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Study Guides > ALGEBRA / TRIG I

Problem Set: Fractions

Representing parts of a whole

Using Models to Represent Fractions

In the following exercises, name the fraction of each figure that is shaded.

Exercise 1

In part

Answer: ⓐ = 14\Large\frac{1}{4} ⓑ = 34\Large\frac{3}{4} ⓒ = 38\Large\frac{3}{8} ⓓ = 59\Large\frac{5}{9}

Exercise 2

In part

Using Models to Represent Fractions

In the following exercises, shade parts of circles or squares to model the following fractions.
  1. 12\Large\frac{1}{2}

    Answer: A circle is shown. It is divided into 2 equal pieces. 1 piece is shaded.

  2. 13\Large\frac{1}{3}
  3. 34\Large\frac{3}{4}

    Answer: A circle is shown. It is divided into 4 equal pieces. 3 pieces are shaded.

  4. 25\Large\frac{2}{5}
  5. 56\Large\frac{5}{6}

    Answer: A circle is shown. It is divided into 6 equal pieces. 5 pieces are shaded.

  6. 78\Large\frac{7}{8}
  7. 58\Large\frac{5}{8}

    Answer: A circle is shown. It is divided into 8 equal pieces. 5 pieces are shaded.

  8. 710\Large\frac{7}{10}

Model Improper Fractions and Mixed Numbers

In the following exercises, use fraction circles to make wholes, if possible, with the following pieces.
  1. 33 thirds

    Answer: A circle is shown. It is divided into 3 equal pieces. All 3 pieces are shaded.

  2. 88 eighths
  3. 77 sixths

    Answer: Two circles are shown. Each is divided into 6 equal pieces. All 6 pieces are shaded in the circle on the left. 1 piece is shaded in the circle on the right.

  4. 44 thirds
  5. 77 fifths

    Answer: Two circles are shown. Each is divided into 5 equal pieces. All 5 pieces are shaded in the circle on the left. 2 pieces are shaded in the circle on the right.

  6. 77 fourths

Model Improper Fractions and Mixed Numbers

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

Exercise 1

In part

Answer: ⓐ 54=114\Large\frac{5}{4}\normalsize =1\Large\frac{1}{4}74=134\Large\frac{7}{4}\normalsize =1\Large\frac{3}{4}118=138\Large\frac{11}{8}\normalsize =1\Large\frac{3}{8}

Exercise 2

In part

Exercise 3

In part

Answer: ⓐ 114=234\Large\frac{11}{4}\normalsize =2\Large\frac{3}{4}198=238\Large\frac{19}{8}\normalsize =2\Large\frac{3}{8}

Model Improper Fractions and Mixed Numbers

In the following exercises, draw fraction circles to model the given fraction.
  1. 33\Large\frac{3}{3}
  2. 44\Large\frac{4}{4}

    Answer: A circle is shown. It is divided into 4 equal pieces. All 4 pieces are shaded.

  3. 74\Large\frac{7}{4}
  4. 53\Large\frac{5}{3}

    Answer: Two circles are shown. Each is divided into 3 equal pieces. All 3 pieces are shaded in the circle on the left. 2 pieces are shaded in the circle on the right.

  5. 116\Large\frac{11}{6}
  6. 138\Large\frac{13}{8}

    Answer: Two circles are shown. Each is divided into 8 equal pieces. All 8 pieces are shaded in the circle on the left. 5 pieces are shaded in the circle on the right.

  7. 103\Large\frac{10}{3}
  8. 94\Large\frac{9}{4}

    Answer: Three circles are shown. Each is divided into 4 equal pieces. All 4 pieces are shaded in the two circles on the left. 1 piece is shaded in the circle on the right.

Converting Between Improper Fractions and Mixed Numbers

Write an Improper Fraction as a Mixed Number

In the following exercises, rewrite the improper fraction as a mixed number.
  1. 32\Large\frac{3}{2}
  2. 53\Large\frac{5}{3}

    Answer: 1231\Large\frac{2}{3}

  3. 114\Large\frac{11}{4}
  4. 135\Large\frac{13}{5}

    Answer: 2352\Large\frac{3}{5}

  5. 256\Large\frac{25}{6}
  6. 289\Large\frac{28}{9}

    Answer: 3193\Large\frac{1}{9}

  7. 4213\Large\frac{42}{13}
  8. 4715\Large\frac{47}{15}

    Answer: 32153\Large\frac{2}{15}

Write a Mixed Number as an Improper Fraction

In the following exercises, rewrite the mixed number as an improper fraction.
  1. 1231\Large\frac{2}{3}
  2. 1251\Large\frac{2}{5}

    Answer: 75\Large\frac{7}{5}

  3. 2142\Large\frac{1}{4}
  4. 2562\Large\frac{5}{6}

    Answer: 176\Large\frac{17}{6}

  5. 2792\Large\frac{7}{9}
  6. 2572\Large\frac{5}{7}

    Answer: 197\Large\frac{19}{7}

  7. 3473\Large\frac{4}{7}
  8. 3593\Large\frac{5}{9}

    Answer: 329\Large\frac{32}{9}

Modeling and Finding Equivalent Fractions

Modeling Equivalent Fractions

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.
  1. How many sixths equal one-third?
  2. How many twelfths equal one-third?

    Answer: 4

  3. How many eighths equal three-fourths?
  4. How many twelfths equal three-fourths?

    Answer: 9

  5. How many fourths equal three-halves?
  6. How many sixths equal three-halves?

    Answer: 9

Finding Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
  1. 14\Large\frac{1}{4}
  2. 13\Large\frac{1}{3}

    Answer: Answers may vary. Correct answers include 26,39,412\Large\frac{2}{6},\Large\frac{3}{9},\Large\frac{4}{12}.

  3. 38\Large\frac{3}{8}
  4. 56\Large\frac{5}{6}

    Answer: Answers may vary. Correct answers include 1012,1518,2024\Large\frac{10}{12},\Large\frac{15}{18},\Large\frac{20}{24}.

  5. 27\Large\frac{2}{7}
  6. 59\Large\frac{5}{9}

    Answer: Answers may vary. Correct answers include 1018,1527,2036\Large\frac{10}{18},\Large\frac{15}{27},\Large\frac{20}{36}.

Locating and Ordering Fractions and Mixed Numbers on the Number Line

Locating Fractions on the Number Line

In the following exercises, plot the numbers on a number line.
  1. 23,54,125\Large\frac{2}{3},\Large\frac{5}{4},\Large\frac{12}{5}
  2. 13,74,135\Large\frac{1}{3},\Large\frac{7}{4},\Large\frac{13}{5}

    Answer: A number line is shown. The numbers 0, 1, 2, 3, 4, 5, and 6 are labeled. Between 0 and 1, 1 third is labeled and shown with a red dot. Between 1 and 2, 7 fourths is labeled and shown with a red dot. Between 2 and 3, 13 fifths is labeled and shown with a red dot.

  3. 14,95,113\Large\frac{1}{4},\Large\frac{9}{5},\Large\frac{11}{3}
  4. 710,52,138,3\Large\frac{7}{10},\Large\frac{5}{2},\Large\frac{13}{8},\normalsize 3

    Answer: A number line is shown. The numbers 0, 1, 2, 3, 4, 5, and 6 are labeled. Between 0 and 1, 7 tenths is labeled and shown with a red dot. Between 1 and 2, 13 eighths is labeled and shown with a red dot. Between 2 and 3, 5 halves is labeled and shown with a red dot. 3 is labeled and shown with a red dot.

  5. 213,2132\Large\frac{1}{3}\normalsize ,-2\Large\frac{1}{3}
  6. 134,1351\Large\frac{3}{4}\normalsize ,-1\Large\frac{3}{5}

    Answer: A number line is shown. The numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4 are labeled. Between negative 3 and negative 2, negative 2 and 1 third is labeled and shown with a red dot. Between 2 and 3, 2 and 1 third is labeled and shown with a red dot.

  7. 34,34,123,123,52,52\Large\frac{3}{4},-\Large\frac{3}{4}\normalsize ,1\Large\frac{2}{3}\normalsize ,-1\Large\frac{2}{3},\Large\frac{5}{2},-\Large\frac{5}{2}
  8. 25,25,134,134,83,83\Large\frac{2}{5},-\Large\frac{2}{5}\normalsize ,1\Large\frac{3}{4}\normalsize ,-1\Large\frac{3}{4},\Large\frac{8}{3},-\Large\frac{8}{3}

    Answer: A number line is shown. The numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4 are labeled. Between negative 3 and negative 2, negative 8 thirds is labeled and shown with a red dot. Between negative 2 and negative 1, negative 1 and 3 fourths is labeled and shown with a red dot. Between negative 1 and 0, negative 2 fifths is labeled and shown with a red dot. Between 0 and 1, 2 fifths is labeled and shown with a red dot. Between 1 and 2, 1 and 3 fourths is labeled and shown with a red dot. Between 2 and 3, 8 thirds is labeled and shown with a red dot.

Ordering Fractions on the Number Line

In the following exercises, order each of the following pairs of numbers, using << or >>.
  1. -1\text{ __}-\Large\frac{1}{4}
  2. -1\text{ __}-\Large\frac{1}{3}

    Answer: <

  3. -2\Large\frac{1}{2}\normalsize\text{ __}-3
  4. -1\Large\frac{3}{4}\normalsize\text{ __}-2

    Answer: >

  5. -\Large\frac{5}{12}\text{ __}-\Large\frac{7}{12}
  6. -\Large\frac{9}{10}\text{ __}-\Large\frac{3}{10}

    Answer: <

  7. -3\text{ __}-\Large\frac{13}{5}
  8. -4\text{ __}-\Large\frac{23}{6}

    Answer: <

Everyday Math

Music Measures

A choreographed dance is broken into counts. A 11\Large\frac{1}{1} count has one step in a count, a 12\Large\frac{1}{2} count has two steps in a count and a 13\Large\frac{1}{3} count has three steps in a count. How many steps would be in a 15\Large\frac{1}{5} count? What type of count has four steps in it?

Music Measures

Fractions are used often in music. In 44\Large\frac{4}{4} time, there are four quarter notes in one measure.
  1. How many measures would eight quarter notes make?

    Answer: 2

  2. The song "Happy Birthday to You" has 2424 quarter notes. How many measures are there in "Happy Birthday to You?"

    Answer: 6

Baking

Nina is making five pans of fudge to serve after a music recital. For each pan, she needs 12\Large\frac{1}{2} cup of walnuts.
  1. How many cups of walnuts does she need for five pans of fudge?
  2. Do you think it is easier to measure this amount when you use an improper fraction or a mixed number? Why?

