Math worksheet on direct and inverse proportions featuring seven word problems.

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?

- Speed is constant → 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 increases → amount of water filled increases.
- Since rate is constant, filling capacity ∝ time.
- ✔ 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

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

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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

Given:
- 35 sheets → 280 g
So, mass per sheet = $ \frac{280}{35} = 8 $ g/sheet

Now, number of sheets in 7200 g:
> $ \frac{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 to days:

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

Expenses are directly proportional to time.

Let the expense for 44 days be ₹x.

$$
\frac{33}{1980} = \frac{44}{x}
\Rightarrow x = \frac{1980 × 44}{33}
$$

Simplify:
- $ \frac{1980}{33} = 60 $
- $ 60 × 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 the required number of workers be $ x $.

Work done is constant:
> $ \text{Workers} × \text{Days} = \text{Constant} $

So,
$$
25 × 40 = x × 25
\Rightarrow x = \frac{25 × 40}{25} = 40
$$

✔ 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

Rice consumption is directly proportional to number of people.

So, more people → rice lasts fewer days.

Let the number of days be $ x $.

Total consumption is fixed (40 kg).

Since consumption per day increases with people:
$$
\text{People} × \text{Days} = \text{Constant}
\Rightarrow 8 × 30 = 10 × x
\Rightarrow x = \frac{8 × 30}{10} = 24
$$

✔ 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 hours per day and number of days.

Let $ x $ be the required hours per day.

Total work = $ 10 × 28 = 280 $ man-hours

Now:
$$
x × 20 = 280
\Rightarrow x = \frac{280}{20} = 14
$$

✔ 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

Cost per notebook = $ \frac{423}{18} = 23.5 $

Cost of 20 notebooks = $ 20 × 23.5 = 470 $

Alternatively:
$$
\frac{18}{423} = \frac{20}{x}
\Rightarrow x = \frac{423 × 20}{18} = \frac{8460}{18} = 470
$$

✔ Answer: ₹470

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✔ Final Answers Summary:

1. Direct variation: (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 a visual explanation or diagram!