Exercise 1.7: Solutions to fraction division problems.

The image contains a math exercise labeled Exercise 1.7, which involves solving a series of division problems with fractions, integers, and negative numbers. The student has provided their solutions, and we will go through each problem step by step to verify correctness and explain the solution process.

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Problem List:

We are asked to solve the following divisions:

1. $ 1 \div \frac{1}{2} $
2. $ 5 \div \left(-\frac{7}{5}\right) $
3. $ -\frac{3}{4} \div \frac{9}{16} $
4. $ -\frac{7}{8} \div \left(-\frac{21}{16}\right) $
5. $ 0 \div \left(-\frac{7}{5}\right) $
6. $ \frac{3}{4} \div (-6) $
7. $ -\frac{3}{4} \div (-6) $
8. $ \frac{8}{3} \div \left(-\frac{7}{12}\right) $
9. $ -4 \div \left(-\frac{3}{5}\right) $
10. $ -\frac{3}{13} \div \left(-\frac{4}{65}\right) $

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Let’s solve each one carefully and compare with the given solutions.

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(i) $ 1 \div \frac{1}{2} $

Solution:
Dividing by a fraction is the same as multiplying by its reciprocal:
$$
1 \div \frac{1}{2} = 1 \times \frac{2}{1} = 2
$$

✔ Correct – Matches the given answer.

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(ii) $ 5 \div \left(-\frac{7}{5}\right) $

Solution:
$$
5 \div \left(-\frac{7}{5}\right) = 5 \times \left(-\frac{5}{7}\right) = -\frac{25}{7}
$$

But the student wrote:
$$
= \frac{-5}{7} \times 7 = -7
$$

This is incorrect.

Error Analysis:
They incorrectly simplified $ 5 \times \left(-\frac{5}{7}\right) $ as $ \frac{-5}{7} \times 7 $. That's not valid.

Correct Answer:
$$
-\frac{25}{7} \quad \text{(or } -3\frac{4}{7} \text{)}
$$

✘ Incorrect – Student made a mistake here.

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(iii) $ -\frac{3}{4} \div \frac{9}{16} $

Solution:
$$
-\frac{3}{4} \div \frac{9}{16} = -\frac{3}{4} \times \frac{16}{9} = -\frac{48}{36} = -\frac{4}{3}
$$

But the student wrote:
$$
= \frac{+3}{4} \times \frac{16}{9} = \frac{4}{3}
$$

They ignored the negative sign.

✔ Sign error – Should be negative.

Correct Answer:
$$
-\frac{4}{3}
$$

✘ Incorrect – Sign missing.

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(iv) $ -\frac{7}{8} \div \left(-\frac{21}{16}\right) $

Solution:
$$
-\frac{7}{8} \div \left(-\frac{21}{16}\right) = -\frac{7}{8} \times \left(-\frac{16}{21}\right)
$$

Multiply:
$$
= \frac{7 \times 16}{8 \times 21} = \frac{112}{168} = \frac{2}{3}
$$

Note: Two negatives make a positive.

Student wrote:
$$
= \frac{-7}{8} \times \frac{16}{-21} = \frac{-2}{-3} = \frac{2}{3}
$$

✔ Correct – Even though notation is slightly off (writing both negatives), result is correct.

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(v) $ 0 \div \left(-\frac{7}{5}\right) $

Solution:
Any number divided by a non-zero number is zero:
$$
0 \div \left(-\frac{7}{5}\right) = 0
$$

✔ Correct

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(vi) $ \frac{3}{4} \div (-6) $

Solution:
$$
\frac{3}{4} \div (-6) = \frac{3}{4} \times \left(-\frac{1}{6}\right) = -\frac{3}{24} = -\frac{1}{8}
$$

Student wrote:
$$
= +\frac{3}{4} \times 6 = \frac{1}{8}
$$

Wait — they multiplied by 6 instead of $-\frac{1}{6}$, and missed the negative sign.

Correct Answer:
$$
-\frac{1}{8}
$$

✘ Incorrect – Wrong sign and wrong operation.

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(vii) $ -\frac{3}{4} \div (-6) $

Solution:
$$
-\frac{3}{4} \div (-6) = -\frac{3}{4} \times \left(-\frac{1}{6}\right) = \frac{3}{24} = \frac{1}{8}
$$

Student wrote:
$$
= -\frac{3}{4} \times 6 = \frac{1}{8}
$$

Wait — they wrote $ \times 6 $, but should be $ \times (-\frac{1}{6}) $. But since both are negative, it becomes positive.

But:
$$
-\frac{3}{4} \times 6 = -\frac{18}{4} = -4.5 \neq \frac{1}{8}
$$

So this is wrong.

Correct:
$$
-\frac{3}{4} \div (-6) = \frac{3}{4} \times \frac{1}{6} = \frac{3}{24} = \frac{1}{8}
$$

✔ Answer is correct, but working is wrong.