Writing Exercises

Give an example from your life experience (outside of school) where it was important to understand fractions. [practice-area rows="4"][/practice-area] Explain how you locate the improper fraction 214\Large\frac{21}{4} on a number line on which only the whole numbers from 00 through 1010 are marked. [practice-area rows="4"][/practice-area]

Multiplying and Dividing Fractions

Simplify Fractions

In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.
  1. 721\Large\frac{7}{21}

    Answer: 13\Large\frac{1}{3}

  2. 824\Large\frac{8}{24}
  3. 1520\Large\frac{15}{20}

    Answer: 34\Large\frac{3}{4}

  4. 1218\Large\frac{12}{18}
  5. 4088-\Large\frac{40}{88}

    Answer: 511-\Large\frac{5}{11}

  6. 6399-\Large\frac{63}{99}
  7. 10863-\Large\frac{108}{63}

    Answer: 127-\Large\frac{12}{7}

  8. 10448-\Large\frac{104}{48}
  9. 120252\Large\frac{120}{252}

    Answer: 1021\Large\frac{10}{21}

  10. 182294\Large\frac{182}{294}
  11. 168192-\Large\frac{168}{192}

    Answer: 78-\Large\frac{7}{8}

  12. 140224-\Large\frac{140}{224}
  13. 11x11y\Large\frac{11x}{11y}

    Answer: xy\Large\frac{x}{y}

  14. 15a15b\Large\frac{15a}{15b}
  15. 3x12y-\Large\frac{3x}{12y}

    Answer: x4y-\Large\frac{x}{4y}

  16. 4x32y-\Large\frac{4x}{32y}
  17. 14x221y\Large\frac{14{x}^{2}}{21y}

    Answer: 2x23y\Large\frac{2{x}^{2}}{3y}

  18. 24a32b2\Large\frac{24a}{32{b}^{2}}

Multiply and Simplify Fractions

In the following exercises, multiply, and write the answer in simplified form.
  1. 2513\Large\frac{2}{5}\cdot\Large\frac{1}{3}

    Answer: 215\Large\frac{2}{15}

  2. 1238\Large\frac{1}{2}\cdot\Large\frac{3}{8}
  3. 34910\Large\frac{3}{4}\cdot\Large\frac{9}{10}

    Answer: 2740\Large\frac{27}{40}

  4. 4527\Large\frac{4}{5}\cdot\Large\frac{2}{7}
  5. 23(38)-\Large\frac{2}{3}\left(-\Large\frac{3}{8}\right)

    Answer: 14\Large\frac{1}{4}

  6. 34(49)-\Large\frac{3}{4}\left(-\Large\frac{4}{9}\right)
  7. 59310-\Large\frac{5}{9}\cdot\Large\frac{3}{10}

    Answer: 16-\Large\frac{1}{6}

  8. 38415-\Large\frac{3}{8}\cdot\Large\frac{4}{15}
  9. 712(821)\Large\frac{7}{12}\left(-\Large\frac{8}{21}\right)

    Answer: 29-\Large\frac{2}{9}

  10. 512(815)\Large\frac{5}{12}\left(-\Large\frac{8}{15}\right)(1415)(920)\left(-\Large\frac{14}{15}\right)\left(\Large\frac{9}{20}\right)

    Answer: 2150-\Large\frac{21}{50}

  11. (910)(2533)\left(-\Large\frac{9}{10}\right)\left(\Large\frac{25}{33}\right)
  12. (6384)(4490)\left(-\Large\frac{63}{84}\right)\left(-\Large\frac{44}{90}\right)

    Answer: 1130\Large\frac{11}{30}

  13. (3360)(4088)\left(-\Large\frac{33}{60}\right)\left(-\Large\frac{40}{88}\right)
  14. 45114\cdot\Large\frac{5}{11}
  15. 2011\Large\frac{20}{11}
  16. 5835\cdot\Large\frac{8}{3}
  17. 3721n\Large\frac{3}{7}\normalsize\cdot 21n

    Answer: 9n

  18. 5630m\Large\frac{5}{6}\normalsize\cdot 30m
  19. 28p(14)-28p\left(-\Large\frac{1}{4}\right)

    Answer: 7p

  20. 51q(13)-51q\left(-\Large\frac{1}{3}\right)
  21. 8(174)-8\Large\left(\frac{17}{4}\right)

    Answer: −34

  22. 145(15)\Large\frac{14}{5}\normalsize\left(-15\right)
  23. 1(38)-1\Large\left(-\frac{3}{8}\right)
  24. 38\Large\frac{3}{8}
  25. (1)(67)\left(-1\right)\Large\left(-\frac{6}{7}\right)
  26. (23)3{\Large\left(\frac{2}{3}\right)}^{3}
  27. 827\Large\frac{8}{27}
  28. (45)2{\Large\left(\frac{4}{5}\right)}^{2}
  29. (65)4{\Large\left(\frac{6}{5}\right)}^{4}
  30. 1296625\Large\frac{1296}{625}
  31. (47)4{\Large\left(\frac{4}{7}\right)}^{4}

Find Reciprocals

In the following exercises, find the reciprocal.
  1. 34\Large\frac{3}{4}

    Answer: 43\Large\frac{4}{3}

  2. 23\Large\frac{2}{3}
  3. 517-\Large\frac{5}{17}

    Answer: 175-\Large\frac{17}{5}

  4. 619-\Large\frac{6}{19}
  5. 118\Large\frac{11}{8}

    Answer: 811\Large\frac{8}{11}

  6. 13-13
  7. 19-19

    Answer: 119-\Large\frac{1}{19}

  8. 1-1

    Answer: 11

Divide and Simplify Fractions

In the following exercises, divide, and write the answer in simplified form.
  1. 12÷14\Large\frac{1}{2}\normalsize\div\Large\frac{1}{4}
  2. 12÷18\Large\frac{1}{2}\normalsize\div\Large\frac{1}{8}

    Answer: 4

  3. 34÷23\Large\frac{3}{4}\normalsize\div\Large\frac{2}{3}
  4. 45÷34\Large\frac{4}{5}\normalsize\div\Large\frac{3}{4}
  5. 1615\Large\frac{16}{15}
  6. 45÷47-\Large\frac{4}{5}\normalsize\div\Large\frac{4}{7}
  7. 34÷35-\Large\frac{3}{4}\normalsize\div\Large\frac{3}{5}
  8. 54-\Large\frac{5}{4}
  9. 79÷(79)-\Large\frac{7}{9}\normalsize\div\Large\left(-\frac{7}{9}\right)
  10. 56÷(56)-\Large\frac{5}{6}\normalsize\div\Large\left(-\frac{5}{6}\right)

    Answer: 1

  11. 34÷x11\Large\frac{3}{4}\normalsize\div\Large\frac{x}{11}
  12. 25÷y9\Large\frac{2}{5}\normalsize\div\Large\frac{y}{9}
  13. 185y\Large\frac{18}{5y}
  14. 58÷a10\Large\frac{5}{8}\normalsize\div\Large\frac{a}{10}
  15. 56÷c15\Large\frac{5}{6}\normalsize\div\Large\frac{c}{15}
  16. 252c\Large\frac{25}{2c}
  17. 518÷(1524)\Large\frac{5}{18}\normalsize\div\Large\left(-\frac{15}{24}\right)
  18. 718÷(1427)\Large\frac{7}{18}\normalsize\div\Large\left(-\frac{14}{27}\right)
  19. 34-\Large\frac{3}{4}
  20. 7p12÷21p8\Large\frac{7p}{12}\normalsize\div\Large\frac{21p}{8}
  21. 5q12÷15q8\Large\frac{5q}{12}\normalsize\div\Large\frac{15q}{8}
  22. 29\Large\frac{2}{9}
  23. 8u15÷12v25\Large\frac{8u}{15}\normalsize\div\Large\frac{12v}{25}
  24. 12r25÷18s35\Large\frac{12r}{25}\normalsize\div\Large\frac{18s}{35}
  25. 14r15s\Large\frac{14r}{15s}
  26. 5÷12-5\normalsize\div\Large\frac{1}{2}
  27. 3÷14-3\normalsize\div\Large\frac{1}{4}

    Answer: −12

  28. 34÷(12)\Large\frac{3}{4}\normalsize\div\left(-12\right)
  29. 25÷(10)\Large\frac{2}{5}\normalsize\div\left(-10\right)
  30. 125-\Large\frac{1}{25}
  31. 18÷(92)-18\normalsize\div\Large\left(-\frac{9}{2}\right)
  32. 15÷(53)-15\normalsize\div\Large\left(-\frac{5}{3}\right)

    Answer: 9

  33. 12÷(34)÷78\Large\frac{1}{2}\normalsize\div\Large\left(-\frac{3}{4}\right)\normalsize\div\Large\frac{7}{8}
  34. 112÷78211\Large\frac{11}{2}\normalsize\div\Large\frac{7}{8}\normalsize\cdot\Large\frac{2}{11}

Everyday Math

Baking

  1. A recipe for chocolate chip cookies calls for 34\Large\frac{3}{4} cup brown sugar. Imelda wants to double the recipe. How much brown sugar will Imelda need? Show your calculation. Write your result as an improper fraction and as a mixed number.
  2. Measuring cups usually come in sets of 18,14,13,12, and 1\Large\frac{1}{8},\Large\frac{1}{4},\Large\frac{1}{3},\Large\frac{1}{2}\normalsize ,\text{ and }1 cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the recipe.

Baking

Nina is making 44 pans of fudge to serve after a music recital. For each pan, she needs 23\Large\frac{2}{3} cup of condensed milk.
  1. How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number.

    Answer: 83=223\Large\frac{8}{3}\normalsize =2\Large\frac{2}{3}

  2. Measuring cups usually come in sets of 18,14,13,12, and 1\Large\frac{1}{8},\Large\frac{1}{4},\Large\frac{1}{3},\Large\frac{1}{2}\normalsize ,\text{ and }1 cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.

Portions

Don purchased a bulk package of candy that weighs 55 pounds. He wants to sell the candy in little bags that hold 14\Large\frac{1}{4} pound. How many little bags of candy can he fill from the bulk package?

Portions

Kristen has 34\Large\frac{3}{4} yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s 66 dolls. How long will each doll’s hair ribbon be?

Answer: 18yard\Large\frac{1}{8}\normalsize\text{yard}

Writing Exercises

Explain how you find the reciprocal of a fraction. [practice-area rows="4"][/practice-area] Explain how you find the reciprocal of a negative fraction. [practice-area rows="4"][/practice-area] Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 66 or 88 slices. Would he prefer 33 out of 66 slices or 44 out of 88 slices? Rafael replied that since he wasn’t very hungry, he would prefer 33 out of 66 slices. Explain what is wrong with Rafael’s reasoning. [practice-area rows="4"][/practice-area] Give an example from everyday life that demonstrates how 1223 is 13\Large\frac{1}{2}\normalsize\cdot\Large\frac{2}{3}\normalsize\text{ is }\Large\frac{1}{3}. [practice-area rows="4"][/practice-area]

Multiplying and Dividing Mixed Numbers and Complex Fractions

Multiply Mixed Numbers

In the following exercises, multiply and write the answer in simplified form.
  1. 4387104\Large\frac{3}{8}\normalsize\cdot\Large\frac{7}{10}
  2. 249672\Large\frac{4}{9}\normalsize\cdot\Large\frac{6}{7}

    Answer: 4421\Large\frac{44}{21}

  3. 1522335\Large\frac{15}{22}\normalsize\cdot 3\Large\frac{3}{5}
  4. 25366310\Large\frac{25}{36}\normalsize\cdot 6\Large\frac{3}{10}

    Answer: 358\Large\frac{35}{8}

  5. 423(118)4\Large\frac{2}{3}\left(\normalsize -1\Large\frac{1}{8}\right)
  6. 225(229)2\Large\frac{2}{5}\left(\normalsize -2\Large\frac{2}{9}\right)

    Answer: 163-\Large\frac{16}{3}

  7. 44951316-4\Large\frac{4}{9}\normalsize\cdot 5\Large\frac{13}{16}
  8. 172021112-1\Large\frac{7}{20}\normalsize\cdot 2\Large\frac{11}{12}

    Answer: 6316-\Large\frac{63}{16}

Divide Mixed Numbers

In the following exercises, divide, and write your answer in simplified form.
  1. 513÷45\Large\frac{1}{3}\normalsize\div 4
  2. 1312÷913\Large\frac{1}{2}\normalsize\div 9

    Answer: 32\Large\frac{3}{2}

  3. 12÷3311-12\Large\normalsize\div 3\Large\frac{3}{11}
  4. 7÷514-7\div 5\Large\frac{1}{4}

    Answer: 43-\Large\frac{4}{3}

  5. 638÷2186\Large\frac{3}{8}\normalsize\div 2\Large\frac{1}{8}
  6. 215÷11102\Large\frac{1}{5}\normalsize\div 1\Large\frac{1}{10}

    Answer: 2

  7. 935÷(135)-9\Large\frac{3}{5}\normalsize\div\Large\left(\normalsize -1\Large\frac{3}{5}\right)
  8. 1834÷(334)-18\Large\frac{3}{4}\normalsize\div\Large\left(\normalsize -3\Large\frac{3}{4}\right)

    Answer: 5

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.
  1. the quotient of 5u5u and 1111
  2. the quotient of 7v7v and 1313

