Rational Numbers worksheet for Grade VII students with five math problems on representing, equivalent forms, absolute values, and ordering rational numbers.

Let's solve each problem step by step from the worksheet on Rational Numbers.

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1. Represent the following rational numbers on a number line:

$$
\frac{-4}{6}, \frac{-2}{6}, 0, \frac{3}{6}
$$

#### Step-by-step:

First, simplify the fractions:
- $\frac{-4}{6} = \frac{-2}{3} \approx -0.67$
- $\frac{-2}{6} = \frac{-1}{3} \approx -0.33$
- $0 = 0$
- $\frac{3}{6} = \frac{1}{2} = 0.5$

Now, plot these on a number line:

```
<---|-----|-----|-----|-----|-----|-----|-----|--->
-1 -0.75 -0.5 -0.25 0 0.25 0.5 1
(-4/6) (-2/6) (0) (3/6)
```

So, mark the points at approximately:
- $-0.67$ → $\frac{-4}{6}$
- $-0.33$ → $\frac{-2}{6}$
- $0$ → $0$
- $0.5$ → $\frac{3}{6}$

> ✔ Answer: Plot these values on a number line in order:
> $-0.67$, $-0.33$, $0$, $0.5$

---

2. Find four equivalent rational numbers of $\frac{5}{6}$

To find equivalent rational numbers, multiply numerator and denominator by the same non-zero integer.

#### Multiply by 2:
$$
\frac{5 \times 2}{6 \times 2} = \frac{10}{12}
$$

#### Multiply by 3:
$$
\frac{5 \times 3}{6 \times 3} = \frac{15}{18}
$$

#### Multiply by 4:
$$
\frac{5 \times 4}{6 \times 4} = \frac{20}{24}
$$

#### Multiply by 5:
$$
\frac{5 \times 5}{6 \times 5} = \frac{25}{30}
$$

> ✔ Answer: Four equivalent rational numbers are:
> $$
> \frac{10}{12}, \frac{15}{18}, \frac{20}{24}, \frac{25}{30}
> $$

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3. Fill in the box to make an equivalent rational number:

#### a. $\frac{3}{4} = \frac{\square}{-24}$

We need to find a number such that when the denominator is $-24$, the fraction equals $\frac{3}{4}$.

Set up proportion:
$$
\frac{3}{4} = \frac{x}{-24}
$$

Cross-multiply:
$$
3 \times (-24) = 4x \Rightarrow -72 = 4x \Rightarrow x = -18
$$

✔ So, $\frac{3}{4} = \frac{-18}{-24}$

> ✔ Answer: -18

---

#### b. $\frac{8}{24} = \frac{\square}{21}$

Simplify $\frac{8}{24} = \frac{1}{3}$

Now find $x$ such that:
$$
\frac{1}{3} = \frac{x}{21}
\Rightarrow x = \frac{21}{3} = 7
$$

✔ So, $\frac{8}{24} = \frac{7}{21}$

> ✔ Answer: 7

---

#### c. $\frac{36}{63} = \frac{-4}{\square}$

Simplify $\frac{36}{63}$:
Divide numerator and denominator by 9:
$$
\frac{36 \div 9}{63 \div 9} = \frac{4}{7}
$$

So $\frac{36}{63} = \frac{4}{7}$

But we want it equal to $\frac{-4}{x}$

Note: $\frac{4}{7} = \frac{-4}{-7}$

So $x = -7$

✔ So, $\frac{36}{63} = \frac{-4}{-7}$

> ✔ Answer: -7

---

4. Find the absolute value of the following:

The absolute value $|x|$ is the distance of $x$ from 0 on the number line, so it's always non-negative.

#### a. $|-25|$

Distance from 0 is 25 → $|-25| = 25$

✔ Answer: 25

#### b. $|71|$

Already positive → $|71| = 71$

✔ Answer: 71

---

5. Arrange the following rational numbers in descending form:

$$
\frac{3}{2}, \frac{1}{6}, \frac{-4}{3}, \frac{6}{-1}, \frac{5}{7}
$$

First, simplify and convert all to decimals for easy comparison:

- $\frac{3}{2} = 1.5$
- $\frac{1}{6} \approx 0.1667$
- $\frac{-4}{3} \approx -1.333$
- $\frac{6}{-1} = -6$
- $\frac{5}{7} \approx 0.7143$

Now list them in descending order (largest to smallest):

$$
1.5 > 0.7143 > 0.1667 > -1.333 > -6
$$

So:
$$
\frac{3}{2} > \frac{5}{7} > \frac{1}{6} > \frac{-4}{3} > \frac{6}{-1}
$$

> ✔ Final Answer (Descending Order):
> $$
> \frac{3}{2}, \frac{5}{7}, \frac{1}{6}, \frac{-4}{3}, \frac{6}{-1}
> $$

---

✔ Summary of Answers:

#### 1. Number Line:
Plot: $-\frac{4}{6} \approx -0.67$, $-\frac{2}{6} \approx -0.33$, $0$, $\frac{3}{6} = 0.5$

#### 2. Equivalent Rational Numbers of $\frac{5}{6}$:
$$
\frac{10}{12}, \frac{15}{18}, \frac{20}{24}, \frac{25}{30}
$$

#### 3. Fill in the boxes:
a. $-18$
b. $7$
c. $-7$

#### 4. Absolute Values:
a. $25$
b. $71$

#### 5. Descending Order:
$$
\frac{3}{2}, \frac{5}{7}, \frac{1}{6}, \frac{-4}{3}, \frac{6}{-1}
$$

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