6th Grade Math Worksheets | Printable PDF Worksheets - Free Printable

Problem Solving and Explanation for the Worksheet

#### Section A: Reciprocals

1. Prove that \(\frac{3}{4} \times \frac{4}{3} = 1\)

- The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). When you multiply a number by its reciprocal, the result is always 1.
- Here, \(\frac{3}{4}\) and \(\frac{4}{3}\) are reciprocals.
- Calculation:
\[
\frac{3}{4} \times \frac{4}{3} = \frac{3 \times 4}{4 \times 3} = \frac{12}{12} = 1
\]
- Answer: Proven.

2. Fill in the blanks:

- a) \(\frac{2}{3} \times \square = 1\)
- The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
- Answer: \(\frac{3}{2}\)

- b) \(\square \times \frac{5}{7} = 1\)
- The reciprocal of \(\frac{5}{7}\) is \(\frac{7}{5}\).
- Answer: \(\frac{7}{5}\)

- c) \(1 = \frac{1}{2} \times \square\)
- The reciprocal of \(\frac{1}{2}\) is \(2\).
- Answer: \(2\)

- d) \(\square \times 8 = 1\)
- The reciprocal of \(8\) is \(\frac{1}{8}\).
- Answer: \(\frac{1}{8}\)

- Any number multiplied by its \_\_\_\_\_\_ is equal to 1.
- The blank should be filled with "reciprocal."
- Answer: reciprocal

3. Find the reciprocal of each of the following numbers:

- a) \(\frac{6}{11}\)
- The reciprocal is \(\frac{11}{6}\).
- Answer: \(\frac{11}{6}\)

- b) \(-\frac{2}{3}\)
- The reciprocal is \(-\frac{3}{2}\).
- Answer: \(-\frac{3}{2}\)

- c) \(5\)
- The reciprocal is \(\frac{1}{5}\).
- Answer: \(\frac{1}{5}\)

- d) \(\frac{1}{2}\)
- The reciprocal is \(2\).
- Answer: \(2\)

- e) \(\frac{8}{19}\)
- The reciprocal is \(\frac{19}{8}\).
- Answer: \(\frac{19}{8}\)

- f) \(4 \frac{2}{3}\)
- Convert the mixed number to an improper fraction: \(4 \frac{2}{3} = \frac{14}{3}\).
- The reciprocal is \(\frac{3}{14}\).
- Answer: \(\frac{3}{14}\)

#### Section B: Dividing integers by fractions

1. Explain how this diagram shows that \(1 \div \frac{1}{3} = 3\):

- The diagram shows a whole (1) divided into three equal parts, each being \(\frac{1}{3}\).
- Dividing 1 by \(\frac{1}{3}\) means finding how many \(\frac{1}{3}\)s fit into 1.
- Since 1 is composed of 3 parts of \(\frac{1}{3}\), \(1 \div \frac{1}{3} = 3\).
- Answer: The diagram shows that 1 can be divided into 3 equal parts of \(\frac{1}{3}\), so \(1 \div \frac{1}{3} = 3\).

2. Calculate the following:

- a) \(2 \div \frac{1}{3}\)
- Dividing by a fraction is equivalent to multiplying by its reciprocal.
- \(2 \div \frac{1}{3} = 2 \times 3 = 6\)
- Answer: \(6\)

- b) \(2 \div \frac{2}{3}\)
- \(2 \div \frac{2}{3} = 2 \times \frac{3}{2} = \frac{6}{2} = 3\)
- Answer: \(3\)

- c) \(10 \div \frac{2}{3}\)
- \(10 \div \frac{2}{3} = 10 \times \frac{3}{2} = \frac{30}{2} = 15\)
- Answer: \(15\)

- d) \(10 \div \frac{2}{5}\)
- \(10 \div \frac{2}{5} = 10 \times \frac{5}{2} = \frac{50}{2} = 25\)
- Answer: \(25\)

- e) \(10 \div \frac{3}{5}\)
- \(10 \div \frac{3}{5} = 10 \times \frac{5}{3} = \frac{50}{3} = 16 \frac{2}{3}\)
- Answer: \(16 \frac{2}{3}\)

- f) \(21 \div 2 \frac{1}{3}\)
- Convert \(2 \frac{1}{3}\) to an improper fraction: \(2 \frac{1}{3} = \frac{7}{3}\).
- \(21 \div \frac{7}{3} = 21 \times \frac{3}{7} = \frac{63}{7} = 9\)
- Answer: \(9\)