    Answer: 7v13\Large\frac{7v}{13}

  3. the quotient of pp and qq
  4. the quotient of aa and bb

    Answer: ab\Large\frac{a}{b}

  5. the quotient of rr and the sum of ss and 1010
  6. the quotient of AA and the difference of 33 and BB

    Answer: A3B\Large\frac{A}{3-B}

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.
  1. 2389\displaystyle\Large\frac{\Large\frac{2}{3}}{\Large\frac{8}{9}}
  2. 45815\displaystyle\Large\frac{\Large\frac{4}{5}}{\Large\frac{8}{15}}

    Answer: 32\displaystyle\frac{3}{2}

  3. 8211235\displaystyle\Large\frac{-\Large\frac{8}{21}}{\Large\frac{12}{35}}
  4. 9163340\displaystyle\Large\frac{-\Large\frac{9}{16}}{\Large\frac{33}{40}}

    Answer: 1522-\Large\frac{15}{22}

  5. 452\displaystyle\frac{-\Large\frac{4}{5}}{2}
  6. 9103\displaystyle\frac{-\Large\frac{9}{10}}{3}

    Answer: 310-\Large\frac{3}{10}

  7. 258\displaystyle\frac{\Large\frac{2}{5}}{8}
  8. 5310\displaystyle\frac{\Large\frac{5}{3}}{10}

    Answer: 16\displaystyle\frac{1}{6}

  9. m3n2\displaystyle\Large\frac{\Large\frac{m}{3}}{\Large\frac{n}{2}}
  10. r5s3\displaystyle\Large\frac{\Large\frac{r}{5}}{\Large\frac{s}{3}}

    Answer: 3r5s\displaystyle\frac{3r}{5s}

  11. x689\displaystyle\Large\frac{-\Large\frac{x}{6}}{-\Large\frac{8}{9}}
  12. 38y12\displaystyle\Large\frac{-\Large\frac{3}{8}}{-\Large\frac{y}{12}}

    Answer: 92y\displaystyle\frac{9}{2y}

  13. 245110\displaystyle\frac{2\Large\frac{4}{5}}{\Large\frac{1}{10}}
  14. 42316\displaystyle\frac{4\Large\frac{2}{3}}{\Large\frac{1}{6}}

    Answer: 28

  15. 79245\displaystyle\Large\frac{\Large\frac{7}{9}}{\normalsize -2\Large\frac{4}{5}}
  16. 38634\displaystyle\Large\frac{\Large\frac{3}{8}}{\normalsize -6\Large\frac{3}{4}}

    Answer: 118-\Large\frac{1}{18}

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.
  1. Which of the following fractions are equivalent to 511?\Large\frac{5}{-11}? 511,511,511,511\Large\frac{-5}{-11},\Large\frac{-5}{11},\Large\frac{5}{11},-\Large\frac{5}{11}
  2. Which of the following fractions are equivalent to 49?\Large\frac{-4}{9}? 49,49,49,49\Large\frac{-4}{-9},\Large\frac{-4}{9},\Large\frac{4}{9},-\Large\frac{4}{9}

    Answer: 49,49\Large\frac{-4}{9},\text{}-\Large\frac{4}{9}

  3. Which of the following fractions are equivalent to 113?-\Large\frac{11}{3}? 113,113,113,113\Large\frac{-11}{3},\Large\frac{11}{3},\Large\frac{-11}{-3},\text{}\Large\frac{11}{-3}
  4. Which of the following fractions are equivalent to 136?-\Large\frac{13}{6}? 136,136,136,136\Large\frac{13}{6},\Large\frac{13}{-6},\Large\frac{-13}{-6},\Large\frac{-13}{6}

    Answer: 136,136\Large\frac{13}{-6},\Large\frac{-13}{6}

Simplify Fractions

In the following exercises, simplify.
  1. 4+118\Large\frac{4+11}{8}
  2. 9+37\Large\frac{9+3}{7}

    Answer: 127\Large\frac{12}{7}

  3. 22+310\Large\frac{22+3}{10}
  4. 1946\Large\frac{19 - 4}{6}

    Answer: 52\Large\frac{5}{2}

  5. 482415\Large\frac{48}{24 - 15}
  6. 464+4\Large\frac{46}{4+4}

    Answer: 234\Large\frac{23}{4}

  7. 6+68+4\Large\frac{-6+6}{8+4}
  8. 6+3178\Large\frac{-6+3}{17 - 8}

    Answer: 13-\Large\frac{1}{3}

  9. 22141913\Large\frac{22 - 14}{19 - 13}
  10. 15+918+12\Large\frac{15+9}{18+12}

    Answer: 45\Large\frac{4}{5}

  11. 5810\Large\frac{5\cdot 8}{-10}
  12. 3424\Large\frac{3\cdot 4}{-24}

    Answer: 12-\Large\frac{1}{2}

  13. 4366\Large\frac{4\cdot 3}{6\cdot 6}
  14. 6692\Large\frac{6\cdot 6}{9\cdot 2}

    Answer: 2

  15. 42125\Large\frac{{4}^{2}-1}{25}
  16. 72+160\Large\frac{{7}^{2}+1}{60}

    Answer: 56\Large\frac{5}{6}

  17. 83+2914+3\Large\frac{8\cdot 3+2\cdot 9}{14+3}
  18. 964722+3\Large\frac{9\cdot 6 - 4\cdot 7}{22+3}

    Answer: 2625\Large\frac{26}{25}

  19. 15552210\Large\frac{15\cdot 5-{5}^{2}}{2\cdot 10}
  20. 12932318\Large\frac{12\cdot 9-{3}^{2}}{3\cdot 18}

    Answer: 116\Large\frac{11}{6}

  21. 56344523\Large\frac{5\cdot 6 - 3\cdot 4}{4\cdot 5 - 2\cdot 3}
  22. 89765692\Large\frac{8\cdot 9 - 7\cdot 6}{5\cdot 6 - 9\cdot 2}

    Answer: 52\Large\frac{5}{2}

  23. 523235\Large\frac{{5}^{2}-{3}^{2}}{3 - 5}
  24. 624246\Large\frac{{6}^{2}-{4}^{2}}{4 - 6}

    Answer: −10

  25. 2+4(3)322\Large\frac{2+4\left(3\right)}{-3-{2}^{2}}
  26. 7+3(5)232\Large\frac{7+3\left(5\right)}{-2-{3}^{2}}

    Answer: −2

  27. 742(85)9.33.5\Large\frac{7\cdot 4 - 2\left(8 - 5\right)}{9.3 - 3.5}
  28. 973(128)8.76.6\Large\frac{9\cdot 7 - 3\left(12 - 8\right)}{8.7 - 6.6}

    Answer: 5120\Large\frac{51}{20}

  29. 9(82)3(157)6(71)3(179)\Large\frac{9\left(8 - 2\right)-3\left(15 - 7\right)}{6\left(7 - 1\right)-3\left(17 - 9\right)}
  30. 8(92)4(149)7(83)3(169)\Large\frac{8\left(9 - 2\right)-4\left(14 - 9\right)}{7\left(8 - 3\right)-3\left(16 - 9\right)}

    Answer: 187\Large\frac{18}{7}

Everyday Math

Baking

A recipe for chocolate chip cookies calls for 2142\Large\frac{1}{4} cups of flour. Graciela wants to double the recipe.
  1.  How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number.
  2. Measuring cups usually come in sets with cups for 18,14,13,12, and 1\Large\frac{1}{8},\Large\frac{1}{4},\Large\frac{1}{3},\Large\frac{1}{2}\normalsize ,\text{ and }1 cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed to double the recipe.

Baking

A booth at the county fair sells fudge by the pound. Their award winning "Chocolate Overdose" fudge contains 2232\Large\frac{2}{3} cups of chocolate chips per pound.
  1. How many cups of chocolate chips are in a half-pound of the fudge

    Answer: 43=113cups\Large\frac{4}{3}\normalsize =1\Large\frac{1}{3}\normalsize\text{cups}

  2. The owners of the booth make the fudge in 1010 -pound batches. How many chocolate chips do they need to make a 1010 -pound batch? Write your results as improper fractions and as a mixed numbers.

    Answer: 803=2623cups\Large\frac{80}{3}\normalsize =26\Large\frac{2}{3}\normalsize\text{cups}

Writing Exercises

Explain how to find the reciprocal of a mixed number. [practice-area rows="4"][/practice-area] Explain how to multiply mixed numbers. [practice-area rows="4"][/practice-area] Randy thinks that 3125143\Large\frac{1}{2}\normalsize\cdot 5\Large\frac{1}{4} is 151815\Large\frac{1}{8}. Explain what is wrong with Randy’s thinking. [practice-area rows="4"][/practice-area] Explain why 12,12-\Large\frac{1}{2},\Large\frac{-1}{2}, and 12\Large\frac{1}{-2} are equivalent. [practice-area rows="4"][/practice-area]

Adding and Subtracting Fractions With Common Denominators

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.
  1. 25+15\Large\frac{2}{5}+\Large\frac{1}{5}
  2. 310+410\Large\frac{3}{10}+\Large\frac{4}{10}

    Answer: A circle is divided into 10 equal pieces. 7 of the pieces are shaded.

  3. 710\Large\frac{7}{10}
  4. 16+36\Large\frac{1}{6}+\Large\frac{3}{6}
  5. 38+38\Large\frac{3}{8}+\Large\frac{3}{8}

    Answer: A circle is divided into 8 equal pieces. 6 of the pieces are shaded.

  6. 34\Large\frac{3}{4}

Add Fractions with a Common Denominator

In the following exercises, find each sum.
  1. 49+19\Large\frac{4}{9}+\Large\frac{1}{9}
  2. 29+59\Large\frac{2}{9}+\Large\frac{5}{9}

    Answer: 79\Large\frac{7}{9}

  3. 613+713\Large\frac{6}{13}+\Large\frac{7}{13}
  4. 915+715\Large\frac{9}{15}+\Large\frac{7}{15}

    Answer: 1615\Large\frac{16}{15}

  5. x4+34\Large\frac{x}{4}+\Large\frac{3}{4}
  6. y3+23\Large\frac{y}{3}+\Large\frac{2}{3}

    Answer: y+23\Large\frac{y+2}{3}

  7. 7p+9p\Large\frac{7}{p}+\Large\frac{9}{p}
  8. 8q+6q\Large\frac{8}{q}+\Large\frac{6}{q}

    Answer: 14q\Large\frac{14}{q}

  9. 8b9+3b9\Large\frac{8b}{9}+\Large\frac{3b}{9}
  10. 5a7+4a7\Large\frac{5a}{7}+\Large\frac{4a}{7}

    Answer: 9a7\Large\frac{9a}{7}

  11. 12y8+3y8\Large\frac{-12y}{8}+\Large\frac{3y}{8}
  12. 11x5+7x5\Large\frac{-11x}{5}+\Large\frac{7x}{5}

    Answer: 4x5\Large\frac{-4x}{5}

  13. 18+(38)-\Large\frac{1}{8}+\Large\left(-\frac{3}{8}\right)
  14. 18+(58)-\Large\frac{1}{8}+\Large\left(-\frac{5}{8}\right)

    Answer: 34-\Large\frac{3}{4}

  15. 316+(716)-\Large\frac{3}{16}+\Large\left(-\frac{7}{16}\right)
  16. 516+(916)-\Large\frac{5}{16}+\Large\left(-\frac{9}{16}\right)

    Answer: 78-\Large\frac{7}{8}

  17. 817+1517-\Large\frac{8}{17}+\Large\frac{15}{17}
  18. 919+1719-\Large\frac{9}{19}+\Large\frac{17}{19}

    Answer: 819\Large\frac{8}{19}

  19. 613+(1013)+(1213)\Large\frac{6}{13}+\Large\left(-\frac{10}{13}\right)+\Large\left(-\frac{12}{13}\right)
  20. 512+(712)+(1112)\Large\frac{5}{12}+\Large\left(-\frac{7}{12}\right)+\Large\left(-\frac{11}{12}\right)

    Answer: 1312-\Large\frac{13}{12}

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.
  1. 5828\Large\frac{5}{8}-\Large\frac{2}{8}
  2. 5626\Large\frac{5}{6}-\Large\frac{2}{6}

    Answer: A circle is divided into eight sections, three of which are shaded.

