Sixth Grade Math Worksheets - Free & Printable | Beestar - Free Printable

Problem Breakdown and Solution

The worksheet provided involves several types of problems:
1. Conversion of mixed numbers to improper fractions
2. Arithmetic operations with fractions (addition, subtraction)
3. Comparison of fractions
4. Division involving fractions

Let's solve each section step by step.

---

Section 1: Convert Mixed Numbers to Improper Fractions

#### Formula for Conversion:
A mixed number \( a \frac{b}{c} \) can be converted to an improper fraction using the formula:
\[
a \frac{b}{c} = \frac{(a \times c) + b}{c}
\]

#### Solutions:
1. 54. \( 7 \frac{2}{3} \)
\[
7 \frac{2}{3} = \frac{(7 \times 3) + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3}
\]

2. 55. \( 2 \frac{1}{5} \)
\[
2 \frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}
\]

3. 56. \( 8 \frac{1}{4} \)
\[
8 \frac{1}{4} = \frac{(8 \times 4) + 1}{4} = \frac{32 + 1}{4} = \frac{33}{4}
\]

4. 57. \( 5 \frac{2}{3} \)
\[
5 \frac{2}{3} = \frac{(5 \times 3) + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}
\]

5. 58. \( 8 \frac{2}{5} \)
\[
8 \frac{2}{5} = \frac{(8 \times 5) + 2}{5} = \frac{40 + 2}{5} = \frac{42}{5}
\]

6. 59. \( 3 \frac{5}{6} \)
\[
3 \frac{5}{6} = \frac{(3 \times 6) + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}
\]

7. 60. \( 7 \frac{1}{3} \)
\[
7 \frac{1}{3} = \frac{(7 \times 3) + 1}{3} = \frac{21 + 1}{3} = \frac{22}{3}
\]

8. 61. \( 2 \frac{1}{4} \)
\[
2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}
\]

9. 62. \( 1 \frac{3}{5} \)
\[
1 \frac{3}{5} = \frac{(1 \times 5) + 3}{5} = \frac{5 + 3}{5} = \frac{8}{5}
\]

10. 63. \( 4 \frac{2}{5} \)
\[
4 \frac{2}{5} = \frac{(4 \times 5) + 2}{5} = \frac{20 + 2}{5} = \frac{22}{5}
\]

11. 64. \( 8 \frac{2}{3} \)
\[
8 \frac{2}{3} = \frac{(8 \times 3) + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3}
\]

12. 65. \( 8 \frac{1}{8} \)
\[
8 \frac{1}{8} = \frac{(8 \times 8) + 1}{8} = \frac{64 + 1}{8} = \frac{65}{8}
\]

13. 66. \( 3 \frac{3}{4} \)
\[
3 \frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}
\]

14. 67. \( 5 \frac{3}{5} \)
\[
5 \frac{3}{5} = \frac{(5 \times 5) + 3}{5} = \frac{25 + 3}{5} = \frac{28}{5}
\]

15. 68. \( 8 \frac{2}{6} \)
\[
8 \frac{2}{6} = \frac{(8 \times 6) + 2}{6} = \frac{48 + 2}{6} = \frac{50}{6} = \frac{25}{3} \quad (\text{simplified})
\]

16. 69. \( 7 \frac{5}{6} \)
\[
7 \frac{5}{6} = \frac{(7 \times 6) + 5}{6} = \frac{42 + 5}{6} = \frac{47}{6}
\]

17. 70. \( 9 \frac{1}{4} \)
\[
9 \frac{1}{4} = \frac{(9 \times 4) + 1}{4} = \frac{36 + 1}{4} = \frac{37}{4}
\]

18. 71. \( 1 \frac{4}{6} \)
\[
1 \frac{4}{6} = \frac{(1 \times 6) + 4}{6} = \frac{6 + 4}{6} = \frac{10}{6} = \frac{5}{3} \quad (\text{simplified})
\]

19. 72. \( 4 \frac{7}{8} \)
\[
4 \frac{7}{8} = \frac{(4 \times 8) + 7}{8} = \frac{32 + 7}{8} = \frac{39}{8}
\]

20. 73. \( 9 \frac{1}{3} \)
\[
9 \frac{1}{3} = \frac{(9 \times 3) + 1}{3} = \frac{27 + 1}{3} = \frac{28}{3}
\]

---

Section 2: Calculate Arithmetic Operations with Fractions

#### Addition and Subtraction:
- For addition/subtraction, ensure the denominators are the same. If not, find the least common denominator (LCD).

