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3B · Chapter 10 · Solutions

Complete worked answer key

Answers and explanations for every original lesson practice set and chapter assessment.

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Equivalent Fractions

Prerequisite

  1. 1/6 — The denominator counts all 6 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/6.
  2. 1/6 — The denominator counts all 6 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/6.
  3. 3/4 — The denominator counts all 4 equal parts; the numerator counts the 3 shaded parts. In simplest form this is 3/4.

Understanding checks

  1. They are equal — Subdivide each of the 6 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 4/6 = 16/24.
  2. They are equal — Subdivide each of the 6 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/6 = 12/18.
  3. They are equal — Subdivide each of the 4 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/4 = 8/16.

Guided

  1. They are equal — Subdivide each of the 8 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 7/8 = 14/16.
  2. They are equal — Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 4/6 = 8/12.
  3. They are equal — Subdivide each of the 12 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 7/12 = 14/24.
  4. They are equal — Subdivide each of the 5 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/5 = 12/15.
  5. They are equal — Subdivide each of the 5 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 1/5 = 3/15.

Independent

  1. They are equal — Subdivide each of the 12 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 8/12 = 24/36.
  2. They are equal — Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 1/8 = 3/24.
  3. They are equal — Subdivide each of the 6 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 3/6 = 9/18.
  4. They are equal — Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/8 = 12/24.
  5. They are equal — Subdivide each of the 6 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 5/6 = 20/24.
  6. They are equal — Subdivide each of the 3 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 2/3 = 6/9.
  7. They are equal — Subdivide each of the 3 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 2/3 = 4/6.
  8. They are equal — Subdivide each of the 10 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 2/10 = 4/20.

Challenge

  1. 4/10 — No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/5 = 4/10.
  2. 2/4 — Grouping numerator and denominator into threes gives 2/4; both represent the same share of the same whole.

Exit

  1. They are equal — Subdivide each of the 5 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 1/5 = 4/20.
  2. They are equal — Subdivide each of the 4 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 2/4 = 4/8.
  3. They are equal — Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 5/8 = 15/24.

Finding Equivalent Fractions

Prerequisite

  1. They are equal — Subdivide each of the 12 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 8/12 = 32/48.
  2. They are equal — Subdivide each of the 10 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 5/10 = 15/30.
  3. They are equal — Subdivide each of the 3 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/3 = 8/12.

Understanding checks

  1. 8 — Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 4/6 = 8/12.
  2. 27 — Subdivide each of the 12 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 9/12 = 27/36.
  3. 15 — Subdivide each of the 6 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 5/6 = 15/18.

Guided

  1. 8 — Subdivide each of the 5 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 4/5 = 8/10.
  2. 8 — Subdivide each of the 4 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/4 = 8/16.
  3. 2 — Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 1/6 = 2/12.
  4. 9 — Subdivide each of the 10 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 3/10 = 9/30.
  5. 4 — Subdivide each of the 8 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 1/8 = 4/32.

Independent

  1. 8 — Subdivide each of the 3 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/3 = 8/12.
  2. 6 — Subdivide each of the 4 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 3/4 = 6/8.
  3. 10 — Subdivide each of the 8 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 5/8 = 10/16.
  4. 8 — Subdivide each of the 5 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/5 = 8/20.
  5. 6 — Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 2/8 = 6/24.
  6. 12 — Subdivide each of the 4 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 3/4 = 12/16.
  7. 28 — Subdivide each of the 10 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 7/10 = 28/40.
  8. 28 — Subdivide each of the 8 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 7/8 = 28/32.

Challenge

  1. 4/12 — No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/6 = 4/12.
  2. 2/5 — Grouping numerator and denominator into threes gives 2/5; both represent the same share of the same whole.

Exit

  1. 14 — Subdivide each of the 10 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 7/10 = 14/20.
  2. 12 — Subdivide each of the 6 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/6 = 12/18.
  3. 6 — Subdivide each of the 10 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 2/10 = 6/30.