  3. 12\Large\frac{1}{2}

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.
  1. 4515\Large\frac{4}{5}-\Large\frac{1}{5}
  2. 4535\Large\frac{4}{5}-\Large\frac{3}{5}

    Answer: 15\Large\frac{1}{5}

  3. 1115715\Large\frac{11}{15}-\Large\frac{7}{15}
  4. 913413\Large\frac{9}{13}-\Large\frac{4}{13}

    Answer: 513\Large\frac{5}{13}

  5. 1112512\Large\frac{11}{12}-\Large\frac{5}{12}
  6. 712512\Large\frac{7}{12}-\Large\frac{5}{12}

    Answer: 16\Large\frac{1}{6}

  7. 4211921\Large\frac{4}{21}-\Large\frac{19}{21}
  8. 89169-\Large\frac{8}{9}-\Large\frac{16}{9}

    Answer: 83-\Large\frac{8}{3}

  9. y17917\Large\frac{y}{17}-\Large\frac{9}{17}
  10. x19819\Large\frac{x}{19}-\Large\frac{8}{19}

    Answer: x819\Large\frac{x - 8}{19}

  11. 5y878\Large\frac{5y}{8}-\Large\frac{7}{8}
  12. 11z13813\Large\frac{11z}{13}-\Large\frac{8}{13}

    Answer: 11z813\Large\frac{11z - 8}{13}

  13. 8d3d-\Large\frac{8}{d}-\Large\frac{3}{d}
  14. 7c7c-\Large\frac{7}{c}-\Large\frac{7}{c}

    Answer: 14c-\Large\frac{14}{c}

  15. 23u15u-\Large\frac{23}{u}-\Large\frac{15}{u}
  16. 29v26v-\Large\frac{29}{v}-\Large\frac{26}{v}

    Answer: 55v-\Large\frac{55}{v}

  17. 6c75c7\Large\frac{6c}{7}-\Large\frac{5c}{7}
  18. 12d119d11\Large\frac{12d}{11}-\Large\frac{9d}{11}

    Answer: 3d11\Large\frac{3d}{11}

  19. 4r135r13\Large\frac{-4r}{13}-\Large\frac{5r}{13}
  20. 7s37s3\Large\frac{-7s}{3}-\Large\frac{7s}{3}

    Answer: 14s3-\Large\frac{14s}{3}

  21. 35(45)-\Large\frac{3}{5}-\Large\left(-\frac{4}{5}\right)
  22. 37(57)-\Large\frac{3}{7}-\Large\left(-\frac{5}{7}\right)

    Answer: 27\Large\frac{2}{7}

  23. 79(59)-\Large\frac{7}{9}-\Large\left(-\frac{5}{9}\right)
  24. 811(511)-\Large\frac{8}{11}-\Large\left(-\frac{5}{11}\right)

    Answer: 311-\Large\frac{3}{11}

Identify and Use Fraction Operations

In the following exercises, perform the indicated operation and write your answers in simplified form.
  1. 518910-\Large\frac{5}{18}\cdot\Large\frac{9}{10}
  2. 314712-\Large\frac{3}{14}\cdot\Large\frac{7}{12}

    Answer: 18-\Large\frac{1}{8}

  3. n545\Large\frac{n}{5}-\Large\frac{4}{5}
  4. 611s11\Large\frac{6}{11}-\Large\frac{s}{11}

    Answer: 6s11\Large\frac{6-s}{11}

  5. 724+224-\Large\frac{7}{24}+\Large\frac{2}{24}
  6. 518+118-\Large\frac{5}{18}+\Large\frac{1}{18}

    Answer: 29-\Large\frac{2}{9}

  7. 815÷125\Large\frac{8}{15}\div\Large\frac{12}{5}
  8. 712÷928\Large\frac{7}{12}\div\Large\frac{9}{28}

    Answer: 4927\Large\frac{49}{27}

Everyday Math

Trail Mix

Jacob is mixing together nuts and raisins to make trail mix. He has 610\Large\frac{6}{10} of a pound of nuts and 310\Large\frac{3}{10} of a pound of raisins. How much trail mix can he make?

Baking

Janet needs 58\Large\frac{5}{8} of a cup of flour for a recipe she is making. She only has 38\Large\frac{3}{8} 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?

Answer: 14cup\Large\frac{1}{4}\normalsize\text{cup}

Writing Exercises

Greg dropped his case of drill bits and three of the bits fell out.  The three bits were 716\Large\frac{7}{16}316\Large\frac{3}{16}, and 14\Large\frac{1}{4}.  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\Large\frac{1}{16} , 18\Large\frac{1}{8} , ___, ___, 516\Large\frac{5}{16} , 38\Large\frac{3}{8} , ___, 12\Large\frac{1}{2} , 916\Large\frac{9}{16} , 58\Large\frac{5}{8} . [practice-area rows="4"][/practice-area]

Answer: Bits that fell out: 716\Large\frac{7}{16} , 316\Large\frac{3}{16} , 14\Large\frac{1}{4} .

After a party, Lupe has 512\Large\frac{5}{12} of a cheese pizza, 412\Large\frac{4}{12} of a pepperoni pizza, and 412\Large\frac{4}{12} of a veggie pizza left. Will all the slices fit into 11 pizza box? Explain your reasoning. [practice-area rows="4"][/practice-area]

Adding and Subtracting Fractions with Different Denominators

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.
  1. 23 and 34\Large\frac{2}{3}\normalsize\text{ and }\Large\frac{3}{4}
  2. 34 and 25\Large\frac{3}{4}\normalsize\text{ and }\Large\frac{2}{5}

    Answer: 20

  3. 712 and 58\Large\frac{7}{12}\normalsize\text{ and }\Large\frac{5}{8}
  4. 916 and 712\Large\frac{9}{16}\normalsize\text{ and }\Large\frac{7}{12}

    Answer: 48

  5. 1330 and 2542\Large\frac{13}{30}\normalsize\text{ and }\Large\frac{25}{42}
  6. 2330 and 548\Large\frac{23}{30}\normalsize\text{ and }\Large\frac{5}{48}

    Answer: 240

  7. 2135 and 3956\Large\frac{21}{35}\normalsize\text{ and }\Large\frac{39}{56}
  8. 1835 and 3349\Large\frac{18}{35}\normalsize\text{ and }\Large\frac{33}{49}

    Answer: 245

  9. 23,16, and 34\Large\frac{2}{3}\normalsize\text{,}\Large\frac{1}{6}\normalsize,\text{ and }\Large\frac{3}{4}
  10. 23,14, and 35\Large\frac{2}{3}\normalsize\text{,}\Large\frac{1}{4}\normalsize,\text{ and }\Large\frac{3}{5}

    Answer: 60

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.
  1. 13 and 14,LCD=12\Large\frac{1}{3}\normalsize\text{ and }\Large\frac{1}{4}\normalsize,\text{LCD}=12
  2. 14 and 15,LCD=20\Large\frac{1}{4}\normalsize\text{ and }\Large\frac{1}{5}\normalsize,\text{LCD}=20

    Answer: 520,420\Large\frac{5}{20},\Large\frac{4}{20}

  3. 512 and 78,LCD=24\Large\frac{5}{12}\normalsize\text{ and }\Large\frac{7}{8}\normalsize,\text{LCD}=24
  4. 712 and 58,LCD=24\Large\frac{7}{12}\normalsize\text{ and }\Large\frac{5}{8}\normalsize,\text{LCD}=24

    Answer: 1424,1524\Large\frac{14}{24},\Large\frac{15}{24}

  5. 1316 and -1112,LCD=48\Large\frac{13}{16}\normalsize\text{ and }\text{-}\Large\frac{11}{12}\normalsize,\text{LCD}=48
  6. 1116 and -512,LCD=48\Large\frac{11}{16}\normalsize\text{ and }\text{-}\Large\frac{5}{12}\normalsize,\text{LCD}=48

    Answer: 3348,2048\Large\frac{33}{48},-\Large\frac{20}{48}

  7. 13,56, and 34,LCD=12\Large\frac{1}{3},\Large\frac{5}{6}\normalsize,\text{ and }\Large\frac{3}{4}\normalsize,\text{LCD}=12
  8. 13,34, and 35,LCD=60\Large\frac{1}{3},\Large\frac{3}{4}\normalsize,\text{ and }\Large\frac{3}{5}\normalsize,\text{LCD}=60

    Answer: 2060,4560,3660\Large\frac{20}{60},\Large\frac{45}{60},\Large\frac{36}{60}

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.
  1. 13+15\Large\frac{1}{3}+\Large\frac{1}{5}
  2. 14+15\Large\frac{1}{4}+\Large\frac{1}{5}

    Answer: 920\Large\frac{9}{20}

  3. 12+17\Large\frac{1}{2}+\Large\frac{1}{7}
  4. 13+18\Large\frac{1}{3}+\Large\frac{1}{8}

    Answer: 1124\Large\frac{11}{24}

  5. 13(19)\Large\frac{1}{3}-\Large\left(-\frac{1}{9}\right)
  6. 14(18)\Large\frac{1}{4}-\Large\left(-\frac{1}{8}\right)

    Answer: 38\Large\frac{3}{8}

  7. 15(110)\Large\frac{1}{5}-\Large\left(-\frac{1}{10}\right)
  8. 12(16)\Large\frac{1}{2}-\Large\left(-\frac{1}{6}\right)

    Answer: 23\Large\frac{2}{3}

  9. 23+34\Large\frac{2}{3}+\Large\frac{3}{4}
  10. 34+25\Large\frac{3}{4}+\Large\frac{2}{5}

    Answer: 2320\Large\frac{23}{20}

  11. 712+58\Large\frac{7}{12}+\Large\frac{5}{8}
  12. 512+38\Large\frac{5}{12}+\Large\frac{3}{8}

    Answer: 1924\Large\frac{19}{24}

  13. 712916\Large\frac{7}{12}-\Large\frac{9}{16}
  14. 716512\Large\frac{7}{16}-\Large\frac{5}{12}

    Answer: 148\Large\frac{1}{48}

  15. 111238\Large\frac{11}{12}-\Large\frac{3}{8}
  16. 58712\Large\frac{5}{8}-\Large\frac{7}{12}

    Answer: 124\Large\frac{1}{24}

  17. 2338\Large\frac{2}{3}-\Large\frac{3}{8}
  18. 5634\Large\frac{5}{6}-\Large\frac{3}{4}

    Answer: 112\Large\frac{1}{12}

  19. 1130+2740-\Large\frac{11}{30}+\Large\frac{27}{40}
  20. 920+1730-\Large\frac{9}{20}+\Large\frac{17}{30}

    Answer: 760\Large\frac{7}{60}

  21. 1330+2542-\Large\frac{13}{30}+\Large\frac{25}{42}
  22. 2330+548-\Large\frac{23}{30}+\Large\frac{5}{48}

    Answer: 5380-\Large\frac{53}{80}

  23. 39562235-\Large\frac{39}{56}-\Large\frac{22}{35}
  24. 33491835-\Large\frac{33}{49}-\Large\frac{18}{35}

    Answer: 291245-\Large\frac{291}{245}

  25. 23(34)-\Large\frac{2}{3}-\Large\left(-\frac{3}{4}\right)
  26. 34(45)-\Large\frac{3}{4}-\Large\left(-\frac{4}{5}\right)

    Answer: 120\Large\frac{1}{20}

  27. 916(45)-\Large\frac{9}{16}-\Large\left(-\frac{4}{5}\right)
  28. 720(58)-\Large\frac{7}{20}-\Large\left(-\frac{5}{8}\right)

    Answer: 1140\Large\frac{11}{40}

  29. 1+781+\Large\frac{7}{8}
  30. 1+561+\Large\frac{5}{6}

    Answer: 116\Large\frac{11}{6}

  31. 1591-\Large\frac{5}{9}
  32. 13101-\Large\frac{3}{10}

    Answer: 710\Large\frac{7}{10}

  33. x3+14\Large\frac{x}{3}+\Large\frac{1}{4}
  34. y2+23\Large\frac{y}{2}+\Large\frac{2}{3}