Student said:
$$
= -\frac{3}{4} \times 6 = \frac{1}{8}
$$

That’s invalid because $ -\frac{3}{4} \times 6 = -\frac{18}{4} = -4.5 $

So the final answer is correct, but the method is incorrect.

⚠️ Partially correct – Answer is right, but reasoning is flawed.

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(viii) $ \frac{8}{3} \div \left(-\frac{7}{12}\right) $

Solution:
$$
\frac{8}{3} \div \left(-\frac{7}{12}\right) = \frac{8}{3} \times \left(-\frac{12}{7}\right) = -\frac{96}{21} = -\frac{32}{7}
$$

Student wrote:
$$
= \frac{8}{3} \times \frac{12}{-7} = \frac{-96}{21} = -\frac{8}{7}
$$

Wait — $ \frac{-96}{21} $ simplifies to:
$$
\frac{-96 \div 3}{21 \div 3} = \frac{-32}{7}
$$

But student says $ -\frac{8}{7} $ — that’s wrong.

Correct: $ -\frac{32}{7} $
Student: $ -\frac{8}{7} $ → ✘ Incorrect

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(ix) $ -4 \div \left(-\frac{3}{5}\right) $

Solution:
$$
-4 \div \left(-\frac{3}{5}\right) = -4 \times \left(-\frac{5}{3}\right) = \frac{20}{3}
$$

Student wrote:
$$
= -4 \times \frac{5}{-3} = \frac{20}{3}
$$

Wait — $ -4 \times \frac{5}{-3} = \frac{-20}{-3} = \frac{20}{3} $

Yes, correct.

✔ Correct

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(x) $ -\frac{3}{13} \div \left(-\frac{4}{65}\right) $

Solution:
$$
-\frac{3}{13} \div \left(-\frac{4}{65}\right) = -\frac{3}{13} \times \left(-\frac{65}{4}\right) = \frac{3 \times 65}{13 \times 4}
$$

Simplify:
- $ 65 \div 13 = 5 $
- So: $ \frac{3 \times 5}{4} = \frac{15}{4} $

Student wrote:
$$
= +\frac{3}{13} \times \frac{65}{4} = \frac{195}{52} = \frac{15}{4}
$$

✔ Correct – even though they dropped the negatives early, final answer is right.

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✔ Final Summary of Corrections:

| Problem | Student's Answer | Correct Answer | Status |
|--------|------------------|----------------|--------|
| (i) | 2 | 2 | ✔ Correct |
| (ii) | -7 | $-\frac{25}{7}$ | ✘ Incorrect |
| (iii) | $\frac{4}{3}$ | $-\frac{4}{3}$ | ✘ Incorrect (sign error) |
| (iv) | $\frac{2}{3}$ | $\frac{2}{3}$ | ✔ Correct |
| (v) | 0 | 0 | ✔ Correct |
| (vi) | $\frac{1}{8}$ | $-\frac{1}{8}$ | ✘ Incorrect (sign) |
| (vii) | $\frac{1}{8}$ | $\frac{1}{8}$ | ⚠️ Answer correct, method wrong |
| (viii) | $-\frac{8}{7}$ | $-\frac{32}{7}$ | ✘ Incorrect |
| (ix) | $\frac{20}{3}$ | $\frac{20}{3}$ | ✔ Correct |
| (x) | $\frac{15}{4}$ | $\frac{15}{4}$ | ✔ Correct |

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🔍 Key Errors Identified:

1. Ignoring signs: Especially in (ii), (iii), (vi).
2. Wrong reciprocals: In (vi) and (vii), multiplying by the divisor instead of its reciprocal.
3. Improper simplification: In (viii), $ \frac{-96}{21} $ reduces to $ -\frac{32}{7} $, not $ -\frac{8}{7} $.

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✔ Corrected Solutions:

1. $ 1 \div \frac{1}{2} = 2 $
2. $ 5 \div \left(-\frac{7}{5}\right) = -\frac{25}{7} $
3. $ -\frac{3}{4} \div \frac{9}{16} = -\frac{4}{3} $
4. $ -\frac{7}{8} \div \left(-\frac{21}{16}\right) = \frac{2}{3} $
5. $ 0 \div \left(-\frac{7}{5}\right) = 0 $
6. $ \frac{3}{4} \div (-6) = -\frac{1}{8} $
7. $ -\frac{3}{4} \div (-6) = \frac{1}{8} $
8. $ \frac{8}{3} \div \left(-\frac{7}{12}\right) = -\frac{32}{7} $
9. $ -4 \div \left(-\frac{3}{5}\right) = \frac{20}{3} $
10. $ -\frac{3}{13} \div \left(-\frac{4}{65}\right) = \frac{15}{4} $

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📌 Final Advice:

Always remember:
- Division by a fraction = multiply by reciprocal.
- Keep track of signs: negative ÷ negative = positive; positive ÷ negative = negative.
- Simplify before multiplying.

Let me know if you'd like these steps explained visually or with diagrams!