Simplifying Fractions

Prerequisite

  1. 9 — Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 3/8 = 9/24.
  2. 3 — Subdivide each of the 5 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 1/5 = 3/15.
  3. 16 — Subdivide each of the 10 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 4/10 = 16/40.

Understanding checks

  1. 1/5 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 1/5.
  2. 2/3 — Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 2/3.
  3. 3/4 — Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 3/4.

Guided

  1. 3/5 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 3/5.
  2. 1/3 — Divide numerator and denominator by their greatest common factor 8. The equivalent simplest fraction is 1/3.
  3. 1/5 — Divide numerator and denominator by their greatest common factor 8. The equivalent simplest fraction is 1/5.
  4. 1/4 — Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 1/4.
  5. 1/5 — Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 1/5.

Independent

  1. 7/8 — Divide numerator and denominator by their greatest common factor 4. The equivalent simplest fraction is 7/8.
  2. 3/8 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 3/8.
  3. 1/6 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 1/6.
  4. 1/2 — Divide numerator and denominator by their greatest common factor 20. The equivalent simplest fraction is 1/2.
  5. 1/3 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 1/3.
  6. 1/2 — Divide numerator and denominator by their greatest common factor 4. The equivalent simplest fraction is 1/2.
  7. 7/12 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 7/12.
  8. 2/3 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 2/3.

Challenge

  1. 4/14 — No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/7 = 4/14.
  2. 2/6 — Grouping numerator and denominator into threes gives 2/6; both represent the same share of the same whole.

Exit

  1. 1/2 — Divide numerator and denominator by their greatest common factor 15. The equivalent simplest fraction is 1/2.
  2. 7/8 — Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 7/8.
  3. 5/8 — Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 5/8.

Comparing Fractions — Part 1

Prerequisite

  1. 2/3 — Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 2/3.
  2. 2/3 — Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 2/3.
  3. 3/5 — Divide numerator and denominator by their greatest common factor 4. The equivalent simplest fraction is 3/5.

Understanding checks

  1. < — Rename as 18/30 and 25/30. Compare numerators: 6/10 < 5/6.
  2. < — Rename as 2/6 and 4/6. Compare numerators: 1/3 < 4/6.
  3. > — Rename as 5/8 and 1/8. Compare numerators: 5/8 > 1/8.

Guided

  1. > — Rename as 3/8 and 2/8. Compare numerators: 3/8 > 2/8.
  2. < — Rename as 12/40 and 20/40. Compare numerators: 3/10 < 4/8.
  3. > — Rename as 10/20 and 2/20. Compare numerators: 2/4 > 1/10.
  4. < — Rename as 6/12 and 10/12. Compare numerators: 2/4 < 10/12.
  5. < — Rename as 5/30 and 27/30. Compare numerators: 1/6 < 9/10.

Independent

  1. < — Rename as 15/24 and 18/24. Compare numerators: 5/8 < 9/12.
  2. > — Rename as 5/12 and 1/12. Compare numerators: 5/12 > 1/12.
  3. > — Rename as 6/12 and 1/12. Compare numerators: 2/4 > 1/12.
  4. > — Rename as 18/24 and 6/24. Compare numerators: 9/12 > 2/8.
  5. < — Rename as 30/60 and 40/60. Compare numerators: 5/10 < 8/12.
  6. > — Rename as 16/20 and 5/20. Compare numerators: 4/5 > 1/4.
  7. < — Rename as 10/20 and 16/20. Compare numerators: 2/4 < 8/10.
  8. > — Rename as 24/30 and 5/30. Compare numerators: 4/5 > 1/6.

Challenge

  1. 5/9 — The pieces are equal-sized, so five of them exceed two.
  2. 3/8 > 3/9 — Each piece is larger when the whole is divided into fewer equal parts; taking three larger pieces gives more.

Exit

  1. > — Rename as 16/20 and 15/20. Compare numerators: 4/5 > 3/4.
  2. < — Rename as 18/60 and 40/60. Compare numerators: 3/10 < 8/12.
  3. > — Rename as 8/12 and 5/12. Compare numerators: 2/3 > 5/12.