    Answer: 3y+46\Large\frac{3y+4}{6}

  35. y435\Large\frac{y}{4}-\Large\frac{3}{5}
  36. x514\Large\frac{x}{5}-\Large\frac{1}{4}

    Answer: 4x520\Large\frac{4x - 5}{20}

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.
  1. 34+16\Large\frac{3}{4}+\Large\frac{1}{6}
  2. 34÷16\Large\frac{3}{4}\div\Large\frac{1}{6}
  3. 23+16\Large\frac{2}{3}+\Large\frac{1}{6}

    Answer: 56\Large\frac{5}{6}

  4. 23÷16\Large\frac{2}{3}\div\Large\frac{1}{6}

    Answer: 44

  5. -2518\text{-}\Large\frac{2}{5}-\Large\frac{1}{8}
  6. -2518\text{-}\Large\frac{2}{5}\cdot\Large\frac{1}{8}
  7. -4518\text{-}\Large\frac{4}{5}-\Large\frac{1}{8}

    Answer: 3740-\Large\frac{37}{40}

  8. -4518\text{-}\Large\frac{4}{5}\cdot\Large\frac{1}{8}

    Answer: 110-\Large\frac{1}{10}

  9. 5n÷815\Large\frac{5}{n}\div\Large\frac{8}{15}
  10. 5n815\Large\frac{5}{n}-\Large\frac{8}{15}
  11. 3a÷712\Large\frac{3}{a}\div\Large\frac{7}{12}

    Answer: 9a14\Large\frac{9a}{14}

  12. 3a712\Large\frac{3}{a}-\Large\frac{7}{12}

    Answer: 9a1424\Large\frac{9a - 14}{24}

  13. 910(11d)\Large\frac{9}{10}\cdot\Large\left(-\frac{11}{d}\right)
  14. 910+(11d)\Large\frac{9}{10}+\Large\left(-\frac{11}{d}\right)
  15. 415(5q)\Large\frac{4}{15}\cdot\Large\left(-\frac{5}{q}\right)

    Answer: 43q-\Large\frac{4}{3q}

  16. 415+(5q)\Large\frac{4}{15}+\Large\left(-\frac{5}{q}\right)

    Answer: 1225q45\Large\frac{12 - 25q}{45}

  17. 38÷(310)-\Large\frac{3}{8}\div\Large\left(-\frac{3}{10}\right)
  18. 512÷(59)-\Large\frac{5}{12}\div\Large\left(-\frac{5}{9}\right)
  19. 38+512-\Large\frac{3}{8}+\Large\frac{5}{12}
  20. 18+712-\Large\frac{1}{8}+\Large\frac{7}{12}

    Answer: 1124\Large\frac{11}{24}

  21. 5619\Large\frac{5}{6}-\Large\frac{1}{9}
  22. 5916\Large\frac{5}{9}-\Large\frac{1}{6}

    Answer: 718\Large\frac{7}{18}

  23. 38(1021)\Large\frac{3}{8}\cdot\Large\left(-\frac{10}{21}\right)
  24. 712(835)\Large\frac{7}{12}\cdot\Large\left(-\frac{8}{35}\right)

    Answer: 215-\Large\frac{2}{15}

  25. 715y4-\Large\frac{7}{15}-\Large\frac{y}{4}
  26. 38x11-\Large\frac{3}{8}-\Large\frac{x}{11}

    Answer: 338x88\Large\frac{-33 - 8x}{88}

  27. 1112a9a16\Large\frac{11}{12a}\cdot\Large\frac{9a}{16}
  28. 10y13815y\Large\frac{10y}{13}\cdot\Large\frac{8}{15y}

    Answer: 1639\Large\frac{16}{39}

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.
  1. (15)22+32\displaystyle\frac{{\Large\left(\frac{1}{5}\right)}^{2}}{2+{3}^{2}}
  2. (13)25+22\displaystyle\frac{{\Large\left(\frac{1}{3}\right)}^{2}}{5+{2}^{2}}

    Answer: 181\displaystyle\frac{1}{81}

  3. 23+42(23)2\displaystyle\frac{{2}^{3}+{4}^{2}}{{\Large\left(\frac{2}{3}\right)}^{2}}
  4. 3332(34)2\displaystyle\frac{{3}^{3}-{3}^{2}}{{\Large\left(\frac{3}{4}\right)}^{2}}

    Answer: 32

  5. (35)2(37)2\displaystyle\frac{{\Large\left(\frac{3}{5}\right)}^{2}}{{\Large\left(\frac{3}{7}\right)}^{2}}
  6. (34)2(58)2\displaystyle\frac{{\Large\left(\frac{3}{4}\right)}^{2}}{{\Large\left(\frac{5}{8}\right)}^{2}}

    Answer: 3625\displaystyle\frac{36}{25}

  7. 213+15\displaystyle\frac{2}{\Large\frac{1}{3}+\frac{1}{5}}
  8. 514+13\displaystyle\frac{5}{\Large\frac{1}{4}+\frac{1}{3}}

    Answer: 607\displaystyle\frac{60}{7}

  9. 23+123423\displaystyle\frac{\Large\frac{2}{3}+\frac{1}{2}}{\Large\frac{3}{4}-\frac{2}{3}}
  10. 34+125623\displaystyle\frac{\Large\frac{3}{4}+\frac{1}{2}}{\Large\frac{5}{6}-\frac{2}{3}}

    Answer: 152\displaystyle\frac{15}{2}

  11. 782312+38\displaystyle\frac{\Large\frac{7}{8}-\frac{2}{3}}{\Large\frac{1}{2}+\frac{3}{8}}
  12. 343514+25\displaystyle\frac{\Large\frac{3}{4}-\frac{3}{5}}{\Large\frac{1}{4}+\frac{2}{5}}

    Answer: 313\displaystyle\frac{3}{13}

Identify and Use Fraction Operations

In the following exercises, simplify.
  1. 12+23512\Large\frac{1}{2}+\frac{2}{3}\cdot \frac{5}{12}
  2. 13+2534\Large\frac{1}{3}+\frac{2}{5}\cdot \frac{3}{4}

    Answer: 1930\Large\frac{19}{30}

  3. 135÷1101-\Large\frac{3}{5}\div \frac{1}{10}
  4. 156÷1121-\Large\frac{5}{6}\div \frac{1}{12}

    Answer: −9

  5. 23+16+34\Large\frac{2}{3}+\frac{1}{6}+\frac{3}{4}
  6. 23+14+35\Large\frac{2}{3}+\frac{1}{4}+\frac{3}{5}

    Answer: 9160\Large\frac{91}{60}

  7. 3816+34\Large\frac{3}{8}-\frac{1}{6}+\frac{3}{4}
  8. 25+5834\Large\frac{2}{5}+\frac{5}{8}-\frac{3}{4}

    Answer: 1140\Large\frac{11}{40}

  9. 12(920415)12\Large\left(\frac{9}{20}-\frac{4}{15}\right)
  10. 8(151656)8\Large\left(\frac{15}{16}-\frac{5}{6}\right)

    Answer: 56\Large\frac{5}{6}

  11. 58+161924\Large\frac{\LARGE\frac{5}{8}+\LARGE\frac{1}{6}}{\LARGE\frac{19}{24}}
  12. 16+3101430\LARGE\frac{\LARGE\frac{1}{6}+\LARGE\frac{3}{10}}{\LARGE\frac{14}{30}}

    Answer: 1

  13. (59+16)÷(2312)\Large\left(\frac{5}{9}+\frac{1}{6}\right)\div \left(\frac{2}{3}-\frac{1}{2}\right)
  14. (34+16)÷(5813)\Large\left(\frac{3}{4}+\frac{1}{6}\right)\div \left(\frac{5}{8}-\frac{1}{3}\right)

    Answer: 227\Large\frac{22}{7}

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

Exercise 1

x+12x+\Large\frac{1}{2} when
  1. x=18x=-\Large\frac{1}{8}
  2. x=12x=-\Large\frac{1}{2}

Exercise 2

x+23x+\Large\frac{2}{3} when
  1. x=16x=-\Large\frac{1}{6}
  2. x=53x=-\Large\frac{5}{3}

Answer:

  1. 12\Large\frac{1}{2}
  2. 1-1

Exercise 3

x+(56)x+\Large\left(-\frac{5}{6}\right) when
  1. x=13x=\Large\frac{1}{3}
  2. x=16x=-\Large\frac{1}{6}

Exercise 4

x+(1112)x+\Large\left(-\frac{11}{12}\right) when
  1. x=1112x=\Large\frac{11}{12}
  2. x=34x=\Large\frac{3}{4}

Answer:

  1. 00
  2. 16-\Large\frac{1}{6}

Exercise 5

x25x-\Large\frac{2}{5} when
  1. x=35x=\Large\frac{3}{5}
  2. x=35x=-\Large\frac{3}{5}

Exercise 6

x13x-\Large\frac{1}{3} when
  1. x=23x=\Large\frac{2}{3}
  2. x=23x=-\Large\frac{2}{3}

Answer:

  1. 13\Large\frac{1}{3}
  2. 1-1

Exercise 7

710w\Large\frac{7}{10}\normalsize-w when
  1. w=12w=\Large\frac{1}{2}
  2. w=12w=-\Large\frac{1}{2}

Exercise 8

512w\Large\frac{5}{12}\normalsize-w when
  1. w=14w=\Large\frac{1}{4}
  2. w=14w=-\Large\frac{1}{4}

Answer:

  1. 16\Large\frac{1}{6}
  2. 23\Large\frac{2}{3}

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.
  1. 4p2q4{p}^{2}q when p=12 and q=59p=-\Large\frac{1}{2}\normalsize\text{ and }q=\Large\frac{5}{9}
  2. 5m2n when m=25 and n=135{m}^{2}n\text{ when }m=-\Large\frac{2}{5}\normalsize\text{ and }n=\Large\frac{1}{3}

    Answer: 415\Large\frac{4}{15}

  3. 2x2y3 when x=23 and y=122{x}^{2}{y}^{3}\text{ when }x=-\Large\frac{2}{3}\normalsize\text{ and }y=-\Large\frac{1}{2}
  4. 8u2v3 when u=34 and v=128{u}^{2}{v}^{3}\text{ when }u=-\Large\frac{3}{4}\normalsize\text{ and }v=-\Large\frac{1}{2}

    Answer: 916-\Large\frac{9}{16}

  5. u+vw when u=4,v=8,w=2\Large\frac{u+v}{w}\normalsize\text{ when }u=-4,v=-8,w=2
  6. m+np when m=6,n=2,p=4\Large\frac{m+n}{p}\normalsize\text{ when }m=-6,n=-2,p=4

    Answer: −2

  7. a+bab when a=3,b=8\Large\frac{a+b}{a-b}\normalsize\text{ when }a=-3,b=8
  8. rsr+s when r=10,s=5\Large\frac{r-s}{r+s}\normalsize\text{ when }r=10,s=-5

    Answer: 3

Everyday Math

Decorating

Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs 316\Large\frac{3}{16} yard of print fabric and 38\Large\frac{3}{8} 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 1141\Large\frac{1}{4} cups of sugar for the chocolate chip cookies, and 1181\Large\frac{1}{8} cups for the oatmeal cookies How much sugar does she need altogether?

Answer: She needs238cups\text{She needs}2\Large\frac{3}{8}\normalsize\text{cups}

Writing Exercises

Explain why it is necessary to have a common denominator to add or subtract fractions. [practice-area rows="4"][/practice-area] Explain how to find the LCD of two fractions. [practice-area rows="4"][/practice-area]

Adding and Subtracting Mixed Numbers

Model Addition of Mixed Numbers

In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.
  1. 115+3151\Large\frac{1}{5}\normalsize+3\Large\frac{1}{5}
  2. 213+1132\Large\frac{1}{3}\normalsize+1\Large\frac{1}{3}

    Answer: Four circles are shown. The first three are shaded. The last circle is divided into 3 equal parts. 2 parts are shaded.

