Multiplying mixed numbers and fractions worksheet for Grade 6 students.

Problem: Multiplying Mixed Numbers and Fractions

The task is to solve the multiplication problems involving mixed numbers and fractions. Here are the steps to solve each problem:

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#### Step 1: Convert Mixed Numbers to Improper Fractions
- A mixed number is of the form \( a \frac{b}{c} \), where \( a \) is the whole number, and \( \frac{b}{c} \) is the fractional part.
- To convert a mixed number to an improper fraction:
\[
a \frac{b}{c} = \frac{(a \times c) + b}{c}
\]

#### Step 2: Multiply the Fractions
- To multiply two fractions, multiply the numerators together and the denominators together:
\[
\frac{p}{q} \times \frac{r}{s} = \frac{p \times r}{q \times s}
\]

#### Step 3: Simplify the Result
- Reduce the resulting fraction to its simplest form by dividing the numerator and denominator by their greatest common divisor (GCD).

#### Step 4: Convert Back to a Mixed Number (if necessary)
- If the result is an improper fraction, convert it back to a mixed number:
\[
\frac{m}{n} = q \frac{r}{n}, \text{ where } q = \left\lfloor \frac{m}{n} \right\rfloor \text{ and } r = m \mod n
\]

---

Solutions to Each Problem

#### Problem 1: \( 6 \frac{5}{9} \times \frac{5}{12} \)
1. Convert \( 6 \frac{5}{9} \) to an improper fraction:
\[
6 \frac{5}{9} = \frac{(6 \times 9) + 5}{9} = \frac{54 + 5}{9} = \frac{59}{9}
\]
2. Multiply the fractions:
\[
\frac{59}{9} \times \frac{5}{12} = \frac{59 \times 5}{9 \times 12} = \frac{295}{108}
\]
3. Simplify (if possible):
- The GCD of 295 and 108 is 1, so the fraction is already in simplest form.
4. Convert to a mixed number:
\[
\frac{295}{108} = 2 \frac{79}{108}
\]
Answer: \( 2 \frac{79}{108} \)

---

#### Problem 2: \( 5 \frac{1}{9} \times \frac{5}{8} \)
1. Convert \( 5 \frac{1}{9} \) to an improper fraction:
\[
5 \frac{1}{9} = \frac{(5 \times 9) + 1}{9} = \frac{45 + 1}{9} = \frac{46}{9}
\]
2. Multiply the fractions:
\[
\frac{46}{9} \times \frac{5}{8} = \frac{46 \times 5}{9 \times 8} = \frac{230}{72}
\]
3. Simplify:
- The GCD of 230 and 72 is 2:
\[
\frac{230 \div 2}{72 \div 2} = \frac{115}{36}
\]
4. Convert to a mixed number:
\[
\frac{115}{36} = 3 \frac{7}{36}
\]
Answer: \( 3 \frac{7}{36} \)

---

#### Problem 3: \( 10 \frac{4}{9} \times \frac{1}{2} \)
1. Convert \( 10 \frac{4}{9} \) to an improper fraction:
\[
10 \frac{4}{9} = \frac{(10 \times 9) + 4}{9} = \frac{90 + 4}{9} = \frac{94}{9}
\]
2. Multiply the fractions:
\[
\frac{94}{9} \times \frac{1}{2} = \frac{94 \times 1}{9 \times 2} = \frac{94}{18}
\]
3. Simplify:
- The GCD of 94 and 18 is 2:
\[
\frac{94 \div 2}{18 \div 2} = \frac{47}{9}
\]
4. Convert to a mixed number:
\[
\frac{47}{9} = 5 \frac{2}{9}
\]
Answer: \( 5 \frac{2}{9} \)

---

#### Problem 4: \( 2 \frac{5}{8} \times \frac{4}{7} \)
1. Convert \( 2 \frac{5}{8} \) to an improper fraction:
\[
2 \frac{5}{8} = \frac{(2 \times 8) + 5}{8} = \frac{16 + 5}{8} = \frac{21}{8}
\]
2. Multiply the fractions:
\[
\frac{21}{8} \times \frac{4}{7} = \frac{21 \times 4}{8 \times 7} = \frac{84}{56}
\]
3. Simplify:
- The GCD of 84 and 56 is 28:
\[
\frac{84 \div 28}{56 \div 28} = \frac{3}{2}
\]
4. Convert to a mixed number:
\[
\frac{3}{2} = 1 \frac{1}{2}
\]
Answer: \( 1 \frac{1}{2} \)