Comparing Fractions — Part 2

Prerequisite

  1. < — Rename as 24/30 and 25/30. Compare numerators: 4/5 < 5/6.
  2. > — Rename as 3/12 and 1/12. Compare numerators: 1/4 > 1/12.
  3. < — Rename as 2/12 and 9/12. Compare numerators: 1/6 < 3/4.

Understanding checks

  1. < — Rename as 36/60 and 45/60. Compare numerators: 3/5 < 9/12.
  2. < — Rename as 3/18 and 14/18. Compare numerators: 1/6 < 7/9.
  3. > — Rename as 10/15 and 3/15. Compare numerators: 2/3 > 1/5.

Guided

  1. > — Rename as 16/24 and 3/24. Compare numerators: 2/3 > 1/8.
  2. > — Rename as 9/36 and 4/36. Compare numerators: 3/12 > 1/9.
  3. < — Rename as 4/12 and 8/12. Compare numerators: 1/3 < 8/12.
  4. < — Rename as 2/12 and 4/12. Compare numerators: 2/12 < 4/12.
  5. > — Rename as 18/36 and 8/36. Compare numerators: 2/4 > 2/9.

Independent

  1. < — Rename as 3/24 and 18/24. Compare numerators: 1/8 < 9/12.
  2. > — Rename as 5/8 and 4/8. Compare numerators: 5/8 > 4/8.
  3. < — Rename as 36/90 and 70/90. Compare numerators: 4/10 < 7/9.
  4. > — Rename as 21/28 and 4/28. Compare numerators: 3/4 > 1/7.
  5. > — Rename as 36/90 and 10/90. Compare numerators: 4/10 > 1/9.
  6. < — Rename as 6/24 and 22/24. Compare numerators: 2/8 < 11/12.
  7. < — Rename as 5/40 and 24/40. Compare numerators: 1/8 < 3/5.
  8. > — Rename as 14/21 and 12/21. Compare numerators: 2/3 > 4/7.

Challenge

  1. 5/10 — The pieces are equal-sized, so five of them exceed two.
  2. 3/9 > 3/10 — Each piece is larger when the whole is divided into fewer equal parts; taking three larger pieces gives more.

Exit

  1. > — Rename as 6/8 and 1/8. Compare numerators: 3/4 > 1/8.
  2. < — Rename as 30/36 and 32/36. Compare numerators: 10/12 < 8/9.
  3. > — Rename as 24/60 and 10/60. Compare numerators: 2/5 > 2/12.

Adding and Subtracting Fractions — Part 1

Prerequisite

  1. > — Rename as 16/24 and 12/24. Compare numerators: 4/6 > 4/8.
  2. > — Rename as 14/21 and 6/21. Compare numerators: 2/3 > 2/7.
  3. > — Rename as 16/24 and 6/24. Compare numerators: 4/6 > 2/8.

Understanding checks

  1. 5/8 — Both fractions use 8ths. 4 + 1 pieces make 5/8 of a whole after simplification.
  2. 0 — Both fractions use 5ths. 1 − 1 pieces make 0 of a whole after simplification.
  3. 1 — Both fractions use 8ths. 4 + 4 pieces make 1 of a whole after simplification.

Guided

  1. 7/10 — Both fractions use 10ths. 8 − 1 pieces make 7/10 of a whole after simplification.
  2. 3/5 — Both fractions use 5ths. 2 + 1 pieces make 3/5 of a whole after simplification.
  3. 3/5 — Both fractions use 5ths. 4 − 1 pieces make 3/5 of a whole after simplification.
  4. 1/3 — Both fractions use 12ths. 3 + 1 pieces make 1/3 of a whole after simplification.
  5. 1/5 — Both fractions use 5ths. 2 − 1 pieces make 1/5 of a whole after simplification.