  3. 3233\Large\frac{2}{3}
  4. 138+1781\Large\frac{3}{8}\normalsize+1\Large\frac{7}{8}
  5. 156+1561\Large\frac{5}{6}\normalsize+1\Large\frac{5}{6}

    Answer: Four circles are shown. The first three are shaded. The last circle is divided into 3 equal parts. 2 parts are shaded.

  6. 3233\Large\frac{2}{3}

Add Mixed Numbers with a Common Denominator

In the following exercises, add.
  1. 513+6135\Large\frac{1}{3}\normalsize+6\Large\frac{1}{3}
  2. 249+5192\Large\frac{4}{9}\normalsize+5\Large\frac{1}{9}

    Answer: 7597\Large\frac{5}{9}

  3. 458+9384\Large\frac{5}{8}\normalsize+9\Large\frac{3}{8}
  4. 7910+31107\Large\frac{9}{10}\normalsize+3\Large\frac{1}{10}

    Answer: 11

  5. 345+6453\Large\frac{4}{5}\normalsize+6\Large\frac{4}{5}
  6. 923+1239\Large\frac{2}{3}\normalsize+1\Large\frac{2}{3}

    Answer: 111311\Large\frac{1}{3}

  7. 6910+83106\Large\frac{9}{10}\normalsize+8\Large\frac{3}{10}
  8. 849+2898\Large\frac{4}{9}\normalsize+2\Large\frac{8}{9}

    Answer: 111311\Large\frac{1}{3}

Model Subtraction of Mixed Numbers

In the following exercises, use a model to find the difference. Draw a picture to illustrate your model.
  1. 116561\Large\frac{1}{6}-\Large\frac{5}{6}
  2. 118581\Large\frac{1}{8}-\Large\frac{5}{8}

    Answer: A circle is shown. It is divided into 8 equal pieces. 4 pieces are shaded.

  3. 12\Large\frac{1}{2}

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.
  1. 2781382\Large\frac{7}{8}\normalsize-1\Large\frac{3}{8}
  2. 271215122\Large\frac{7}{12}\normalsize-1\Large\frac{5}{12}

    Answer: 1161\Large\frac{1}{6}

  3. 8172049208\Large\frac{17}{20}\normalsize-4\Large\frac{9}{20}
  4. 1913151371519\Large\frac{13}{15}\normalsize-13\Large\frac{7}{15}

    Answer: 6256\Large\frac{2}{5}

  5. 8374478\Large\frac{3}{7}\normalsize-4\Large\frac{4}{7}
  6. 5293495\Large\frac{2}{9}\normalsize-3\Large\frac{4}{9}

    Answer: 1791\Large\frac{7}{9}

  7. 2581782\Large\frac{5}{8}\normalsize-1\Large\frac{7}{8}
  8. 251217122\Large\frac{5}{12}\normalsize-1\Large\frac{7}{12}

    Answer: 56\Large\frac{5}{6}

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.
  1. 314+6133\Large\frac{1}{4}\normalsize+6\Large\frac{1}{3}
  2. 216+5342\Large\frac{1}{6}\normalsize+5\Large\frac{3}{4}

    Answer: 711127\Large\frac{11}{12}

  3. 158+4121\Large\frac{5}{8}\normalsize+4\Large\frac{1}{2}
  4. 723+8127\Large\frac{2}{3}\normalsize+8\Large\frac{1}{2}

    Answer: 161616\Large\frac{1}{6}

  5. 97102139\Large\frac{7}{10}\normalsize-2\Large\frac{1}{3}
  6. 6451146\Large\frac{4}{5}\normalsize-1\Large\frac{1}{4}

    Answer: 511205\Large\frac{11}{20}

  7. 2233122\Large\frac{2}{3}\normalsize-3\Large\frac{1}{2}
  8. 2784132\Large\frac{7}{8}\normalsize-4\Large\frac{1}{3}

    Answer: 11124-1\Large\frac{11}{24}

Identify and Use Fraction Operations

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.
  1. 2581342\Large\frac{5}{8}\normalsize\cdot 1\Large\frac{3}{4}
  2. 1234161\Large\frac{2}{3}\normalsize\cdot 4\Large\frac{1}{6}

    Answer: 617186\Large\frac{17}{18}

  3. 27+47\Large\frac{2}{7}+\Large\frac{4}{7}
  4. 29+59\Large\frac{2}{9}+\Large\frac{5}{9}

    Answer: 79\Large\frac{7}{9}

  5. 1512÷1121\Large\frac{5}{12}\div\Large\frac{1}{12}
  6. 2310÷1102\Large\frac{3}{10}\div\Large\frac{1}{10}

    Answer: 23

  7. 13512971213\Large\frac{5}{12}\normalsize-9\Large\frac{7}{12}
  8. 155867815\Large\frac{5}{8}\normalsize-6\Large\frac{7}{8}

    Answer: 8348\Large\frac{3}{4}

  9. 5949\Large\frac{5}{9}-\Large\frac{4}{9}
  10. 1115715\Large\frac{11}{15}-\Large\frac{7}{15}

    Answer: 415\Large\frac{4}{15}

  11. 4344-\Large\frac{3}{4}
  12. 6256-\Large\frac{2}{5}

    Answer: 5355\Large\frac{3}{5}

  13. 920÷34\Large\frac{9}{20}\div\Large\frac{3}{4}
  14. 724÷143\Large\frac{7}{24}\div\Large\frac{14}{3}

    Answer: 116\Large\frac{1}{16}

  15. 9611+710119\Large\frac{6}{11}\normalsize+7\Large\frac{10}{11}
  16. 8513+49138\Large\frac{5}{13}\normalsize+4\Large\frac{9}{13}

    Answer: 1311313\Large\frac{1}{13}

  17. 325+5343\Large\frac{2}{5}\normalsize+5\Large\frac{3}{4}
  18. 256+4152\Large\frac{5}{6}\normalsize+4\Large\frac{1}{5}

    Answer: 71307\Large\frac{1}{30}

  19. 8151019\Large\frac{8}{15}\cdot\Large\frac{10}{19}
  20. 51289\Large\frac{5}{12}\cdot\Large\frac{8}{9}

    Answer: 1027\Large\frac{10}{27}

  21. 6782136\Large\frac{7}{8}\normalsize-2\Large\frac{1}{3}
  22. 6594256\Large\frac{5}{9}\normalsize-4\Large\frac{2}{5}

    Answer: 27452\Large\frac{7}{45}

  23. 5294455\Large\frac{2}{9}\normalsize-4\Large\frac{4}{5}
  24. 4383234\Large\frac{3}{8}\normalsize-3\Large\frac{2}{3}

    Answer: 1724\Large\frac{17}{24}

Everyday Math

Sewing

Renata is sewing matching shirts for her husband and son. According to the patterns she will use, she needs 2382\Large\frac{3}{8} yards of fabric for her husband’s shirt and 1181\Large\frac{1}{8} yards of fabric for her son’s shirt. How much fabric does she need to make both shirts?

Sewing

Pauline has 3143\Large\frac{1}{4} yards of fabric to make a jacket. The jacket uses 2232\Large\frac{2}{3} yards. How much fabric will she have left after making the jacket?

Answer: 712yards\Large\frac{7}{12}\normalsize\text{yards}

Printing

Nishant is printing invitations on his computer. The paper is 8128\Large\frac{1}{2} inches wide, and he sets the print area to have a 1121\Large\frac{1}{2} -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 88 inches wide. The picture frame is 2582\Large\frac{5}{8} inches wide on each side. How wide will the framed picture be?

Answer: 1314inches13\Large\frac{1}{4}\normalsize\text{inches}

Writing Exercises

Draw a diagram and use it to explain how to add 158+2781\Large\frac{5}{8}\normalsize+2\Large\frac{7}{8}. [practice-area rows="4"][/practice-area] Edgar will have to pay $3.75\$3.75 in tolls to drive to the city.
  1. Explain how he can make change from a $10\$10 bill before he leaves so that he has the exact amount he needs.
  2. How is Edgar’s situation similar to how you subtract 10334?10 - 3\Large\frac{3}{4}?
[practice-area rows="4"][/practice-area] Add 4512+3784\Large\frac{5}{12}\normalsize+3\Large\frac{7}{8} twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why? [practice-area rows="4"][/practice-area] Subtract 37845123\Large\frac{7}{8}\normalsize-4\Large\frac{5}{12} twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why? [practice-area rows="4"][/practice-area]

Solving Equations That Contain Fractions

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

Exercise 1

x25=110:x-\Large\frac{2}{5}=\Large\frac{1}{10}\normalsize\text{:}
  1. x=1x=1
  2. x=12x=\Large\frac{1}{2}
  3. x=12x=-\Large\frac{1}{2}

Exercise 2

y13=512:y-\Large\frac{1}{3}=\Large\frac{5}{12}\normalsize\text{:}
  1. y=1y=1
  2. y=34y=\Large\frac{3}{4}
  3. y=34y=-\Large\frac{3}{4}

Answer:

  1. no
  2. yes
  3. no

Exercise 3

h+34=25:h+\Large\frac{3}{4}=\Large\frac{2}{5}\normalsize\text{:}
  1. h=1h=1
  2. h=720h=\Large\frac{7}{20}
  3. h=720h=-\Large\frac{7}{20}

Exercise 4

k+25=56:k+\Large\frac{2}{5}=\Large\frac{5}{6}\normalsize\text{:}
  1. k=1k=1
  2. k=1330k=\Large\frac{13}{30}
  3. k=1330k=-\Large\frac{13}{30}

Answer:

  1. no
  2. yes
  3. no

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.
  1. y+13=43y+\Large\frac{1}{3}=\Large\frac{4}{3}
  2. m+38=78m+\Large\frac{3}{8}=\Large\frac{7}{8}

    Answer: m=12m=\Large\frac{1}{2}

  3. f+910=25f+\Large\frac{9}{10}=\Large\frac{2}{5}
  4. h+56=16h+\Large\frac{5}{6}=\Large\frac{1}{6}

    Answer: h=23h=-\Large\frac{2}{3}

  5. a58=78a-\Large\frac{5}{8}=-\Large\frac{7}{8}
  6. c14=54c-\Large\frac{1}{4}=-\Large\frac{5}{4}

    Answer: c = −1

  7. x(320)=1120x-\Large\left(-\frac{3}{20}\right)=-\Large\frac{11}{20}
  8. z(512)=712z-\Large\left(-\frac{5}{12}\right)=-\Large\frac{7}{12}

    Answer: z = −1

  9. n16=34n-\Large\frac{1}{6}=\Large\frac{3}{4}
  10. p310=58p-\Large\frac{3}{10}=\Large\frac{5}{8}

    Answer: p=3740p=\Large\frac{37}{40}

  11. s+(12)=89s+\Large\left(-\frac{1}{2}\right)=-\Large\frac{8}{9}
  12. k+(13)=45k+\Large\left(-\frac{1}{3}\right)=-\Large\frac{4}{5}

    Answer: k=715k=-\Large\frac{7}{15}

  13. 5j=175j=17
  14. 7k=187k=18

    Answer: k=187k=\Large\frac{18}{7}

  15. 4w=26-4w=26
  16. 9v=33-9v=33

    Answer: v=113v=-\Large\frac{11}{3}

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.
  1. f4=20\Large\frac{f}{4}\normalsize=-20
  2. b3=9\Large\frac{b}{3}\normalsize=-9

    Answer: b = −27

  3. y7=21\Large\frac{y}{7}\normalsize=-21
  4. x8=32\Large\frac{x}{8}\normalsize=-32

    Answer: x = −256

  5. p5=40\Large\frac{p}{-5}\normalsize=-40
  6. q4=40\Large\frac{q}{-4}\normalsize=-40

    Answer: q = 160

  7. r12=6\Large\frac{r}{-12}\normalsize=-6
  8. s15=3\Large\frac{s}{-15}\normalsize=-3

