Direct and Inverse Proportions Class 8 Worksheets - Free Printable

Let’s solve each problem step by step with clear explanations.

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1. Complete the equivalent ratio.

> \(\frac{1}{3} = \frac{2}{\square} = \frac{3}{\square} = \frac{\square}{12}\)

We are finding equivalent fractions to \(\frac{1}{3}\).

- \(\frac{1}{3} = \frac{2}{x}\) → Cross-multiply: \(1 \cdot x = 3 \cdot 2\) → \(x = 6\)
- \(\frac{1}{3} = \frac{3}{x}\) → \(1 \cdot x = 3 \cdot 3\) → \(x = 9\)
- \(\frac{1}{3} = \frac{x}{12}\) → \(1 \cdot 12 = 3 \cdot x\) → \(12 = 3x\) → \(x = 4\)

✔ Answer:
\(\frac{1}{3} = \frac{2}{\boxed{6}} = \frac{3}{\boxed{9}} = \frac{\boxed{4}}{12}\)

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

> a. 10, 49
> b. 405, 36

a. 10 : 49

Check for common factors:
GCF of 10 and 49 is 1 (since 49 = 7², 10 = 2×5).
So, it's already in simplest form.

✔ Answer: 10 : 49

b. 405 : 36

Find GCF of 405 and 36.

- Prime factorization:
- 405 = 5 × 81 = 5 × 3⁴
- 36 = 6² = (2×3)² = 2² × 3²
- Common factor: 3² = 9

Divide both by 9:

- 405 ÷ 9 = 45
- 36 ÷ 9 = 4

✔ Answer: 45 : 4

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3. Express the following quantities as ratios.

> a. 12 kg to 6 kg
> b. 2 h to 90 min

a. 12 kg to 6 kg

Same units → just write as ratio and simplify:

12 : 6 → divide both by 6 → 2 : 1

✔ Answer: 2 : 1

b. 2 h to 90 min

Convert to same unit. Let’s convert hours to minutes.

2 hours = 2 × 60 = 120 minutes

So, ratio = 120 min : 90 min

Simplify by dividing both by 30:

120 ÷ 30 = 4
90 ÷ 30 = 3

✔ Answer: 4 : 3

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4. Find the value of \(m\) in the given proportions.

> a. \(\frac{5}{6} = \frac{30}{m}\)
> b. \(m : 4 :: 5 : 2\)
> c. \(\frac{6}{9} = \frac{2}{m + 1}\)

a. \(\frac{5}{6} = \frac{30}{m}\)

Cross-multiply:

\(5 \cdot m = 6 \cdot 30\)
\(5m = 180\)
\(m = 36\)

✔ Answer: 36

b. \(m : 4 :: 5 : 2\)

This means \(\frac{m}{4} = \frac{5}{2}\)

Cross-multiply:

\(2m = 4 \cdot 5 = 20\)
\(m = 10\)

✔ Answer: 10

c. \(\frac{6}{9} = \frac{2}{m + 1}\)

Simplify \(\frac{6}{9} = \frac{2}{3}\), so:

\(\frac{2}{3} = \frac{2}{m + 1}\)

Since numerators are equal, denominators must be equal:

\(3 = m + 1\)
→ \(m = 2\)

✔ Answer: 2

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5. If y varies directly as z² and y = 4 when z = 5, then find y when z = 15.

Direct variation: \(y = k \cdot z^2\)

Given: When \(z = 5\), \(y = 4\)

Find \(k\):

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

Now, find \(y\) when \(z = 15\):

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

\(225 ÷ 25 = 9\), so \(4 × 9 = 36\)

✔ Answer: 36

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6. Using the scale 1 cm = 65 km, find the length of the map that represents 520 km.

Scale: 1 cm → 65 km

So, to find how many cm represent 520 km:

\(\text{Length on map} = \frac{520}{65} = 8\)

✔ Answer: 8 cm

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7. If x varies inversely as y and y = 30 when x = 5, then the value of y when x = 3 is:

Inverse variation: \(x \cdot y = k\) (constant)

Given: x = 5, y = 30 → \(k = 5 × 30 = 150\)

Now, when x = 3:

\(3 \cdot y = 150\)
→ \(y = \frac{150}{3} = 50\)

✔ Answer: 50

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✔ Final Answers Summary:

1. \(\frac{1}{3} = \frac{2}{\boxed{6}} = \frac{3}{\boxed{9}} = \frac{\boxed{4}}{12}\)
2. a. 10 : 49; b. 45 : 4
3. a. 2 : 1; b. 4 : 3
4. a. 36; b. 10; c. 2
5. 36
6. 8 cm
7. 50

Let me know if you’d like these explained further or need help with more problems!