Direct and Inverse Proportions Class 8 Worksheets - Free Printable

Let's solve each problem step by step from the worksheet titled "Direct and Inverse Proportions".

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1. Complete the equivalent ratio.

$$
\frac{1}{3} = \frac{2}{\boxed{6}} = \frac{3}{\boxed{9}} = \frac{\boxed{4}}{12}
$$

Explanation:
We are finding equivalent fractions of $\frac{1}{3}$.

- $\frac{1}{3} = \frac{2}{6}$ → multiply numerator and denominator by 2.
- $\frac{1}{3} = \frac{3}{9}$ → multiply by 3.
- $\frac{1}{3} = \frac{4}{12}$ → multiply by 4.

✔ So, the missing values are:
→ $6$, $9$, and $4$

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2. Represent the ratio of the first number to the second in the simplest form.

#### a. $10, 49$

$$
\frac{10}{49}
$$

Check if they have common factors:
- 10 = $2 \times 5$
- 49 = $7^2$

No common factors → already in simplest form.

✔ Answer: $\boxed{10:49}$

#### b. $405, 36$

$$
\frac{405}{36}
$$

Find GCF of 405 and 36.

- Prime factorization:
- $405 = 5 \times 81 = 5 \times 3^4$
- $36 = 2^2 \times 3^2$

Common factor: $3^2 = 9$

Divide both by 9:

$$
\frac{405 \div 9}{36 \div 9} = \frac{45}{4}
$$

✔ Answer: $\boxed{45:4}$

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3. Express the following quantities as ratios.

#### a. 12 kg to 6 kg

$$
\frac{12}{6} = 2:1
$$

✔ Answer: $\boxed{2:1}$

#### b. 2 h to 90 min

Convert both to same unit (minutes):

- 2 hours = 120 minutes

So,
$$
\frac{120}{90} = \frac{4}{3} \quad \text{(divide numerator and denominator by 30)}
$$

✔ Answer: $\boxed{4:3}$

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4. Find the value of $m$ in the given proportions.

#### a. $\frac{5}{6} = \frac{30}{m}$

Cross-multiply:
$$
5m = 6 \times 30 = 180 \\
m = \frac{180}{5} = 36
$$

✔ $m = \boxed{36}$

#### b. $m : 4 :: 5 : 2$

This means:
$$
\frac{m}{4} = \frac{5}{2}
$$

Cross-multiply:
$$
2m = 20 \Rightarrow m = 10
$$

✔ $m = \boxed{10}$

#### c. $\frac{6}{9} = \frac{2}{m+1}$

Simplify left side:
$$
\frac{6}{9} = \frac{2}{3}
$$

So:
$$
\frac{2}{3} = \frac{2}{m+1}
$$

Since numerators are equal, denominators must be equal:
$$
3 = m + 1 \Rightarrow m = 2
$$

✔ $m = \boxed{2}$

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5. If $y$ varies directly as $z^2$ and $y = 4$ when $z = 5$, then find $y$ when $z = 15$.

Direct variation: $y = k z^2$

Use given values to find $k$:

$$
4 = k \cdot (5)^2 = k \cdot 25 \Rightarrow k = \frac{4}{25}
$$

Now find $y$ when $z = 15$:

$$
y = \frac{4}{25} \cdot (15)^2 = \frac{4}{25} \cdot 225 = 4 \cdot 9 = 36
$$

✔ $y = \boxed{36}$

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6. Using the scale $1\,\text{cm} = 65\,\text{km}$, find the length of the map that represents $520\,\text{km}$.

Set up proportion:

$$
\frac{1\,\text{cm}}{65\,\text{km}} = \frac{x\,\text{cm}}{520\,\text{km}}
$$

Solve for $x$:

$$
x = \frac{520}{65} = 8
$$

✔ Map length = $\boxed{8\,\text{cm}}$

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7. If $x$ varies inversely as $y$ and $y = 30$ when $x = 5$, then find $y$ when $x = 3$.

Inverse variation: $x \propto \frac{1}{y}$ → $xy = k$

Find $k$:
$$
k = x \cdot y = 5 \cdot 30 = 150
$$

Now when $x = 3$:
$$
3 \cdot y = 150 \Rightarrow y = \frac{150}{3} = 50
$$

✔ $y = \boxed{50}$

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✔ Final Answers Summary:

1. $\frac{1}{3} = \frac{2}{6} = \frac{3}{9} = \frac{4}{12}$
2. a. $10:49$ b. $45:4$
3. a. $2:1$ b. $4:3$
4. a. $m = 36$ b. $m = 10$ c. $m = 2$
5. $y = 36$
6. $8\,\text{cm}$
7. $y = 50$

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