Class 8 Maths Direct and Inverse Proportion Worksheet - Free Printable

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

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1. Which of the following are in Direct or Inverse proportion?

#### (a)
Given:
| x | 4 | 5 | 6 |
|---|---|---|---|
| y |10 |12 |14 |

Check if $ \frac{y}{x} $ is constant (Direct Proportion) or $ x \cdot y $ is constant (Inverse Proportion).

- $ \frac{10}{4} = 2.5 $
- $ \frac{12}{5} = 2.4 $
- $ \frac{14}{6} \approx 2.33 $

Not constant → Not direct proportion.

Now check product $ x \cdot y $:
- $ 4 \times 10 = 40 $
- $ 5 \times 12 = 60 $
- $ 6 \times 14 = 84 $

Not constant → Not inverse proportion.

✔ Answer: Neither direct nor inverse proportion

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#### (b)
| x | 4 | 5 | 6 |
|---|---|---|---|
| y |12 |10 | 8 |

Check $ \frac{y}{x} $:
- $ \frac{12}{4} = 3 $
- $ \frac{10}{5} = 2 $
- $ \frac{8}{6} \approx 1.33 $

Not constant → Not direct.

Check $ x \cdot y $:
- $ 4 \times 12 = 48 $
- $ 5 \times 10 = 50 $
- $ 6 \times 8 = 48 $

Not constant → Not inverse.

But notice: as $ x $ increases, $ y $ decreases, but not consistently.

Wait — let’s look more closely:

$ x $: 4 → 5 → 6
$ y $: 12 → 10 → 8

So $ y $ decreases as $ x $ increases.

Try checking $ x \cdot y $ again:
- 4×12 = 48
- 5×10 = 50
- 6×8 = 48

Not constant → Not inverse.

So again, neither.

But wait — maybe it's close to inverse?

No, since products aren't equal.

✔ Answer: Neither direct nor inverse proportion

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#### (c) Distance travelled and petrol consumed by a car

As distance increases, petrol consumed increases (assuming same efficiency). So they increase together.

→ Direct proportion ✔

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#### (d) Area of a piece of land and price of land

Larger area → higher price (generally). So directly related.

→ Direct proportion ✔

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#### (e) Speed of a car and time taken to cover a fixed distance

For fixed distance:
$ \text{Speed} \times \text{Time} = \text{Distance} $ (constant)

So speed ∝ $ \frac{1}{\text{time}} $

→ Inverse proportion ✔

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✔ Summary for Q1:

- (a): Neither
- (b): Neither
- (c): Direct
- (d): Direct
- (e): Inverse

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2. Find the value of $ x $ and $ y $ if $ x : y = 2 : 3 $ and $ 2 : x = 1 : 2 $

We have two ratios:

1. $ x : y = 2 : 3 $ → $ \frac{x}{y} = \frac{2}{3} $ → $ y = \frac{3}{2}x $

2. $ 2 : x = 1 : 2 $ → $ \frac{2}{x} = \frac{1}{2} $

Cross-multiply:
$ 2 \times 2 = 1 \times x $ → $ x = 4 $

Now use $ y = \frac{3}{2}x = \frac{3}{2} \times 4 = 6 $

✔ Answer: $ x = 4 $, $ y = 6 $

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3. A car is moving at a uniform speed of 45 km/h

(a) How far will it travel in 5 hours?

Use: $ \text{Distance} = \text{Speed} \times \text{Time} $

$ = 45 \times 5 = 225 $ km

✔ Answer: 225 km

(b) Find the distance covered in 40 minutes

Convert 40 min to hours: $ \frac{40}{60} = \frac{2}{3} $ hours

Distance = $ 45 \times \frac{2}{3} = 30 $ km

✔ Answer: 30 km

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4. In a students' hostel with 32 students, food lasts 45 days. If 16 more students join, how many days will the food last?

This is an inverse proportion problem: more students → less time food lasts.

Total food = number of students × days

Initial total food = $ 32 \times 45 = 1440 $ student-days

New number of students = $ 32 + 16 = 48 $

Let $ d $ be number of days food lasts:

$ 48 \times d = 1440 $

$ d = \frac{1440}{48} = 30 $

✔ Answer: 30 days

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5. A gardener can plant 30 plants in 2 hours. How many more gardeners should be appointed so that 150 plants can be planted in 2 hours?

One gardener plants 30 plants in 2 hours.

So, rate per gardener = 30 plants / 2 h = 15 plants/hour

In 2 hours, one gardener plants 30 plants.

To plant 150 plants in 2 hours, number of gardeners needed:

$ \frac{150}{30} = 5 $ gardeners

Currently 1 gardener → need $ 5 - 1 = 4 $ more

✔ Answer: 4 more gardeners

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6. If $ 2 : 3 = x : 51 $, find $ x $

Cross-multiply:

$ 2 \times 51 = 3 \times x $

$ 102 = 3x $

$ x = \frac{102}{3} = 34 $

✔ Answer: $ x = 34 $

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7. If $ x $ and $ y $ are in inverse proportion. Find values of $ a, b, c $ in the table:

| x | 25 | 15 | b | 10 |
|---|----|----|---|----|
| y | 3 | a | 4 | c |

Since $ x $ and $ y $ are inversely proportional:
$ x \cdot y = \text{constant} $

First, find constant using first pair:

$ 25 \times 3 = 75 $

So $ x \cdot y = 75 $ always.

Now:

- For $ x = 15 $, $ y = a $:
$ 15 \times a = 75 $ → $ a = \frac{75}{15} = 5 $

- For $ x = b $, $ y = 4 $:
$ b \times 4 = 75 $ → $ b = \frac{75}{4} = 18.75 $

- For $ x = 10 $, $ y = c $:
$ 10 \times c = 75 $ → $ c = \frac{75}{10} = 7.5 $

✔ Answer:
- $ a = 5 $
- $ b = 18.75 $
- $ c = 7.5 $

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8. If $ A : B = 2 : 3 $ and $ B : C = 4 : 5 $, find $ A : C $

We need to make $ B $ common in both ratios.

$ A : B = 2 : 3 $ → Multiply both sides by 4 → $ A : B = 8 : 12 $

$ B : C = 4 : 5 $ → Multiply both sides by 3 → $ B : C = 12 : 15 $

Now $ A : B : C = 8 : 12 : 15 $

So $ A : C = 8 : 15 $

✔ Answer: $ A : C = 8 : 15 $

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✔ Final Answers Summary:

1.
(a) Neither
(b) Neither
(c) Direct
(d) Direct
(e) Inverse

2. $ x = 4 $, $ y = 6 $
3. (a) 225 km, (b) 30 km
4. 30 days
5. 4 more gardeners
6. $ x = 34 $
7. $ a = 5 $, $ b = 18.75 $, $ c = 7.5 $
8. $ A : C = 8 : 15 $

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