# Adding and Subtracting Mixed Numbers

Adding and subtracting mixed numbers to simplest form

6 2/3 - 5 3/9 =
6 8/12 - 5 1/3 =
5 5/9 + 3 4/6 =
9 5/9 - 6 ½ =
14 ¾ - 8 5/6 =
5 2/9 + 3 3/6 =
7 3/12 + 4 1/8 =
6 7/8 - 4 2/9 =
8 1/3 + 6 7/6 =
5 2/7 - 3 5/6 =

 Dec 16, 2012 Mixed Numbers by: Staff Answer Part I There are three widely used methods of adding or subtracting mixed numbers: (1) add or subtract the whole numbers and the fractional components separately (5 + 5/9) + (3 + 4/6); (2) convert the two numbers to their decimal equivalents, and then add the decimal equivalents (5.555556 + 3.666667); (3) convert any mixed numbers to improper fractions, and then add the fractions. I’m going to demonstrate the third method. Problem 1: 6 2/3 - 5 3/9 Begin by converting the mixed numbers 6 ⅔ and 5 3/9 into improper fractions. This will make the subtraction of the two fractions much easier. Converting 6 ⅔ to an improper fraction 6*3 + 2 = 20, so the improper fraction will be 20/3 Converting 5 3/9 to an improper fraction First, reduce the fraction 3/9 to its lowest terms: 3/9 = ⅓ 5 3/9 becomes 5 ⅓. Next, convert 5 ⅓ to an improper fraction 5*3 + 1 = 16, so the improper fraction will be 16/3 Last, subtract the two improper fractions 6 2/3 - 5 3/9 = 20/3 - 16/3 As you can see, this is exactly the same problem as problem 1. Since the two fractions have the same denominator, they can be subtracted easily. (Remember, only the numerators are subtracted. The denominator of 3 remains the same) = (20 - 16)/3 = 4/3 This fraction cannot be reduced, but it can be converted into a mixed number. = 4/3 To convert 4/3 to a mixed number, divide 4 by 3 = 4÷3 = 1, remainder 1 = 1 ⅓ The final answer is: 1 ⅓ (or 4/3 or 1.33333…) Check the answer with a calculator: 6 2/3 - 5 3/9 = ????? = (6.6666…) - (5.3333…) = 1.3333… Since 1.3333 is the same answer obtained by converting 1 ⅓ to its decimal equivalent, the answer of 1 ⅓ is correct. ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part II Problem 2: 6 8/12 - 5 1/3 Begin by converting the mixed numbers 6 8/12 and 5 1/3 into improper fractions. This will make the subtraction much easier. Converting 6 8/12 to an improper fraction First, reduce the fraction 8/12 to its lowest terms: 8/12 = ⅔ 6 8/12 can be rewritten as 6 ⅔ 6*3 + 2 = 20, so the improper fraction will be 20/3 Converting 5 1/3 to an improper fraction 5*3 + 1 = 16, so the improper fraction will be 16/3 Last, subtract the two improper fractions 6 2/3 - 5 3/9 = 20/3 - 16/3 Since the two fractions have the same denominator, they can be subtracted easily. (Remember, only the numerators are subtracted. The denominator of 3 remains the same) = (20 - 16)/3 = 4/3 This fraction cannot be reduced, but it can be converted into a mixed number. = 4/3 To convert 4/3 to a mixed number, divide 4 by 3 = 4÷3 = 1, remainder 1 = 1 ⅓ The final answer is (the same as problem 1): 1 ⅓ (or 4/3 or 1.33333…) Problem 3: 5 5/9 + 3 4/6 Convert the mixed numbers 5 5/9 and 3 4/6 into improper fractions. Converting 5 5/9 to an improper fraction 5*9 + 5 = 50, so the improper fraction will be 50/9 Converting 3 4/6 to an improper fraction First, reduce the fraction 4/6 to its lowest terms: 4/6 = ⅔ 3 4/6 is equivalent to 3 ⅔. Next, convert 3 ⅔ to an improper fraction 3*3 + 2 = 11, so the improper fraction will be 11/3 Last, add the two improper fractions 5 5/9 + 3 4/6 = 50/9 + 11/3 As you can see, this presents a new problem. Since the two fractions DO NOT have the same denominator, they cannot be added. (Only fractions with like denominators can be added.) Before these fractions can be added, they must be rewritten so that each fraction