Independent

  1. 1/3 — Both fractions use 6ths. 3 − 1 pieces make 1/3 of a whole after simplification.
  2. 1 — Both fractions use 6ths. 1 + 5 pieces make 1 of a whole after simplification.
  3. 2/3 — Both fractions use 12ths. 9 − 1 pieces make 2/3 of a whole after simplification.
  4. 1 — Both fractions use 12ths. 7 + 5 pieces make 1 of a whole after simplification.
  5. 1/2 — Both fractions use 8ths. 5 − 1 pieces make 1/2 of a whole after simplification.
  6. 4/5 — Both fractions use 5ths. 3 + 1 pieces make 4/5 of a whole after simplification.
  7. 1/6 — Both fractions use 6ths. 3 − 2 pieces make 1/6 of a whole after simplification.
  8. 4/5 — Both fractions use 5ths. 1 + 3 pieces make 4/5 of a whole after simplification.

Challenge

  1. 3/10 — Rename 1/5 as 2/10; then add 2/10 + 1/10 = 3/10.
  2. 3/10 — The number of equal parts in the whole stays 10; only the count of selected parts changes from 4 to 3.

Exit

  1. 4/5 — Both fractions use 5ths. 2 + 2 pieces make 4/5 of a whole after simplification.
  2. 1/2 — Both fractions use 10ths. 7 − 2 pieces make 1/2 of a whole after simplification.
  3. 3/4 — Both fractions use 8ths. 5 + 1 pieces make 3/4 of a whole after simplification.

Adding and Subtracting Fractions — Part 2

Prerequisite

  1. 1/6 — Both fractions use 6ths. 2 − 1 pieces make 1/6 of a whole after simplification.
  2. 2/3 — Both fractions use 6ths. 1 + 3 pieces make 2/3 of a whole after simplification.
  3. 1/5 — Both fractions use 5ths. 3 − 2 pieces make 1/5 of a whole after simplification.

Understanding checks

  1. 5/6 — First rename 1/3 as 2/6. Add 3/6 to get 5/6 = 5/6.
  2. 3/5 — First rename 2/5 as 4/10. Add 2/10 to get 6/10 = 3/5.
  3. 3/10 — First rename 1/5 as 2/10. Add 1/10 to get 3/10 = 3/10.

Guided

  1. 7/10 — First rename 2/5 as 4/10. Add 3/10 to get 7/10 = 7/10.
  2. 1 — First rename 1/2 as 2/4. Add 2/4 to get 4/4 = 1.
  3. 3/2 — First rename 2/3 as 4/6. Add 5/6 to get 9/6 = 3/2.
  4. 9/10 — First rename 2/5 as 4/10. Add 5/10 to get 9/10 = 9/10.
  5. 6/5 — First rename 4/5 as 8/10. Add 4/10 to get 12/10 = 6/5.

Independent

  1. 1 — First rename 2/4 as 4/8. Add 4/8 to get 8/8 = 1.
  2. 7/8 — First rename 2/4 as 4/8. Add 3/8 to get 7/8 = 7/8.
  3. 11/10 — First rename 1/5 as 2/10. Add 9/10 to get 11/10 = 11/10.
  4. 7/8 — First rename 1/4 as 2/8. Add 5/8 to get 7/8 = 7/8.
  5. 9/8 — First rename 3/4 as 6/8. Add 3/8 to get 9/8 = 9/8.
  6. 7/6 — First rename 1/3 as 2/6. Add 5/6 to get 7/6 = 7/6.
  7. 13/8 — First rename 3/4 as 6/8. Add 7/8 to get 13/8 = 13/8.
  8. 1/2 — First rename 1/3 as 2/6. Add 1/6 to get 3/6 = 1/2.

Challenge

  1. 3/12 — Rename 1/6 as 2/12; then add 2/12 + 1/12 = 3/12.
  2. 3/12 — The number of equal parts in the whole stays 12; only the count of selected parts changes from 4 to 3.

Exit

  1. 1/2 — First rename 1/5 as 2/10. Add 3/10 to get 5/10 = 1/2.
  2. 5/8 — First rename 2/4 as 4/8. Add 1/8 to get 5/8 = 5/8.
  3. 9/8 — First rename 2/4 as 4/8. Add 5/8 to get 9/8 = 9/8.