    Answer: s = 45

  9. x=23-x=23
  10. y=42-y=42

    Answer: y = −42

  11. h=512-h=-\Large\frac{5}{12}
  12. k=1720-k=-\Large\frac{17}{20}

    Answer: k=1720k=\Large\frac{17}{20}

  13. 45n=20\Large\frac{4}{5}\normalsize n=20
  14. 310p=30\Large\frac{3}{10}\normalsize p=30

    Answer: p = 100

  15. 38q=48\Large\frac{3}{8}\normalsize q=-48
  16. 52m=40\Large\frac{5}{2}\normalsize m=-40

    Answer: m = −16

  17. 29a=16-\Large\frac{2}{9}\normalsize a=16
  18. 37b=9-\Large\frac{3}{7}\normalsize b=9

    Answer: b = −21

  19. 611u=24-\Large\frac{6}{11}\normalsize u=-24
  20. 512v=15-\Large\frac{5}{12}\normalsize v=-15

    Answer: v = 36

Mixed Practice

In the following exercises, solve.
  1. 3x=03x=0
  2. 8y=08y=0

    Answer: y = 0

  3. 4f=454f=\Large\frac{4}{5}
  4. 7g=797g=\Large\frac{7}{9}

    Answer: g=19g=\Large\frac{1}{9}

  5. p+23=112p+\Large\frac{2}{3}=\Large\frac{1}{12}
  6. q+56=112q+\Large\frac{5}{6}=\Large\frac{1}{12}

    Answer: q=34q=-\Large\frac{3}{4}

  7. 78m=110\Large\frac{7}{8}\normalsize m=\Large\frac{1}{10}
  8. 14n=710\Large\frac{1}{4}\normalsize n=\Large\frac{7}{10}

    Answer: n=145n=\Large\frac{14}{5}

  9. 25=x+34-\Large\frac{2}{5}\normalsize=x+\Large\frac{3}{4}
  10. 23=y+38-\Large\frac{2}{3}\normalsize=y+\Large\frac{3}{8}

    Answer: y=2524y=-\Large\frac{25}{24}

  11. 1120=-f\Large\frac{11}{20}\normalsize=\text{-}\mathit{\text{f}}
  12. 815=-d\Large\frac{8}{15}\normalsize=\text{-}\mathit{\text{d}}

    Answer: d=815d=-\Large\frac{8}{15}

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.
  1. nn divided by eight is 16-16.
  2. nn divided by six is 24-24.

    Answer: n6=24;n=144\Large\frac{n}{6}\normalsize=-24;n=-144

  3. mm divided by 9-9 is 7-7.
  4. mm divided by 7-7 is 8-8.

    Answer: m7=8;m=56\Large\frac{m}{-7}\normalsize=-8;m=56

  5. The quotient of ff and 3-3 is 18-18.
  6. The quotient of ff and 4-4 is 20-20.

    Answer: f4=20;f=80\Large\frac{f}{-4}\normalsize=-20;f=80

  7. The quotient of gg and twelve is 88.
  8. The quotient of gg and nine is 1414.

    Answer: g9=14;g=126\Large\frac{g}{9}\normalsize=14;g=126

  9. Three-fourths of qq is 1212.
  10. Two-fifths of qq is 2020.

    Answer: 25q=20;q=50\Large\frac{2}{5}\normalsize q=20;q=50

  11. Seven-tenths of pp is 63-63.
  12. Four-ninths of pp is 28-28.

    Answer: 49p=28;p=63\Large\frac{4}{9}\normalsize p=-28;p=-63

  13. mm divided by 44 equals negative 66.
  14. The quotient of hh and 22 is 4343.

    Answer: h2=43\Large\frac{h}{2}\normalsize=43

  15. Three-fourths of zz is the same as 1515.
  16. The quotient of aa and 23\Large\frac{2}{3} is 34\Large\frac{3}{4}.

    Answer: a23=34\Large\frac{a}{\LARGE\frac{2}{3}}=\Large\frac{3}{4}

  17. The sum of five-sixths and xx is 12\Large\frac{1}{2}.
  18. The sum of three-fourths and xx is 18\Large\frac{1}{8}.

    Answer: 34+x=18;x=58\Large\frac{3}{4}\normalsize+x=\Large\frac{1}{8}\normalsize ;x=-\Large\frac{5}{8}

  19. The difference of yy and one-fourth is 18-\Large\frac{1}{8}.
  20. The difference of yy and one-third is 16-\Large\frac{1}{6}.

    Answer: y13=16;y=16y-\Large\frac{1}{3}=-\Large\frac{1}{6}\normalsize ;y=\Large\frac{1}{6}

Everyday Math

Shopping

Teresa bought a pair of shoes on sale for $48\$48. The sale price was 23\Large\frac{2}{3} of the regular price. Find the regular price of the shoes by solving the equation 23p=48\Large\frac{2}{3}\normalsize p=48

Playhouse

The table in a child’s playhouse is 35\Large\frac{3}{5} of an adult-size table. The playhouse table is 1818 inches high. Find the height of an adult-size table by solving the equation 35h=18\Large\frac{3}{5}\normalsize h=18.

Answer: 30 inches

Writing Exercises

There are three methods to solve the equation y=15-y=15. Which method do you prefer? Why? [practice-area rows="4"][/practice-area] Richard thinks the solution to the equation 34x=24\Large\frac{3}{4}\normalsize x=24 is 1616. Explain why Richard is wrong. [practice-area rows="4"][/practice-area]

Chapter Review Exercises

Visualize Fractions

In the following exercises, name the fraction of each figure that is shaded.

Using Models to Represent Fractions

Exercise 1
A circle is shown. It is divided into 8 equal pieces. 5 pieces are shaded.
Exercise 2
A square is shown. It is divided into 9 equal pieces. 5 pieces are shaded.

Answer: 59\Large\frac{5}{9}

Using Models to Represent Mixed Numbers

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.
Exercise 1
Two squares are shown. Both are divided into four equal pieces. The square on the left has all 4 pieces shaded. The square on the right has one piece shaded.
Exercise 2
Two circles are shown. Both are divided into two equal pieces. The circle on the left has both pieces shaded. The circle on the right has one piece shaded.

Answer: 32\Large\frac{3}{2}

Write an Improper Fraction as a Mixed Number

In the following exercises, convert the improper fraction to a mixed number.
  1. 5815\Large\frac{58}{15}
  2. 6311\Large\frac{63}{11}

    Answer: 58115\Large\frac{8}{11}

Write a Mixed Number as an Improper Fraction

In the following exercises, convert the mixed number to an improper fraction.
  1. 121412\Large\frac{1}{4}
  2. 9459\Large\frac{4}{5}

    Answer: 495\Large\frac{49}{5}

  3. Find three fractions equivalent to 25\Large\frac{2}{5}. Show your work, using figures or algebra.
  4. Find three fractions equivalent to 43-\Large\frac{4}{3}. Show your work, using figures or algebra.

Locating Fractions on the Number Line

In the following exercises, locate the numbers on a number line.
  1. 58,43,334,4\Large\frac{5}{8},\Large\frac{4}{3},\normalsize 3\Large\frac{3}{4},\normalsize4
  2. 14,14,113,113,72,72\Large\frac{1}{4},-\Large\frac{1}{4},\normalsize 1\Large\frac{1}{3},\normalsize -1\Large\frac{1}{3},\Large\frac{7}{2},-\Large\frac{7}{2}

    Answer: A number line is shown. Integers from negative 4 to 4 are labeled. Between negative 4 and negative 3, negative 7 halves is labeled and marked with a red dot. Between negative 2 and negative 1, negative 1 and 1 third is labeled and marked with a red dot. Between negative 1 and 0, negative 1 fourth is labeled and marked with a red dot. Between 0 and 1, 1 fourth is labeled and marked with a red dot. Between 1 and 2, 1 and 1 third is labeled and marked with a red dot. Between 3 and 4, 7 halves is labeled and marked with a red dot.

Ordering Fractions on the Number Line

In the following exercises, order each pair of numbers, using << or >>.
  1. -1\text{ ___}-\Large\frac{2}{5}
  2. -2\Large\frac{1}{2}\normalsize\text{ ___}- 3

Multiply and Divide Fractions

Simplify Fractions

In the following exercises, simplify.
  1. 6384-\Large\frac{63}{84}
  2. 90120-\Large\frac{90}{120}

    Answer: 34-\Large\frac{3}{4}

  3. 14a14b-\Large\frac{14a}{14b}
  4. 8x8y-\Large\frac{8x}{8y}

    Answer: xy-\Large\frac{x}{y}

Multiply Fractions

In the following exercises, multiply.
  1. 25813\Large\frac{2}{5}\cdot\Large\frac{8}{13}
  2. 13127-\Large\frac{1}{3}\cdot\Large\frac{12}{7}

    Answer: 47-\Large\frac{4}{7}

  3. 29(4532)\Large\frac{2}{9}\cdot\Large\left(-\Large\frac{45}{32}\right)
  4. 6m4116m\cdot\Large\frac{4}{11}

    Answer: 2411m\Large\frac{24}{11}\normalsize m

  5. 14(32)-\Large\frac{1}{4}\normalsize\left(-32\right)
  6. 3151783\Large\frac{1}{5}\normalsize\cdot 1\Large\frac{7}{8}

    Answer: 6

Find Reciprocals

In the following exercises, find the reciprocal.
  1. 29\Large\frac{2}{9}
  2. 154\Large\frac{15}{4}

    Answer: 415\Large\frac{4}{15}

  3. 33
  4. 14-\Large\frac{1}{4}

    Answer: −4

Exercise 5
Fill in the chart.
Opposite Absolute Value Reciprocal
513-\Large\frac{5}{13}
310\Large\frac{3}{10}
94\Large\frac{9}{4}
12-12

Divide Fractions

In the following exercises, divide.
  1. 23÷16\Large\frac{2}{3}\div\Large\frac{1}{6}

    Answer: 4

  2. (3x5)÷(2y3)\Large\left(-\frac{3x}{5}\right)\div\Large\left(-\frac{2y}{3}\right)
  3. 45÷3\Large\frac{4}{5}\normalsize\div 3

    Answer: 415\Large\frac{4}{15}

  4. 8÷2238\div 2\Large\frac{2}{3}
  5. 823÷11128\Large\frac{2}{3}\normalsize\div 1\Large\frac{1}{12}

    Answer: 8

Multiply and Divide Mixed Numbers and Complex Fractions

In the following exercises, perform the indicated operation.
  1. 3151783\Large\frac{1}{5}\normalsize\cdot 1\Large\frac{7}{8}
  2. 57124411-5\Large\frac{7}{12}\normalsize\cdot 4\Large\frac{4}{11}

    Answer: 26811-\Large\frac{268}{11}

  3. 8÷2238\div 2\Large\frac{2}{3}
  4. 823÷11128\Large\frac{2}{3}\normalsize\div 1\Large\frac{1}{12}

    Answer: 8

Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.
  1. the quotient of 88 and yy
  2. the quotient of VV and the difference of hh and 66

    Answer: Vh6\Large\frac{V}{h - 6}

Simplify Complex Fractions

In the following exercises, simplify the complex fraction
  1. 5845\Large\frac{\LARGE\frac{5}{8}}{\LARGE\frac{4}{5}}
  2. 894\Large\frac{\LARGE\frac{8}{9}}{-4}

    Answer: 29-\Large\frac{2}{9}

  3. n438\Large\frac{\LARGE\frac{n}{4}}{\LARGE\frac{3}{8}}
  4. 156112\Large\frac{\normalsize-1\LARGE\frac{5}{6}}{-\LARGE\frac{1}{12}}

    Answer: 22

Simplify Fractions

In the following exercises, simplify.
  1. 5+165\Large\frac{5+16}{5}
  2. 8452312\Large\frac{8\cdot 4-{5}^{2}}{3\cdot 12}

