Class 8 Maths Rational Numbers Worksheet - Free Printable

Absolutely! Let’s solve each problem step by step and explain the reasoning clearly.

---

1. Fill in the boxes:

> $\frac{6}{8} = \frac{-18}{\square} = \frac{30}{\square}$

We need to find two missing denominators such that all three fractions are equivalent.

First, simplify $\frac{6}{8}$:

$$
\frac{6}{8} = \frac{3}{4}
$$

So we want:

$$
\frac{-18}{x} = \frac{3}{4} \quad \text{and} \quad \frac{30}{y} = \frac{3}{4}
$$

---

Solve for first box ($\frac{-18}{x} = \frac{3}{4}$):

Cross-multiply:

$$
-18 \cdot 4 = 3 \cdot x \\
-72 = 3x \\
x = -24
$$

✔ First box: -24

---

Solve for second box ($\frac{30}{y} = \frac{3}{4}$):

Cross-multiply:

$$
30 \cdot 4 = 3 \cdot y \\
120 = 3y \\
y = 40
$$

✔ Second box: 40

---

Final Answer for #1:

> $\frac{6}{8} = \frac{-18}{\boxed{-24}} = \frac{30}{\boxed{40}}$

---

2. Locate the points on a number line:

> $\frac{3}{6}, -\frac{1}{6}, 1, -\frac{5}{6}, 0$

First, simplify where possible:

- $\frac{3}{6} = \frac{1}{2} = 0.5$
- $-\frac{1}{6} \approx -0.166...$
- $1 = 1.0$
- $-\frac{5}{6} \approx -0.833...$
- $0 = 0$

Now, arrange them from least to greatest:

$$
-\frac{5}{6} < -\frac{1}{6} < 0 < \frac{3}{6} < 1
$$

On a number line, mark these positions approximately:

```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->

-1 -5/6 -1/6 0 1/2 1
```

You would place dots at:
- $-\frac{5}{6}$ (just left of -0.8)
- $-\frac{1}{6}$ (just left of 0)
- $0$
- $\frac{1}{2}$ (midway between 0 and 1)
- $1$

✔ You can sketch this on paper or visualize it as above.

---

3. Determine which of the two rational numbers is greater:

> $-\frac{7}{12}, -\frac{8}{23}$

Both are negative. The *greater* number is the one *closer to zero* (less negative).

Compare their absolute values:

- $|\frac{7}{12}| = \frac{7}{12} \approx 0.583$
- $|\frac{8}{23}| \approx 0.348$

Since $0.348 < 0.583$, then $-\frac{8}{23}$ is *closer to zero* than $-\frac{7}{12}$.

Therefore:

✔ $-\frac{8}{23} > -\frac{7}{12}$

---

4. Arrange the following in descending order:

> $-\frac{5}{4}, \frac{2}{3}, -\frac{7}{10}, \frac{4}{5}$

Descending order = from largest to smallest.

Convert to decimals for easier comparison:

- $-\frac{5}{4} = -1.25$
- $\frac{2}{3} \approx 0.666...$
- $-\frac{7}{10} = -0.7$
- $\frac{4}{5} = 0.8$

Now order from largest to smallest:

$$
0.8 > 0.666... > -0.7 > -1.25
$$

Which corresponds to:

✔ $\frac{4}{5} > \frac{2}{3} > -\frac{7}{10} > -\frac{5}{4}$

---

5. Verify that $a + (b + c) = (a + b) + c$ using:

> $a = \frac{1}{2}, b = \frac{3}{4}, c = -\frac{2}{3}$

This is verifying the Associative Property of Addition.

---

Left Side: $a + (b + c)$

First compute $b + c$:

$$
\frac{3}{4} + (-\frac{2}{3}) = \frac{3}{4} - \frac{2}{3}
$$

Find common denominator (LCM of 4 and 3 is 12):

$$
= \frac{9}{12} - \frac{8}{12} = \frac{1}{12}
$$

Now add $a$:

$$
\frac{1}{2} + \frac{1}{12} = \frac{6}{12} + \frac{1}{12} = \frac{7}{12}
$$

✔ Left side = $\frac{7}{12}$

---

Right Side: $(a + b) + c$

First compute $a + b$:

$$
\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}
$$

Now add $c$:

$$
\frac{5}{4} + (-\frac{2}{3}) = \frac{5}{4} - \frac{2}{3}
$$

Common denominator (LCM of 4 and 3 is 12):

$$
= \frac{15}{12} - \frac{8}{12} = \frac{7}{12}
$$

✔ Right side = $\frac{7}{12}$

---

Conclusion: Both sides equal $\frac{7}{12}$ → Associative property verified.

✔ Verified: $a + (b + c) = (a + b) + c$

---

6. Simplify:

> $\frac{-4}{11} + \frac{-2}{3} - \left(\frac{-5}{9}\right)$

First, simplify the expression:

Note: subtracting a negative is adding a positive:

$$
= \frac{-4}{11} + \frac{-2}{3} + \frac{5}{9}
$$

Now, we need a common denominator to combine these fractions.

Denominators: 11, 3, 9

Prime factors:
- 11 → 11
- 3 → 3
- 9 → 3²

LCM = 11 × 3² = 11 × 9 = 99

Convert each fraction:

- $\frac{-4}{11} = \frac{-4 \times 9}{99} = \frac{-36}{99}$
- $\frac{-2}{3} = \frac{-2 \times 33}{99} = \frac{-66}{99}$
- $\frac{5}{9} = \frac{5 \times 11}{99} = \frac{55}{99}$

Now add:

$$
\frac{-36}{99} + \frac{-66}{99} + \frac{55}{99} = \frac{-36 -66 + 55}{99} = \frac{-47}{99}
$$

✔ Final Answer: $\boxed{\frac{-47}{99}}$

---

## ✔ Final Answers Summary:

1. $\frac{6}{8} = \frac{-18}{\boxed{-24}} = \frac{30}{\boxed{40}}$

2. Number line points: $-\frac{5}{6}, -\frac{1}{6}, 0, \frac{1}{2}, 1$ (from left to right)

3. Greater number: $\boxed{-\frac{8}{23}}$

4. Descending order: $\boxed{\frac{4}{5} > \frac{2}{3} > -\frac{7}{10} > -\frac{5}{4}}$

5. Verified: $a + (b + c) = (a + b) + c = \frac{7}{12}$

6. Simplified: $\boxed{\frac{-47}{99}}$

Let me know if you’d like a visual sketch for #2 or more explanation on any part!