Practice A

  1. They are equal — Subdivide each of the 10 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 3/10 = 9/30.
  2. 12 — Subdivide each of the 4 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 3/4 = 12/16.
  3. 1/8 — Divide numerator and denominator by their greatest common factor 4. The equivalent simplest fraction is 1/8.
  4. > — Rename as 15/30 and 12/30. Compare numerators: 3/6 > 4/10.
  5. > — Rename as 21/28 and 8/28. Compare numerators: 3/4 > 2/7.
  6. They are equal — Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 2/6 = 4/12.
  7. 6 — Subdivide each of the 4 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 2/4 = 6/12.
  8. 2/5 — Divide numerator and denominator by their greatest common factor 4. The equivalent simplest fraction is 2/5.
  9. > — Rename as 30/40 and 4/40. Compare numerators: 6/8 > 1/10.
  10. > — Rename as 45/90 and 30/90. Compare numerators: 5/10 > 3/9.
  11. They are equal — Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 5/6 = 10/12.
  12. 8 — Subdivide each of the 5 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/5 = 8/20.
  13. 3/4 — Divide numerator and denominator by their greatest common factor 9. The equivalent simplest fraction is 3/4.
  14. < — Rename as 4/10 and 8/10. Compare numerators: 2/5 < 8/10.
  15. < — Rename as 36/45 and 40/45. Compare numerators: 4/5 < 8/9.
  16. They are equal — Subdivide each of the 10 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 1/10 = 4/40.
  17. 2/7 — Grouping numerator and denominator into threes gives 2/7; both represent the same share of the same whole.
  18. 4/14 — No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/7 = 4/14.

Practice B

  1. They are equal — Subdivide each of the 5 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 4/5 = 16/20.
  2. 16 — Subdivide each of the 12 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 4/12 = 16/48.
  3. 2/3 — Divide numerator and denominator by their greatest common factor 4. The equivalent simplest fraction is 2/3.
  4. > — Rename as 10/30 and 3/30. Compare numerators: 2/6 > 1/10.
  5. > — Rename as 7/10 and 4/10. Compare numerators: 7/10 > 2/5.
  6. 2/3 — Both fractions use 6ths. 2 + 2 pieces make 2/3 of a whole after simplification.
  7. 5/6 — First rename 1/3 as 2/6. Add 3/6 to get 5/6 = 5/6.
  8. They are equal — Subdivide each of the 8 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 7/8 = 14/16.
  9. 12 — Subdivide each of the 5 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/5 = 12/15.
  10. 1/5 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 1/5.
  11. < — Rename as 4/8 and 6/8. Compare numerators: 2/4 < 6/8.
  12. < — Rename as 6/18 and 12/18. Compare numerators: 2/6 < 6/9.
  13. 2/5 — Both fractions use 5ths. 3 − 1 pieces make 2/5 of a whole after simplification.
  14. 1 — First rename 1/2 as 2/4. Add 2/4 to get 4/4 = 1.
  15. They are equal — Subdivide each of the 4 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/4 = 8/16.
  16. 10 — Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 5/6 = 10/12.
  17. 2/5 — Grouping numerator and denominator into threes gives 2/5; both represent the same share of the same whole.
  18. 4/10 — No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/5 = 4/10.

