. Multiply. The product will be negative. Simplify. Remove the common factors. Simplify. Evaluate. 3ab2 when a=−23 and b=−12. −12 Evaluate. 4c3d when c=−12 and d=−43. 23 Evaluate p+qr when p=−4,q=−2, and r=8. We substitute the values into the expression and simplify. p+qr Add in the numerator first. −68 Simplify. −34 Evaluate: a+bc when a=−8,b=−7, and c=6. −52 Evaluate: x+yz when x=9,y=−18, and z=−6. 32

Find the least common denominator (LCD) of two fractions.Factor each denominator into its primes. List the primes, matching primes in columns when possible. Bring down the columns. Multiply the factors. The product is the LCM of the denominators. The LCM of the denominators is the LCD of the fractions. Equivalent Fractions PropertyIf a, b, and c are whole numbers where b≠0, c≠0 then ab = a⋅cb⋅c and a⋅cb⋅c=ab Convert two fractions to equivalent fractions with their LCD as the common denominator.Find the LCD. For each fraction, determine the number needed to multiply the denominator to get the LCD. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2. Simplify the numerator and denominator.Add or subtract fractions with different denominators.Find the LCD. Convert each fraction to an equivalent form with the LCD as the denominator. Add or subtract the fractions. Write the result in simplified form.Summary of Fraction OperationsFraction multiplication: Multiply the numerators and multiply the denominators. ab⋅cd=acbd Fraction division: Multiply the first fraction by the reciprocal of the second.ab+cd=ab⋅dc Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.ac+bc=a+bc Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.ac-bc=a-bc Simplify complex fractions.Simplify the numerator. Simplify the denominator. Divide the numerator by the denominator. Simplify if possible.

Find the Least Common Denominator (LCD) In the following exercises, find the least common denominator (LCD) for each set of fractions. 23 and 34 34 and 25 20 712 and 58 916 and 712 48 1330 and 2542 2330 and 548 240 2135 and 3956 1835 and 3349 245 23,16, and 34 23,14, and 35 60 Convert Fractions to Equivalent Fractions with the LCD In the following exercises, convert to equivalent fractions using the LCD. 13 and 14, LCD =12 14 and 15, LCD =20 520,420 512 and 78, LCD =24 712 and 58, LCD =24 1424,1524 1316 and -1112, LCD =48 1116 and -512, LCD =48 3348,−2048 13,56, and 34, LCD =12 13,34, and 35, LCD =60 2060,4560,3660 Add and Subtract Fractions with Different Denominators In the following exercises, add or subtract. Write the result in simplified form. 13+15 14+15 920 12+17 13+18 1124 13−(−19) 14−(−18) 38 15−(−110) 12−(−16) 23 23+34 34+25 2320 712+58 512+38 1924 712−916 716−512 148 1112−38 58−712 124 23−38 56−34 112 −1130+2740 −920+1730 760 −1330+2542 −2330+548 −5380 −3956−2235 −3349−1835 −291245 −23−(−34) −34−(−45) 120 −916−(−45) −720−(−58) 1140 1+78 1+56 116 1−59 1−310 710 x3+14 y2+23 3y+46 y4−35 x5−14 4x−520 Identify and Use Fraction Operations In the following exercises, perform the indicated operations. Write your answers in simplified form. ⓐ 34+16 ⓑ 34÷16 ⓐ 23+16 ⓑ 23÷16 ⓐ 56 ⓑ 4 ⓐ -25−18 ⓑ -25·18 ⓐ -45−18 ⓑ -45·18 ⓐ −3740 ⓑ −110 ⓐ 5n6÷815 ⓑ 5n6−815 ⓐ 3a8÷712 ⓑ 3a8−712 ⓐ 9a14 ⓑ 9a−1424 ⓐ 910·(−11d12) ⓑ 910+(−11d12) ⓐ 415·(−5q9) ⓑ 415+(−5q9) ⓐ −4q27 ⓑ 12−25q45 −38÷(−310) −512÷(−59) 34 −38+512 −18+712 1124 56−19 59−16 718 38·(−1021) 712·(−835) −215 −715−y4 −38−x11 −33−8x88 1112a·9a16 10y13·815y 1639 Use the Order of Operations to Simplify Complex Fractions In the following exercises, simplify. (15)22+32 (13)25+22 181 23+42(23)2 33−32(34)2 32 (35)2(37)2 (34)2(58)2 3625 213+15 514+13 607 23+1234−23 34+1256−23 152 78−2312+38 34−3514+25 313 Mixed Practice In the following exercises, simplify. 12+23·512 13+25·34 1930 1−35÷110 1−56÷112 −9 23+16+34 23+14+35 9160 38−16+34 25+58−34 1140 12(920−415) 8(1516−56) 56 58+161924 16+3101430 1 (59+16)÷(23−12) (34+16)÷(58−13) 227 In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. x+12 when ⓐ x=−18 ⓑ x=−12 x+23 when ⓐ x=−16 ⓑ x=−53 ⓐ 12 ⓑ −1 x+(−56) when ⓐ x=13 ⓑ x=−16 x+(−1112) when ⓐ x=1112 ⓑ x=34 ⓐ 0 ⓑ −16 x−25 when ⓐ x=35 ⓑ x=−35 x−13 when ⓐ x=23 ⓑ x=−23 ⓐ 13 ⓑ −1 710−w when ⓐ w=12 ⓑ w=−12 512−w when ⓐ w=14 ⓑ w=−14 ⓐ 16 ⓑ 23 4p2q when p=−12 and q=59 5m2n when m=−25 and n=13 415 2x2y3 when x=−23 and y=−12 8u2v3 when u=−34 and v=−12 −916 u+vw when u=−4,v=−8,w=2 m+np when m=−6,n=−2,p=4 −2 a+ba−b when a=−3,b=8 r−sr+s when r=10,s=−5 3

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs 316 yard of print fabric and 38 yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover? Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs 114 cups of sugar for the chocolate chip cookies, and 118 cups for the oatmeal cookies How much sugar does she need altogether? She needs 238 cups

Explain why it is necessary to have a common denominator to add or subtract fractions. Explain how to find the LCD of two fractions. Answers will vary.

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

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

least common denominator (LCD)The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.