#### Solutions:
1. 74. \( 9 \frac{2}{5} - 8 \frac{4}{5} \)
\[
9 \frac{2}{5} = \frac{47}{5}, \quad 8 \frac{4}{5} = \frac{44}{5}
\]
\[
\frac{47}{5} - \frac{44}{5} = \frac{47 - 44}{5} = \frac{3}{5}
\]

2. 75. \( 5 \frac{1}{10} + 5 \frac{5}{10} \)
\[
5 \frac{1}{10} = \frac{51}{10}, \quad 5 \frac{5}{10} = \frac{55}{10}
\]
\[
\frac{51}{10} + \frac{55}{10} = \frac{51 + 55}{10} = \frac{106}{10} = \frac{53}{5}
\]

3. 76. \( 7 \frac{9}{18} + 4 \frac{1}{18} \)
\[
7 \frac{9}{18} = \frac{135}{18}, \quad 4 \frac{1}{18} = \frac{73}{18}
\]
\[
\frac{135}{18} + \frac{73}{18} = \frac{135 + 73}{18} = \frac{208}{18} = \frac{104}{9}
\]

4. 77. \( 7 \frac{1}{9} - 1 \frac{5}{9} \)
\[
7 \frac{1}{9} = \frac{64}{9}, \quad 1 \frac{5}{9} = \frac{14}{9}
\]
\[
\frac{64}{9} - \frac{14}{9} = \frac{64 - 14}{9} = \frac{50}{9}
\]

5. 78. \( 6 \frac{2}{21} + 8 \frac{14}{21} \)
\[
6 \frac{2}{21} = \frac{128}{21}, \quad 8 \frac{14}{21} = \frac{182}{21}
\]
\[
\frac{128}{21} + \frac{182}{21} = \frac{128 + 182}{21} = \frac{310}{21}
\]

6. 79. \( 2 \frac{3}{4} + 1 \frac{3}{4} \)
\[
2 \frac{3}{4} = \frac{11}{4}, \quad 1 \frac{3}{4} = \frac{7}{4}
\]
\[
\frac{11}{4} + \frac{7}{4} = \frac{11 + 7}{4} = \frac{18}{4} = \frac{9}{2}
\]

7. 80. \( 4 \frac{3}{10} + 5 \frac{8}{10} \)
\[
4 \frac{3}{10} = \frac{43}{10}, \quad 5 \frac{8}{10} = \frac{58}{10}
\]
\[
\frac{43}{10} + \frac{58}{10} = \frac{43 + 58}{10} = \frac{101}{10}
\]

8. 81. \( 7 \frac{5}{7} + 7 \frac{6}{7} \)
\[
7 \frac{5}{7} = \frac{54}{7}, \quad 7 \frac{6}{7} = \frac{55}{7}
\]
\[
\frac{54}{7} + \frac{55}{7} = \frac{54 + 55}{7} = \frac{109}{7}
\]

9. 82. \( 7 \frac{12}{23} - 3 \frac{14}{23} \)
\[
7 \frac{12}{23} = \frac{163}{23}, \quad 3 \frac{14}{23} = \frac{83}{23}
\]
\[
\frac{163}{23} - \frac{83}{23} = \frac{163 - 83}{23} = \frac{80}{23}
\]

10. 83. \( 3 \frac{8}{11} + 8 \frac{4}{11} \)
\[
3 \frac{8}{11} = \frac{41}{11}, \quad 8 \frac{4}{11} = \frac{92}{11}
\]
\[
\frac{41}{11} + \frac{92}{11} = \frac{41 + 92}{11} = \frac{133}{11}
\]