has “exactly” the same denominator. Convert the fraction 11/3 to a fraction with a denominator of 9 (the same as the denominator of the fraction 50/9) To accomplish this, multiply the fraction 11/3 by another fraction: 3/3. Note that 3/3 = 1, so we are actually multiplying by 1. The decimal value of the fraction 11/3 will not change. = (11/3)*(3/3) Multiply the numerators together, and multiply the denominators together. = (11*3)/(3*3) = 33/9 11/3 is now 33/9 ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part III Since the two fractions now have the same denominator, they can now be added. (Remember, only the numerators are added. The denominator of 9 remains the same) 5 5/9 + 3 4/6 = 50/9 + 11/3 = 50/9 + 33/9 = (50 + 33)/9 = 83/9 This fraction cannot be reduced, but it can be converted into a mixed number. = 83/9 To convert 83/9 to a mixed number, divide 83 by 9 = 83 ÷ 9 = 9, remainder 2 = 9 2/9 The final answer is (the same as problem 1): 9 2/9 (or 83/9 or 9.22222…) Check the answer with a calculator: 5 5/9 + 3 4/6 = ????? = (5.55555…) + (3.66666…) = 9.2222… Since 9.2222 is the same answer obtained by converting 83/9 to its decimal equivalent, the answer of 9 2/9 is correct. Problem 4: 9 5/9 - 6 ½ Convert the mixed numbers 9 5/9 and 6 ½ into improper fractions. Converting 9 5/9 to an improper fraction 9*9 + 5 = 86, so the improper fraction will be 86/9 Converting 6 ½ to an improper fraction 6*2 + 1 = 13, so the improper fraction will be 13/2 Last, add the two improper fractions 9 5/9 - 6 ½ = 86/9 - 13/2 Since the two fractions DO NOT have the same denominator, they cannot be subtracted. (Only fractions with like denominators can be subtracted.) Before these fractions can be subtracted, they must be rewritten so that each fraction has “exactly” the same denominator. Convert both fractions so that they each have a denominator of 18. To accomplish this, multiply the fraction 86/9 by the fraction 2/2, and multiply the fraction 13/2 by the fraction 9/9. Note that 2/2 = 1 and 9/9 = 1, so we are actually multiplying by 1. The decimal value of both fractions will not change. = (86/9)*(2/2) - (13/2)*(9/9) Multiply the numerators together, and multiply the denominators together. = (86/9)*(2/2) - (13/2)*(9/9) = (86*2)/(9*2) - (13*9)/(2*9) = (172)/(18) - (117)/(18) Since the two fractions now have the same denominator, they can now be subtracted. (Remember, only the numerators are subtracted. The denominator of 18 remains the same) 9 5/9 - 6 ½ = 86/9 - 13/2 = (172)/(18) - (117)/(18) = (172 - 117)/18 = 55/18 ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part IV This fraction cannot be reduced, but it can be converted into a mixed number. = 55/18 To convert 55/18 to a mixed number, divide 55 by 18 = 55 ÷ 18 = 3, remainder 1 = 3 1/18 The final answer is: 3 1/18 (or 55/18 or 3.0555…) Check the answer with a calculator: 9 5/9 - 6 ½ = ????? = (9.55555…) - (6.5) = 3.05555… Since 3.05555… is the same answer obtained by converting 55/18 to its decimal equivalent, the answer of 3 1/18 is correct. Problem 5: 14 ¾ - 8 5/6 Convert the mixed numbers 14 ¾ and 8 5/6 into improper fractions. Converting 14 ¾ to an improper fraction 14*4 + 3 = 59, so the improper fraction will be 59/4 Converting 8 5/6 to an improper fraction 8*6 + 5 = 53, so the improper fraction will be 53/6 Last, add the two improper fractions 14 ¾ - 8 5/6 = 59/4 - 53/6 Since the two fractions DO NOT have the same denominator, they cannot be subtracted. (Only fractions with like denominators can be subtracted.) Before these fractions can be subtracted, they must be rewritten so that each fraction has “exactly” the same