CBSE Class 8 Mathematics Worksheet - Direct and Inverse Variations ... - Free Printable

Let's solve each problem from the worksheet step by step, explaining the concepts of direct and inverse proportion where applicable.

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1. In which of the following cases is there direct variation between the two given quantities? (Do not carry out any calculations)

We need to identify direct variation — this means when one quantity increases, the other also increases proportionally.

#### (a) 10 workers finish a job in 6 days. In how many days will 20 workers finish the same job?

- More workers → less time to finish the job.
- This is inverse proportion (more workers reduce time).
- ✘ Not direct variation

#### (b) A car runs at a uniform speed. If it covers 135 km in 3 hours, how much distance will it cover in 2 hours?

- At constant speed: Distance ∝ Time
- More time → more distance covered.
- ✔ Direct variation

#### (c) A tap can completely fill a tank in 1½ hours. How much of it can be filled in ¾ hours?

- Time taken ∝ Amount of water filled (at constant rate)
- More time → more water filled.
- ✔ Direct variation

#### (d) Curtain cloth was purchased at Rs. 230 per meter. How much will 12 meters of cloth cost?

- Cost ∝ Length of cloth
- More cloth → higher cost.
- ✔ Direct variation

✔ So, (b), (c), and (d) show direct variation.

> Answer: Direct variation exists in (b), (c), and (d).

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2. If the mass of 35 sheets of paper is 280 g, how many sheets will weigh 7.2 kg?

First, convert 7.2 kg to grams:
> 7.2 kg = 7200 g

Now, find weight per sheet:
> 280 g / 35 sheets = 8 g per sheet

Now, number of sheets in 7200 g:
> 7200 ÷ 8 = 900 sheets

✔ Answer: 900 sheets

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3. A family’s expenses for 4 weeks 5 days amount to ₹1980. What is their expense at the same rate for 6 weeks 2 days?

Convert weeks and days into total days:

- 4 weeks 5 days = (4×7) + 5 = 28 + 5 = 33 days
- 6 weeks 2 days = (6×7) + 2 = 42 + 2 = 44 days

Assuming expenses ∝ time, so use direct proportion:

Let expense for 44 days be ₹x

So,
> 33 days → ₹1980
> 44 days → ₹x

$$
\frac{33}{1980} = \frac{44}{x}
\Rightarrow x = \frac{1980 \times 44}{33}
= 60 \times 44 = 2640
$$

✔ Answer: ₹2640

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4. If 25 workers can finish a job in 40 days, how many workers will complete the same job in 25 days?

This is inverse proportion: more workers → fewer days.

Let number of workers required be $ x $

Work is constant, so:
> Workers × Days = Constant

$$
25 \times 40 = x \times 25
\Rightarrow x = \frac{25 \times 40}{25} = 40
$$

✔ Answer: 40 workers

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5. 40 kg of rice lasts 30 days in a family of 8 persons. If 2 guests stay with the family, how many days will 40 kg of rice last?

Now, total people = 8 + 2 = 10 persons

Rice consumption is proportional to number of people.

More people → rice lasts fewer days → inverse proportion

Let number of days be $ x $

Total consumption (in person-days) is constant:
> 8 people × 30 days = 10 people × $ x $ days

$$
8 \times 30 = 10 \times x
\Rightarrow x = \frac{240}{10} = 24
$$

✔ Answer: 24 days

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6. If Ram works 10 hours per day, he can finish a job in 28 days. If he has to finish the job in 20 days, how many hours should he work per day?

Again, inverse proportion between days and hours per day.

Let hours per day be $ x $

Total work = hours per day × number of days

$$
10 \times 28 = x \times 20
\Rightarrow x = \frac{280}{20} = 14
$$

✔ Answer: 14 hours per day

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7. The cost of 18 notebooks is ₹423. Find the cost of 20 notebooks.

Cost ∝ Number of notebooks → Direct proportion

Let cost of 20 notebooks be ₹x

$$
\frac{18}{423} = \frac{20}{x}
\Rightarrow x = \frac{423 \times 20}{18}
= \frac{8460}{18} = 470
$$

✔ Answer: ₹470

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✔ Final Answers Summary:

1. Direct variation in: (b), (c), (d)
2. 900 sheets
3. ₹2640
4. 40 workers
5. 24 days
6. 14 hours per day
7. ₹470

Let me know if you'd like these explained in a different way or want diagrams!