---

Section 3: Compare Fractions

#### Comparison:
- Convert fractions to have a common denominator or compare their decimal equivalents.

#### Solutions:
1. 84. \( \frac{1}{4} \_ \frac{1}{3} \)
\[
\frac{1}{4} = 0.25, \quad \frac{1}{3} \approx 0.333
\]
\[
\frac{1}{4} < \frac{1}{3}
\]

2. 85. \( \frac{1}{4} \_ \frac{1}{6} \)
\[
\frac{1}{4} = 0.25, \quad \frac{1}{6} \approx 0.167
\]
\[
\frac{1}{4} > \frac{1}{6}
\]

3. 86. \( \frac{3}{8} \_ \frac{3}{5} \)
\[
\frac{3}{8} = 0.375, \quad \frac{3}{5} = 0.6
\]
\[
\frac{3}{8} < \frac{3}{5}
\]

4. 87. \( \frac{1}{3} \_ \frac{1}{4} \)
\[
\frac{1}{3} \approx 0.333, \quad \frac{1}{4} = 0.25
\]
\[
\frac{1}{3} > \frac{1}{4}
\]

5. 88. \( \frac{5}{6} \_ \frac{1}{5} \)
\[
\frac{5}{6} \approx 0.833, \quad \frac{1}{5} = 0.2
\]
\[
\frac{5}{6} > \frac{1}{5}
\]

6. 89. \( \frac{5}{8} \_ \frac{2}{3} \)
\[
\frac{5}{8} = 0.625, \quad \frac{2}{3} \approx 0.667
\]
\[
\frac{5}{8} < \frac{2}{3}
\]

7. 90. \( \frac{2}{8} \_ \frac{2}{6} \)
\[
\frac{2}{8} = \frac{1}{4} = 0.25, \quad \frac{2}{6} = \frac{1}{3} \approx 0.333
\]
\[
\frac{2}{8} < \frac{2}{6}
\]

8. 91. \( \frac{2}{4} \_ \frac{1}{5} \)
\[
\frac{2}{4} = \frac{1}{2} = 0.5, \quad \frac{1}{5} = 0.2
\]
\[
\frac{2}{4} > \frac{1}{5}
\]

9. 92. \( \frac{3}{4} \_ \frac{1}{3} \)
\[
\frac{3}{4} = 0.75, \quad \frac{1}{3} \approx 0.333
\]
\[
\frac{3}{4} > \frac{1}{3}
\]

10. 93. \( \frac{5}{8} \_ \frac{2}{6} \)
\[
\frac{5}{8} = 0.625, \quad \frac{2}{6} = \frac{1}{3} \approx 0.333
\]
\[
\frac{5}{8} > \frac{2}{6}
\]

11. 94. \( \frac{3}{5} \_ \frac{5}{8} \)
\[
\frac{3}{5} = 0.6, \quad \frac{5}{8} = 0.625
\]
\[
\frac{3}{5} < \frac{5}{8}
\]

12. 95. \( \frac{2}{4} \_ \frac{3}{6} \)
\[
\frac{2}{4} = \frac{1}{2} = 0.5, \quad \frac{3}{6} = \frac{1}{2} = 0.5
\]
\[
\frac{2}{4} = \frac{3}{6}
\]

---

Section 4: Division Involving Fractions

#### Division Rule:
\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}
\]

#### Solutions:
1. 96. \( \frac{2}{8} \div 1 \)
\[
\frac{2}{8} \div 1 = \frac{2}{8} = \frac{1}{4}
\]

2. 97. \( 4 \div \frac{1}{5} \)
\[
4 \div \frac{1}{5} = 4 \times 5 = 20
\]

3. 98. \( \frac{5}{6} \div 5 \)
\[
\frac{5}{6} \div 5 = \frac{5}{6} \times \frac{1}{5} = \frac{5 \cdot 1}{6 \cdot 5} = \frac{1}{6}
\]

4. 99. \( \frac{2}{4} \div 5 \)
\[
\frac{2}{4} \div 5 = \frac{1}{2} \div 5 = \frac{1}{2} \times \frac{1}{5} = \frac{1 \cdot 1}{2 \cdot 5} = \frac{1}{10}
\]