#### Section C: Dividing any pair of fractions

1. Calculate:

- a) \(\frac{1}{3} \div \frac{1}{3}\)
- \(\frac{1}{3} \div \frac{1}{3} = \frac{1}{3} \times \frac{3}{1} = 1\)
- Answer: \(1\)

- b) \(\frac{2}{3} \div \frac{1}{2}\)
- \(\frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3} = 1 \frac{1}{3}\)
- Answer: \(1 \frac{1}{3}\)

- c) \(4 \frac{2}{3} \div \frac{1}{2}\)
- Convert \(4 \frac{2}{3}\) to an improper fraction: \(4 \frac{2}{3} = \frac{14}{3}\).
- \(\frac{14}{3} \div \frac{1}{2} = \frac{14}{3} \times \frac{2}{1} = \frac{28}{3} = 9 \frac{1}{3}\)
- Answer: \(9 \frac{1}{3}\)

- d) \(\frac{5}{7} \div \frac{5}{12}\)
- \(\frac{5}{7} \div \frac{5}{12} = \frac{5}{7} \times \frac{12}{5} = \frac{60}{35} = \frac{12}{7} = 1 \frac{5}{7}\)
- Answer: \(1 \frac{5}{7}\)

- e) \(-\frac{5}{12} \div \frac{4}{9}\)
- \(-\frac{5}{12} \div \frac{4}{9} = -\frac{5}{12} \times \frac{9}{4} = -\frac{45}{48} = -\frac{15}{16}\)
- Answer: \(-\frac{15}{16}\)

- f) \(2 \frac{1}{8} \div \frac{9}{10}\)
- Convert \(2 \frac{1}{8}\) to an improper fraction: \(2 \frac{1}{8} = \frac{17}{8}\).
- \(\frac{17}{8} \div \frac{9}{10} = \frac{17}{8} \times \frac{10}{9} = \frac{170}{72} = \frac{85}{36} = 2 \frac{13}{36}\)
- Answer: \(2 \frac{13}{36}\)

- g) \(\frac{9}{11} \div \frac{9}{11}\)
- \(\frac{9}{11} \div \frac{9}{11} = \frac{9}{11} \times \frac{11}{9} = 1\)
- Answer: \(1\)

- h) \(\frac{7}{12} \div \frac{3}{4} \div \frac{1}{2}\)
- First, \(\frac{7}{12} \div \frac{3}{4} = \frac{7}{12} \times \frac{4}{3} = \frac{28}{36} = \frac{7}{9}\).
- Then, \(\frac{7}{9} \div \frac{1}{2} = \frac{7}{9} \times \frac{2}{1} = \frac{14}{9} = 1 \frac{5}{9}\)
- Answer: \(1 \frac{5}{9}\)

- i) \(3 \frac{1}{7} \div 5 \frac{1}{2}\)
- Convert \(3 \frac{1}{7}\) to an improper fraction: \(3 \frac{1}{7} = \frac{22}{7}\).
- Convert \(5 \frac{1}{2}\) to an improper fraction: \(5 \frac{1}{2} = \frac{11}{2}\).
- \(\frac{22}{7} \div \frac{11}{2} = \frac{22}{7} \times \frac{2}{11} = \frac{44}{77} = \frac{4}{7}\)
- Answer: \(\frac{4}{7}\)

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Section A:} \\
1. \text{Proven} \\
2. \text{a) } \frac{3}{2}, \text{ b) } \frac{7}{5}, \text{ c) } 2, \text{ d) } \frac{1}{8}, \text{ reciprocal} \\
3. \text{a) } \frac{11}{6}, \text{ b) } -\frac{3}{2}, \text{ c) } \frac{1}{5}, \text{ d) } 2, \text{ e) } \frac{19}{8}, \text{ f) } \frac{3}{14} \\
\text{Section B:} \\
1. \text{Explanation provided} \\
2. \text{a) } 6, \text{ b) } 3, \text{ c) } 15, \text{ d) } 25, \text{ e) } 16 \frac{2}{3}, \text{ f) } 9 \\
\text{Section C:} \\
1. \text{a) } 1, \text{ b) } 1 \frac{1}{3}, \text{ c) } 9 \frac{1}{3}, \text{ d) } 1 \frac{5}{7}, \text{ e) } -\frac{15}{16}, \text{ f) } 2 \frac{13}{36}, \text{ g) } 1, \text{ h) } 1 \frac{5}{9}, \text{ i) } \frac{4}{7}
\end{array}
}
\]