RD Sharma Solutions for Class 8 Chapter 1 - Rational Numbers ... - Free Printable

The image you've uploaded contains solutions from RD Sharma Class 8 Maths, Chapter 1 – Rational Numbers, specifically focusing on the distributive property of multiplication over addition and subtraction for rational numbers.

Let’s go through each problem carefully, explain the solution steps, and verify whether they are correct.

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Problem (iv): Verification of Distributive Property

Given:
$$
x = -\frac{3}{4}, \quad y = -\frac{5}{2}, \quad z = \frac{7}{6}
$$

We need to verify:
$$
x \times (y + z) = x \times y + x \times z
$$

#### Left-hand side (LHS):
$$
x \times (y + z) = -\frac{3}{4} \times \left(-\frac{5}{2} + \frac{7}{6}\right)
$$

First, compute $ y + z $:
$$
-\frac{5}{2} + \frac{7}{6} = \frac{-15 + 7}{6} = \frac{-8}{6} = -\frac{4}{3}
$$

Now multiply:
$$
-\frac{3}{4} \times -\frac{4}{3} = \frac{12}{12} = 1
$$

So, LHS = 1

#### Right-hand side (RHS):
$$
x \times y + x \times z = \left(-\frac{3}{4} \times -\frac{5}{2}\right) + \left(-\frac{3}{4} \times \frac{7}{6}\right)
$$

Compute each term:
- $ -\frac{3}{4} \times -\frac{5}{2} = \frac{15}{8} $
- $ -\frac{3}{4} \times \frac{7}{6} = -\frac{21}{24} = -\frac{7}{8} $

Now add:
$$
\frac{15}{8} - \frac{7}{8} = \frac{8}{8} = 1
$$

So, RHS = 1

✔ Hence, LHS = RHS, so the distributive property is verified.

> ✔ Correct!

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Question 4: Use distributivity to simplify

#### (i) $ \frac{3}{5} \times \left(\frac{35}{24} + \frac{10}{1}\right) $

Apply distributivity:
$$
= \frac{3}{5} \times \frac{35}{24} + \frac{3}{5} \times 10
$$

Calculate each:
- $ \frac{3}{5} \times \frac{35}{24} = \frac{105}{120} = \frac{7}{8} $
- $ \frac{3}{5} \times 10 = \frac{30}{5} = 6 $

Now add:
$$
\frac{7}{8} + 6 = \frac{7}{8} + \frac{48}{8} = \frac{55}{8}
$$

✔ So, answer is $ \boxed{\frac{55}{8}} $

> ✔ Correct!

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#### (ii) $ -\frac{5}{4} \times \left(\frac{8}{5} + \frac{16}{5}\right) $

Apply distributivity:
$$
= -\frac{5}{4} \times \frac{8}{5} + (-\frac{5}{4}) \times \frac{16}{5}
$$

Simplify:
- $ -\frac{5}{4} \times \frac{8}{5} = -\frac{40}{20} = -2 $
- $ -\frac{5}{4} \times \frac{16}{5} = -\frac{80}{20} = -4 $

Add:
$$
-2 + (-4) = -6
$$

✔ Answer: $ \boxed{-6} $

> ✔ Correct!

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#### (iii) $ \frac{2}{7} \times \left(\frac{7}{16} - \frac{21}{4}\right) $

Apply distributivity:
$$
= \frac{2}{7} \times \frac{7}{16} - \frac{2}{7} \times \frac{21}{4}
$$

Simplify:
- $ \frac{2}{7} \times \frac{7}{16} = \frac{14}{112} = \frac{1}{8} $
- $ \frac{2}{7} \times \frac{21}{4} = \frac{42}{28} = \frac{3}{2} $

Now subtract:
$$
\frac{1}{8} - \frac{3}{2} = \frac{1}{8} - \frac{12}{8} = -\frac{11}{8}
$$

But in the given solution, it says:
> $ \frac{1}{8} - \frac{3}{2} = \frac{1 \times 1 - 3 \times 4}{8} = \frac{1 - 12}{8} = -\frac{11}{8} $

Wait — there's a typo in the original solution!

It says:
> $ \frac{1}{8} - \frac{3}{2} = (1×1 - 3×4)/8 $

This is incorrect reasoning. It should be:

$$
\frac{1}{8} - \frac{3}{2} = \frac{1}{8} - \frac{12}{8} = \frac{-11}{8}
$$

But the way they wrote $ (1×1 - 3×4)/8 $ suggests they did $ \frac{1}{8} - \frac{3}{2} = \frac{1 - 12}{8} $, which is correct numerically, but the explanation is flawed.

Actually, $ \frac{3}{2} = \frac{12}{8} $, so yes, $ \frac{1}{8} - \frac{12}{8} = -\frac{11}{8} $

So the final answer is correct, but the intermediate step has a misleading notation.

✔ Final answer: $ \boxed{-\frac{11}{8}} $

> ✘ Incorrect reasoning in steps, but ✔ correct final answer

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✔ Summary of Corrections & Explanations

| Problem | Correctness | Notes |
|--------|-------------|-------|
| (iv) Verification | ✔ Correct | Well done. LHS = RHS = 1 |
| 4(i) | ✔ Correct | $ \frac{55}{8} $ |
| 4(ii) | ✔ Correct | $ -6 $ |
| 4(iii) | ⚠️ Partially Incorrect | Final answer $ -\frac{11}{8} $ is correct, but the simplification step shows $ \frac{1}{8} - \frac{3}{2} = (1×1 - 3×4)/8 $, which is not standard. Should be: convert $ \frac{3}{2} $ to $ \frac{12}{8} $, then subtract. |

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✔ Final Answers:

1. (iv) Verified: $ x(y+z) = xy + xz $ → Both sides = 1 ✔️
2. (i) $ \frac{3}{5} \times \left(\frac{35}{24} + 10\right) = \frac{55}{8} $ ✔️
3. (ii) $ -\frac{5}{4} \times \left(\frac{8}{5} + \frac{16}{5}\right) = -6 $ ✔️
4. (iii) $ \frac{2}{7} \times \left(\frac{7}{16} - \frac{21}{4}\right) = -\frac{11}{8} $ ✔️ (Answer correct, but method slightly misleading)

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📌 Key Concept: Distributive Property

For rational numbers:
$$
a \times (b + c) = a \times b + a \times c
$$
and
$$
a \times (b - c) = a \times b - a \times c
$$

This property allows us to simplify expressions by distributing multiplication across addition or subtraction.

Always remember:
- Convert fractions to like denominators when adding/subtracting.
- Simplify each multiplication step carefully.
- Check signs carefully (negative × negative = positive).

Let me know if you'd like this explained with diagrams or visual examples!