5. 100. \( \frac{6}{8} \div 8 \)
\[
\frac{6}{8} \div 8 = \frac{3}{4} \div 8 = \frac{3}{4} \times \frac{1}{8} = \frac{3 \cdot 1}{4 \cdot 8} = \frac{3}{32}
\]

6. 101. \( 5 \div \frac{3}{6} \)
\[
5 \div \frac{3}{6} = 5 \div \frac{1}{2} = 5 \times 2 = 10
\]

7. 102. \( \frac{2}{4} \div 8 \)
\[
\frac{2}{4} \div 8 = \frac{1}{2} \div 8 = \frac{1}{2} \times \frac{1}{8} = \frac{1 \cdot 1}{2 \cdot 8} = \frac{1}{16}
\]

8. 103. \( 9 \div \frac{2}{3} \)
\[
9 \div \frac{2}{3} = 9 \times \frac{3}{2} = \frac{9 \cdot 3}{2} = \frac{27}{2}
\]

9. 104. \( 6 \div \frac{2}{5} \)
\[
6 \div \frac{2}{5} = 6 \times \frac{5}{2} = \frac{6 \cdot 5}{2} = \frac{30}{2} = 15
\]

10. 105. \( \frac{1}{4} \div 4 \)
\[
\frac{1}{4} \div 4 = \frac{1}{4} \times \frac{1}{4} = \frac{1 \cdot 1}{4 \cdot 4} = \frac{1}{16}
\]

11. 106. \( \frac{1}{8} \div 8 \)
\[
\frac{1}{8} \div 8 = \frac{1}{8} \times \frac{1}{8} = \frac{1 \cdot 1}{8 \cdot 8} = \frac{1}{64}
\]

12. 107. \( \frac{4}{6} \div 3 \)
\[
\frac{4}{6} \div 3 = \frac{2}{3} \div 3 = \frac{2}{3} \times \frac{1}{3} = \frac{2 \cdot 1}{3 \cdot 3} = \frac{2}{9}
\]

---

Final Answer

\[
\boxed{
\begin{array}{l}
\text{Convert:} \\
54. \frac{23}{3}, \, 55. \frac{11}{5}, \, 56. \frac{33}{4}, \, 57. \frac{17}{3}, \, 58. \frac{42}{5}, \, 59. \frac{23}{6}, \, 60. \frac{22}{3}, \, 61. \frac{9}{4}, \, 62. \frac{8}{5}, \, 63. \frac{22}{5}, \, 64. \frac{26}{3}, \, 65. \frac{65}{8}, \, 66. \frac{15}{4}, \, 67. \frac{28}{5}, \, 68. \frac{25}{3}, \, 69. \frac{47}{6}, \, 70. \frac{37}{4}, \, 71. \frac{5}{3}, \, 72. \frac{39}{8}, \, 73. \frac{28}{3} \\
\text{Calculate:} \\
74. \frac{3}{5}, \, 75. \frac{53}{5}, \, 76. \frac{104}{9}, \, 77. \frac{50}{9}, \, 78. \frac{310}{21}, \, 79. \frac{9}{2}, \, 80. \frac{101}{10}, \, 81. \frac{109}{7}, \, 82. \frac{80}{23}, \, 83. \frac{133}{11} \\
\text{Compare:} \\
84. <, \, 85. >, \, 86. <, \, 87. >, \, 88. >, \, 89. <, \, 90. <, \, 91. >, \, 92. >, \, 93. >, \, 94. <, \, 95. = \\
\text{Divide:} \\
96. \frac{1}{4}, \, 97. 20, \, 98. \frac{1}{6}, \, 99. \frac{1}{10}, \, 100. \frac{3}{32}, \, 101. 10, \, 102. \frac{1}{16}, \, 103. \frac{27}{2}, \, 104. 15, \, 105. \frac{1}{16}, \, 106. \frac{1}{64}, \, 107. \frac{2}{9}
\end{array}
}
\]