Chapter Review

  1. They are equal — Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 3/6 = 6/12.
  2. 12 — Subdivide each of the 10 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/10 = 12/30.
  3. 7/12 — Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 7/12.
  4. > — Rename as 7/8 and 6/8. Compare numerators: 7/8 > 3/4.
  5. < — Rename as 6/24 and 14/24. Compare numerators: 2/8 < 7/12.
  6. 2/3 — Both fractions use 6ths. 1 + 3 pieces make 2/3 of a whole after simplification.
  7. 5/4 — First rename 1/2 as 2/4. Add 3/4 to get 5/4 = 5/4.
  8. They are equal — Subdivide each of the 5 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 3/5 = 9/15.
  9. 4 — Subdivide each of the 3 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 1/3 = 4/12.
  10. 3/4 — Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 3/4.
  11. > — Rename as 4/6 and 2/6. Compare numerators: 2/3 > 2/6.
  12. < — Rename as 9/36 and 28/36. Compare numerators: 1/4 < 7/9.
  13. 1/10 — Both fractions use 10ths. 5 − 4 pieces make 1/10 of a whole after simplification.
  14. 3/4 — First rename 1/2 as 2/4. Add 1/4 to get 3/4 = 3/4.
  15. They are equal — Subdivide each of the 8 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 7/8 = 28/32.
  16. 2 — Subdivide each of the 3 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 1/3 = 2/6.
  17. 3/4 — Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 3/4.
  18. < — Rename as 1/8 and 5/8. Compare numerators: 1/8 < 5/8.
  19. > — Rename as 28/35 and 15/35. Compare numerators: 4/5 > 3/7.
  20. 4/5 — Both fractions use 5ths. 2 + 2 pieces make 4/5 of a whole after simplification.
  21. 3/2 — First rename 2/3 as 4/6. Add 5/6 to get 9/6 = 3/2.
  22. They are equal — Subdivide each of the 5 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 4/5 = 8/10.
  23. 4 — Subdivide each of the 8 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 1/8 = 4/32.
  24. 2/3 — Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 2/3.
  25. 2/8 — Grouping numerator and denominator into threes gives 2/8; both represent the same share of the same whole.
  26. 4/16 — No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/8 = 4/16.

Practice Exam

  1. > — The suggested answer is not correct. Rename as 6/9 and 1/9. Compare numerators: 2/3 > 1/9. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  2. 8 — Subdivide each of the 10 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/10 = 8/40. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  3. 1/4 — Both fractions use 8ths. 3 − 1 pieces make 1/4 of a whole after simplification. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  4. 1/3 — Divide numerator and denominator by their greatest common factor 3. The equivalent simplest fraction is 1/3. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  5. 8 — The suggested answer is not correct. Subdivide each of the 8 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/8 = 8/32. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  6. 3/4 — Both fractions use 8ths. 7 − 1 pieces make 3/4 of a whole after simplification. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  7. < — Rename as 6/8 and 7/8. Compare numerators: 3/4 < 7/8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  8. < — Rename as 16/24 and 21/24. Compare numerators: 2/3 < 7/8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  9. 5/8 — The suggested answer is not correct. Divide numerator and denominator by their greatest common factor 2. The equivalent simplest fraction is 5/8. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  10. They are equal — Subdivide each of the 5 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 1/5 = 4/20. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  11. 21 — Subdivide each of the 10 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 7/10 = 21/30. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  12. < — Rename as 20/30 and 24/30. Compare numerators: 4/6 < 4/5. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  13. > — The suggested answer is not correct. Rename as 5/8 and 3/8. Compare numerators: 5/8 > 3/8. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  14. 3/10 — First rename 1/5 as 2/10. Add 1/10 to get 3/10 = 3/10. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  15. They are equal — Subdivide each of the 12 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 8/12 = 16/24. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  16. 20 — Subdivide each of the 8 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 5/8 = 20/32. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  17. 11/10 — The suggested answer is not correct. First rename 3/5 as 6/10. Add 5/10 to get 11/10 = 11/10. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  18. 1 — Both fractions use 5ths. 3 + 2 pieces make 1 of a whole after simplification. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  19. 1/2 — Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 1/2. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  20. They are equal — Subdivide each of the 6 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 5/6 = 20/24. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  21. 4/5 — The suggested answer is not correct. Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 4/5. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  22. < — Rename as 2/10 and 7/10. Compare numerators: 2/10 < 7/10. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  23. 2/4 — Grouping numerator and denominator into threes gives 2/4; both represent the same share of the same whole. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  24. 4/18 — No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/9 = 4/18. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)