    Answer: 736\Large\frac{7}{36}

  3. 87+5(810)9364\Large\frac{8\cdot 7+5\left(8 - 10\right)}{9\cdot 3 - 6\cdot 4}

Add and Subtract Fractions with Common Denominators

Add Fractions with a Common Denominator

In the following exercises, add.
  1. 38+28\Large\frac{3}{8}+\Large\frac{2}{8}

    Answer: 58\Large\frac{5}{8}

  2. 45+15\Large\frac{4}{5}+\Large\frac{1}{5}
  3. 25+15\Large\frac{2}{5}+\Large\frac{1}{5}

    Answer: 35\Large\frac{3}{5}

  4. 1532+932\Large\frac{15}{32}+\Large\frac{9}{32}
  5. x10+710\Large\frac{x}{10}+\Large\frac{7}{10}

    Answer: x+710\Large\frac{x+7}{10}

Subtract Fractions with a Common Denominator

In the following exercises, subtract.
  1. 811611\Large\frac{8}{11}-\Large\frac{6}{11}
  2. 1112512\Large\frac{11}{12}-\Large\frac{5}{12}

    Answer: 12\Large\frac{1}{2}

  3. 45y5\Large\frac{4}{5}-\frac{y}{5}
  4. 3130730-\Large\frac{31}{30}-\Large\frac{7}{30}

    Answer: 1915-\Large\frac{19}{15}

  5. 32(32)\Large\frac{3}{2}-\Large\left(\frac{3}{2}\right)
  6. 1115515(215)\Large\frac{11}{15}-\Large\frac{5}{15}-\Large\left(-\frac{2}{15}\right)

    Answer: 815\Large\frac{8}{15}

Add and Subtract Fractions with Different Denominators

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator.
  1. 13 and 112\Large\frac{1}{3}\normalsize\text{ and }\Large\frac{1}{12}
  2. 13 and 45\Large\frac{1}{3}\normalsize\text{ and }\Large\frac{4}{5}

    Answer: 15

  3. 815 and 1120\Large\frac{8}{15}\normalsize\text{ and }\Large\frac{11}{20}
  4. 34,16, and 510\Large\frac{3}{4},\Large\frac{1}{6},\normalsize\text{ and }\Large\frac{5}{10}

    Answer: 60

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, change to equivalent fractions using the given LCD.
  1. 13 and 15,LCD=15\Large\frac{1}{3}\normalsize\text{ and }\Large\frac{1}{5},\normalsize\text{LCD}=15
  2. 38 and 56,LCD=24\Large\frac{3}{8}\normalsize\text{ and }\Large\frac{5}{6},\normalsize\text{LCD}=24

    Answer: 924 and 2024\Large\frac{9}{24}\normalsize\text{ and }\Large\frac{20}{24}

  3. 916 and 512,LCD=48-\Large\frac{9}{16}\normalsize\text{ and }\Large\frac{5}{12},\normalsize\text{LCD}=48
  4. 13,34 and 45,LCD=60\Large\frac{1}{3}\normalsize\text{,}\Large\frac{3}{4}\normalsize\text{ and }\Large\frac{4}{5},\normalsize\text{LCD}=60

    Answer: 2060,1560 and 4860\Large\frac{20}{60},\Large\frac{15}{60}\normalsize\text{ and }\Large\frac{48}{60}

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations and simplify.
  1. 15+23\Large\frac{1}{5}+\Large\frac{2}{3}
  2. 111223\Large\frac{11}{12}-\Large\frac{2}{3}

    Answer: 14\Large\frac{1}{4}

  3. 91034-\Large\frac{9}{10}-\Large\frac{3}{4}
  4. 11361120-\Large\frac{11}{36}-\Large\frac{11}{20}

    Answer: 7790-\Large\frac{77}{90}

  5. 2225+940-\Large\frac{22}{25}+\Large\frac{9}{40}
  6. y1013\Large\frac{y}{10}-\Large\frac{1}{3}

    Answer: 3y1030\Large\frac{3y - 10}{30}

  7. 25+(59)\Large\frac{2}{5}+\Large\left(-\frac{5}{9}\right)
  8. 411÷27d\Large\frac{4}{11}\div\Large\frac{2}{7d}

    Answer: 14d11\Large\frac{14d}{11}

  9. 25+(3n8)(29n)\Large\frac{2}{5}+\Large\left(-\frac{3n}{8}\right)\Large\left(-\frac{2}{9n}\right)
  10. (23)2(58)2\Large\frac{{\left(\Large\frac{2}{3}\right)}^{2}}{{\Large\left(\frac{5}{8}\right)}^{2}}

    Answer: 256225\Large\frac{256}{225}

  11. (1112+38)÷(56110)\Large\left(\frac{11}{12}+\Large\frac{3}{8}\right)\div\Large\left(\Large\frac{5}{6}-\Large\frac{1}{10}\right)

Mixed Practice

In the following exercises, evaluate.
Exercise 1
y45y-\Large\frac{4}{5} when
  1. y=45y=-\Large\frac{4}{5}
  2. y=14y=\Large\frac{1}{4}

Answer:

  1. 85-\Large\frac{8}{5}
  2. 1120-\Large\frac{11}{20}

Exercise 2
6mn2[/latex]when[latex]m=34 and n=136m{n}^{2}[/latex] when [latex]m=\Large\frac{3}{4}\normalsize\text{ and }n=-\Large\frac{1}{3}

Add and Subtract Mixed Numbers

In the following exercises, perform the indicated operation.
  1. 413+9134\Large\frac{1}{3}\normalsize+9\Large\frac{1}{3}

    Answer: 132313\Large\frac{2}{3}

  2. 625+7356\Large\frac{2}{5}\normalsize+7\Large\frac{3}{5}
  3. 5811+24115\Large\frac{8}{11}\normalsize+2\Large\frac{4}{11}

    Answer: 81118\Large\frac{1}{11}

  4. 358+3783\Large\frac{5}{8}\normalsize+3\Large\frac{7}{8}
  5. 91320411209\Large\frac{13}{20}\normalsize-4\Large\frac{11}{20}

    Answer: 51105\Large\frac{1}{10}

  6. 231019102\Large\frac{3}{10}\normalsize-1\Large\frac{9}{10}
  7. 2111217122\Large\frac{11}{12}\normalsize-1\Large\frac{7}{12}

    Answer: 103\Large\frac{10}{3}

  8. 861129118\Large\frac{6}{11}\normalsize-2\Large\frac{9}{11}

Solve Equations with Fractions

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether the each number is a solution of the given equation.
Exercise 1
x12=16:x-\Large\frac{1}{2}=\Large\frac{1}{6}\normalsize\text{:}
  1. x=1x=1
  2. x=23x=\Large\frac{2}{3}
  3. x=13x=-\Large\frac{1}{3}

Answer:

  1. no
  2. yes
  3. no

Exercise 2
y+35=59:y+\Large\frac{3}{5}=\Large\frac{5}{9}\normalsize\text{:}
  1. y=12y=\Large\frac{1}{2}
  2. y=5245y=\Large\frac{52}{45}
  3. y=245y=-\Large\frac{2}{45}

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve the equation.
  1. n+911=411n+\Large\frac{9}{11}=\Large\frac{4}{11}

    Answer: n=511n=-\Large\frac{5}{11}

  2. x16=76x-\Large\frac{1}{6}=\Large\frac{7}{6}
  3. h(78)=25h-\Large\left(-\frac{7}{8}\right)=-\Large\frac{2}{5}

    Answer: h=5140h=-\Large\frac{51}{40}

  4. x5=10\Large\frac{x}{5}\normalsize=-10
  5. z=23-z=23

    Answer: z = −23

Translate Sentences to Equations and Solve

In the following exercises, translate and solve.
  1. The sum of two-thirds and nn is 35-\Large\frac{3}{5}.
  2. The difference of qq and one-tenth is 12\Large\frac{1}{2}.

    Answer: q110=12;q=35q-\Large\frac{1}{10}=\Large\frac{1}{2}\normalsize;q=\Large\frac{3}{5}

  3. The quotient of pp and 4-4 is 8-8.
  4. Three-eighths of yy is 2424.

    Answer: 38y=24;y=64\Large\frac{3}{8}\normalsize y=24;y=64

Chapter Practice Test

Convert the improper fraction to a mixed number.
  1. 195\Large\frac{19}{5}
Convert the mixed number to an improper fraction.
  1. 3273\Large\frac{2}{7}

    Answer: 237\Large\frac{23}{7}

Locate the numbers on a number line.
  1. 12,123,234, and 94\Large\frac{1}{2}\normalsize ,1\Large\frac{2}{3}\normalsize ,-2\Large\frac{3}{4}\normalsize ,\text{ and }\Large\frac{9}{4}
In the following exercises, simplify.
  1. 520\Large\frac{5}{20}

    Answer: 14\Large\frac{1}{4}

  2. 18r27s\Large\frac{18r}{27s}
  3. 1334\Large\frac{1}{3}\cdot\Large\frac{3}{4}

    Answer: 14\Large\frac{1}{4}

  4. 3515\Large\frac{3}{5}\normalsize\cdot 15
  5. 36u(49)-36u\Large\left(-\frac{4}{9}\right)

    Answer: 16u

  6. 57124411-5\Large\frac{7}{12}\normalsize\cdot 4\Large\frac{4}{11}
  7. 56÷512-\Large\frac{5}{6}\div\Large\frac{5}{12}

    Answer: −2

  8. 711÷(711)\Large\frac{7}{11}\div\Large\left(-\frac{7}{11}\right)
  9. 9a10÷15a8\Large\frac{9a}{10}\div\Large\frac{15a}{8}

    Answer: 1225\Large\frac{12}{25}

  10. 625÷4-6\Large\frac{2}{5}\normalsize\div 4
  11. (1556)÷(316)\left(-15\Large\frac{5}{6}\right)\normalsize\div\left(-3\Large\frac{1}{6}\right)

    Answer: 5

  12. 6611\Large\frac{-6}{\LARGE\frac{6}{11}}
  13. p2q5\Large\frac{\LARGE\frac{p}{2}}{\LARGE\frac{q}{5}}

    Answer: 5p2q\Large\frac{5p}{2q}

  14. 415223\Large\frac{-\LARGE\frac{4}{15}}{-2\LARGE\frac{2}{3}}
  15. 924294\Large\frac{{9}^{2}-{4}^{2}}{9 - 4}

    Answer: 13

  16. 2d+9d\Large\frac{2}{d}+\Large\frac{9}{d}
  17. 313+(413)-\Large\frac{3}{13}+\Large\left(-\frac{4}{13}\right)

    Answer: 713-\Large\frac{7}{13}

  18. 2225+940-\Large\frac{22}{25}+\Large\frac{9}{40}
  19. 25+(75)\Large\frac{2}{5}+\Large\left(-\frac{7}{5}\right)

    Answer: −1

  20. 310+(58)-\Large\frac{3}{10}+\Large\left(-\frac{5}{8}\right)
  21. 34÷x3-\Large\frac{3}{4}\div\Large\frac{x}{3}

    Answer: 94x-\Large\frac{9}{4x}

  22. 2322(34)2\Large\frac{{2}^{3}-{2}^{2}}{{\LARGE\left(\frac{3}{4}\right)}^{2}}
  23. 514+18956\Large\frac{\LARGE\frac{5}{14}+\LARGE\frac{1}{8}}{\LARGE\frac{9}{56}}

    Answer: 3

Evaluate x+13x+\Large\frac{1}{3} when
  1. x=23x=\Large\frac{2}{3}
  2. x=56x=-\Large\frac{5}{6}
In the following exercises, solve the equation.
  1. y+35=75y+\Large\frac{3}{5}=\Large\frac{7}{5}

    Answer: y=45y=\Large\frac{4}{5}

  2. a310=910a-\Large\frac{3}{10}=-\Large\frac{9}{10}
  3. f+(23)=512f+\Large\left(-\frac{2}{3}\right)=\Large\frac{5}{12}

    Answer: f=1312f=\Large\frac{13}{12}

  4. m2=16\Large\frac{m}{-2}\normalsize=-16
  5. 23c=18-\Large\frac{2}{3}\normalsize c=18

    Answer: c = −27

Translate and solve: The quotient of pp and 4-4 is 8-8. Solve for [latex]p[/late

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