---

#### Problem 5: \( 4 \frac{1}{3} \times \frac{3}{4} \)
1. Convert \( 4 \frac{1}{3} \) to an improper fraction:
\[
4 \frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}
\]
2. Multiply the fractions:
\[
\frac{13}{3} \times \frac{3}{4} = \frac{13 \times 3}{3 \times 4} = \frac{39}{12}
\]
3. Simplify:
- The GCD of 39 and 12 is 3:
\[
\frac{39 \div 3}{12 \div 3} = \frac{13}{4}
\]
4. Convert to a mixed number:
\[
\frac{13}{4} = 3 \frac{1}{4}
\]
Answer: \( 3 \frac{1}{4} \)

---

#### Problem 6: \( 9 \frac{1}{3} \times \frac{1}{3} \)
1. Convert \( 9 \frac{1}{3} \) to an improper fraction:
\[
9 \frac{1}{3} = \frac{(9 \times 3) + 1}{3} = \frac{27 + 1}{3} = \frac{28}{3}
\]
2. Multiply the fractions:
\[
\frac{28}{3} \times \frac{1}{3} = \frac{28 \times 1}{3 \times 3} = \frac{28}{9}
\]
3. Simplify (already in simplest form).
4. Convert to a mixed number:
\[
\frac{28}{9} = 3 \frac{1}{9}
\]
Answer: \( 3 \frac{1}{9} \)

---

#### Problem 7: \( 1 \frac{6}{11} \times \frac{1}{6} \)
1. Convert \( 1 \frac{6}{11} \) to an improper fraction:
\[
1 \frac{6}{11} = \frac{(1 \times 11) + 6}{11} = \frac{11 + 6}{11} = \frac{17}{11}
\]
2. Multiply the fractions:
\[
\frac{17}{11} \times \frac{1}{6} = \frac{17 \times 1}{11 \times 6} = \frac{17}{66}
\]
3. Simplify (already in simplest form).
4. No need to convert to a mixed number since it is a proper fraction.
Answer: \( \frac{17}{66} \)

---

#### Problem 8: \( 8 \frac{6}{8} \times \frac{6}{8} \)
1. Simplify \( 8 \frac{6}{8} \):
\[
8 \frac{6}{8} = 8 \frac{3}{4} \quad (\text{since } \frac{6}{8} = \frac{3}{4})
\]
2. Convert \( 8 \frac{3}{4} \) to an improper fraction:
\[
8 \frac{3}{4} = \frac{(8 \times 4) + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4}
\]
3. Multiply the fractions:
\[
\frac{35}{4} \times \frac{6}{8} = \frac{35 \times 6}{4 \times 8} = \frac{210}{32}
\]
4. Simplify:
- The GCD of 210 and 32 is 2:
\[
\frac{210 \div 2}{32 \div 2} = \frac{105}{16}
\]
5. Convert to a mixed number:
\[
\frac{105}{16} = 6 \frac{9}{16}
\]
Answer: \( 6 \frac{9}{16} \)

---

#### Problem 9: \( 9 \frac{3}{8} \times \frac{1}{7} \)
1. Convert \( 9 \frac{3}{8} \) to an improper fraction:
\[
9 \frac{3}{8} = \frac{(9 \times 8) + 3}{8} = \frac{72 + 3}{8} = \frac{75}{8}
\]
2. Multiply the fractions:
\[
\frac{75}{8} \times \frac{1}{7} = \frac{75 \times 1}{8 \times 7} = \frac{75}{56}
\]
3. Simplify (already in simplest form).
4. Convert to a mixed number:
\[
\frac{75}{56} = 1 \frac{19}{56}
\]
Answer: \( 1 \frac{19}{56} \)

---

#### Problem 10: \( 7 \frac{3}{8} \times \frac{3}{10} \)
1. Convert \( 7 \frac{3}{8} \) to an improper fraction:
\[
7 \frac{3}{8} = \frac{(7 \times 8) + 3}{8} = \frac{56 + 3}{8} = \frac{59}{8}
\]
2. Multiply the fractions:
\[
\frac{59}{8} \times \frac{3}{10} = \frac{59 \times 3}{8 \times 10} = \frac{177}{80}
\]
3. Simplify (already in simplest form).
4. Convert to a mixed number:
\[
\frac{177}{80} = 2 \frac{17}{80}
\]
Answer: \( 2 \frac{17}{80} \)

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Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 2 \frac{79}{108} \\
2. & \ 3 \frac{7}{36} \\
3. & \ 5 \frac{2}{9} \\
4. & \ 1 \frac{1}{2} \\
5. & \ 3 \frac{1}{4} \\
6. & \ 3 \frac{1}{9} \\
7. & \ \frac{17}{66} \\
8. & \ 6 \frac{9}{16} \\
9. & \ 1 \frac{19}{56} \\
10. & \ 2 \frac{17}{80}
\end{aligned}
}
\]