denominator. Convert both fractions so that they each have a denominator of 12. To accomplish this, multiply the fraction 59/4 by the fraction 3/3, and multiply the fraction 53/6 by the fraction 2/2. Note that 3/3 = 1 and 2/2 = 1, so we are actually multiplying by 1. The decimal value of both fractions will not change. = (59/4)*(3/3) - (53/6)*(2/2) Multiply the numerators together, and multiply the denominators together. = (59/4)*(3/3) - (53/6)*(2/2) = (59*3)/(4*3) - (53*2)/(6*2) = (177)/(12) - (106)/(12) Since the two fractions now have the same denominator, they can now be subtracted. (Remember, only the numerators are subtracted. The denominator of 12 remains the same) 14 ¾ - 8 5/6 = 59/4 - 53/6 = (177)/(12) - (106)/(12) = (177 - 106)/12 = 71/12 This fraction cannot be reduced, but it can be converted into a mixed number. = 71/12 To convert 71/12 to a mixed number, divide 71 by 12 = 71 ÷ 12 = 5, remainder 11 = 5 11/12 The final answer is: 5 11/12 (or 71/12 or 5.91666…) Check the answer with a calculator: 14 ¾ - 8 5/6 = ????? = (14.75) - (8.83333…) = 5.91666… Since 5.91666… is the same answer obtained by converting 71/12 to its decimal equivalent, the answer of 5 11/12 is correct. ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part V Problem 6: 5 2/9 + 3 3/6 Convert the mixed numbers 5 2/9 and 3 3/6 into improper fractions. Converting 5 2/9 to an improper fraction 5*9 + 2 = 47, so the improper fraction will be 47/9 Converting 3 3/6 to an improper fraction First, reduce the fraction 3/6 to its lowest terms (reducing this fraction to its lowest terms is not necessary for this particular problem, but it is a generally a good idea): 3/6 = ½ 3 3/6 can be rewritten as 3 ½ 3*2 + 1 = 7, so the improper fraction will be 7/2 Add the two improper fractions 5 2/9 + 3 3/6 = 47/9 + 7/2 Since the two fractions DO NOT have the same denominator, they cannot be added. (Only fractions with like denominators can be added.) Before these fractions can be subtracted, they must be rewritten so that each fraction has “exactly” the same denominator. Convert both fractions so that they each have a denominator of 18. To accomplish this, multiply the fraction 47/9 by the fraction 2/2, and multiply the fraction 7/2 by the fraction 9/9. Note that 2/2 = 1 and 9/9 = 1, so we are actually multiplying by 1. The decimal value of both fractions will not change. = (47/9)*(2/2) + (7/2)*(9/9) Multiply the numerators together, and multiply the denominators together. = (47/9)*(2/2) + (7/2)*(9/9) = (47*2)/(9*2) + (7*9)/(2*9) = (94)/(18) + (63)/(18) Since the two fractions now have the same denominator, they can now be subtracted. (Remember, only the numerators are subtracted. The denominator of 12 remains the same) 5 2/9 + 3 3/6 = 47/9 + 7/2 = (94)/(18) + (63)/(18) = (94 + 63)/18 = 157/18 This fraction cannot be reduced, but it can be converted into a mixed number. = 157/18 To convert 157/18 to a mixed number, divide 157 by 18 = 157 ÷ 18 = 8, remainder 13 = 8 13/18 The final answer is: 8 13/18 (or 157/18 or 8.72222…) Check the answer with a calculator: 5 2/9 + 3 3/6 = ????? = (5.2222…) + (3.5) = 8.7222… Since 8.7222… is the same answer obtained by converting 157/18 to its decimal equivalent, the answer of 8 13/18 is correct. ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part VI Problem 7: 7 3/12 + 4 1/8 Convert the mixed numbers 7 3/12 and 4 1/8 into improper fractions. Converting 7 3/12 to an improper fraction First, reduce the fraction 3/12 to its lowest terms (this will simplify to problem to some extent): 3/12 = ¼ 7 3/12 can be rewritten as 7 ¼ 7*4 + 1 = 29, so the improper fraction will be 29/4 Converting 4 1/8 to an improper fraction 4*8 + 1 = 33, so the improper fraction will be 33/8 Add the two improper fractions 7 3/12 + 4 1/8 = 7 ¼ + 4 ⅛ = 29/4 + 33/8 Since the two fractions DO NOT have the same denominator, they cannot be added. (Only fractions with like denominators can be added.) Before these fractions can be subtracted, they must be rewritten so that each fraction has “exactly” the same denominator. Convert the fraction 29/4 so that it has the denominator of 8 (the same as the second fraction, 33/8). To accomplish this, multiply the fraction 29/4 by the fraction 2/2. Note that 2/2 = 1, so we are actually multiplying by 1. The decimal value the fraction will not change. = (29/4)*(2/2) + (33/8) Multiply the numerators together, and multiply the denominators together. = (29/4)*(2/2) + (33/8) = (29*2)/(4*2) + (33/8) = (58/8) + (33/8) Since the two fractions now have the same denominator, they can now be added. (Remember, only the numerators are added. The denominator of 8 remains the same) 7 ¼ + 4 ⅛ = 29/4 + 33/8 = (58/8) + (33/8) = (58 + 33)/8 = 91/8 This fraction cannot be reduced, but it can be converted into a mixed number. = 91/8 To convert 157/18 to a mixed number, divide 157 by 18 = 91 ÷ 8 = 11, remainder 3 = 11 ⅜ The final answer is: 11 ⅜ (or 91/8 or 11.375) Check the answer with a calculator: 7 3/12 + 4 1/8 = ????? = (7.25) + (4.125) = 11.375 Since 11.375 is the same answer obtained by converting 91/8 to its decimal equivalent, the answer of 11 ⅜ is correct. ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part VII Problem 8: 6 7/8 - 4 2/9 Convert the mixed numbers 6 7/8 and 4 2/9 into improper fractions. Converting 6 7/8 to an improper fraction 6*8 + 7 = 55, so the improper fraction will be 55/8 Converting 4 2/9 to an improper fraction 4*9 + 2 = 38, so the improper fraction will be 38/9 Last, add the two improper fractions 6 7/8 - 4 2/9 = 55/8 - 38/9 Since the two fractions DO NOT have the same denominator, they cannot be subtracted. (Only fractions with like denominators can be subtracted.) Before these fractions can be subtracted, they must be rewritten so that each fraction has “exactly” the same denominator. Convert both fractions so that they each have a denominator of 72. To accomplish this, multiply the fraction 55/8 by the fraction 9/9, and multiply the fraction 38/9 by the fraction 8/8. Note that 9/9 = 1 and 8/8 = 1, so we are actually multiplying by 1. The decimal value of both fractions will not change. = (55/8)*(9/9) - (38/9)*(8/8) Multiply the numerators together, and multiply the denominators together. = (55/8)*(9/9) - (38/9)*(8/8) = (55*9)/(8*9) - (38*8)/(9*8) = (495)/(72) - (304)/(72) Since the two fractions now have the same denominator, they can now be subtracted. (Remember, only the numerators are subtracted. The denominator of 72 remains the same) 6 7/8 - 4 2/9 = 55/8 - 38/9 = (495)/(72) - (304)/(72) = (495 - 304)/72 = 191/72 This fraction cannot be reduced, but it can be converted into a mixed number. = 191/72 To convert 191/72 to a mixed number, divide 191 by 72 = 191 ÷ 72 = 2, remainder 47 = 2 47/72 The final answer is: 2 47/72 (or 191/72 or 2.6527777…) Check the answer with a calculator: 6 7/8 - 4 2/9 = ????? = (6.875) - (4.22222…) = 2.6527777… Since 2.6527777… is the same answer obtained by converting 191/72 to its decimal equivalent, the answer of 2 47/72 is correct. ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part VIII Problem 9: 8 1/3 + 6 7/6 (NOTE: you can leave the mixed number 6 7/6 as it is, or reduce it to 7 1/6. Either way, the final answer will be the same.) Convert the mixed numbers 8 1/3 and 6 7/6 into improper fractions. Converting 8 1/3 to an improper fraction 8*3 + 1 = 25, so the improper fraction will be 25/3 Converting 6 7/6 to an improper fraction 6*6 + 7 = 43, so the improper fraction will be 43/6 Add the two improper fractions 8 1/3 + 6 7/6 = 25/3 + 43/6 Since the two fractions DO NOT have the same denominator, they cannot be added. (Only fractions with like denominators can be added.) Before these fractions can be subtracted, they must be rewritten so that each fraction has “exactly” the same denominator. Convert the fraction 25/3 so that it has the denominator of 6. To accomplish this, multiply the fraction 25/3 by the fraction 2/2. Note that 2/2 = 1, so we are actually multiplying by 1. The decimal value the fraction will not change. = (25/3)*(2/2) + (43/6) Multiply the numerators together, and multiply the denominators together. = (25/3)*(2/2) + (43/6) = (25*2)/(3*2) + (43/6) = (50)/(6) + (43)/(6) Since the two fractions now have the same denominator, they can now be subtracted. (Remember, only the numerators are subtracted. The denominator of 6 remains the same) 8 1/3 + 6 7/6 = 25/3 + 43/6 = (50)/(6) + (43)/(6) = (50 + 43)/6 = 93/6 This fraction CAN BE REDUCED 93/6 = 31/2 This fraction 31/2 cannot be reduced, but it can be converted into a mixed number. = 31/2 To convert 31/2 to a mixed number, divide 31 by 2 = 31 ÷ 2 = 15, remainder 1 = 15 ½ The final answer is: 15 ½ (or 31/2 or 15.5) Check the answer with a calculator: 8 1/3 + 6 7/6= ????? = (8.3333…) + (7.16666…) = 15.5 Since 15.5 is the same answer obtained by converting 157/18 to its decimal equivalent, the answer of 15 ½ is correct. ------------------------------------------------

 Dec 16, 2012 Mixed Numbers by: Staff ------------------------------------------------ Part IX Problem 10: 5 2/7 - 3 5/6 Convert the mixed numbers 5 2/7 and 3 5/6 into improper fractions. Converting 5 2/7 to an improper fraction 5*7 + 2 = 37, so the improper fraction will be 37/7 Converting 3 5/6 to an improper fraction 3*6 + 5 = 23, so the improper fraction will be 23/6 Last, add the two improper fractions 5 2/7 - 3 5/6 = 37/7 - 23/6 Since the two fractions DO NOT have the same denominator, they cannot be subtracted. (Only fractions with like denominators can be subtracted.) Before these fractions can be subtracted, they must be rewritten so that each fraction has “exactly” the same denominator. Convert both fractions so that they each have a denominator of 42. To accomplish this, multiply the fraction 37/7 by the fraction 6/6, and multiply the fraction 23/6 by the fraction 7/7. Note that 6/6 = 1 and 7/7 = 1, so we are actually multiplying by 1. The decimal value of both fractions will not change. = (37/7)*(6/6) - (23/6)*(7/7) Multiply the numerators together, and multiply the denominators together. = (37/7)*(6/6) - (23/6)*(7/7) = (37*6)/(7*6) - (23*7)/(6*7) = (222)/(42) - (161)/(42) Since the two fractions now have the same denominator, they can now be subtracted. (Remember, only the numerators are subtracted. The denominator of 42 remains the same) 5 2/7 - 3 5/6 = 37/7 - 23/6 = (222)/(42) - (161)/(42) = (222 - 161)/(42) = 61/42 This fraction cannot be reduced, but it can be converted into a mixed number. = 61/42 To convert 61/42 to a mixed number, divide 61 by 42 = 61 ÷ 42 = 1, remainder 19 = 1 19/42 The final answer is: 1 19/42 (or 61/42 or 1.452380952381) Check the answer with a calculator: 5 2/7 - 3 5/6 = ????? = (5.2857142857143) - (3.8333333333333…) = 1.452380952381 Since 1.452380952381 is the same answer obtained by converting 61/42 to its decimal equivalent, the answer of 1 19/42 is correct. Thanks for writing. Staff www.solving-math-problems.com