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

Worksheet Analysis and Solutions

The worksheet focuses on Direct and Inverse Proportions. Let's solve each problem step by step.

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#### Question 1: Direct Variation
Identify which of the following cases show direct variation between the two given quantities.

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

- Analysis: This is an example of inverse proportion because as the number of workers increases, the time taken to complete the job decreases. The product of workers and days remains constant.
- Answer: 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?

- Analysis: Distance covered is directly proportional to time when speed is constant. As time increases, distance also increases.
- Answer: Direct variation.

(c) A tap can completely fill a tank in \( \frac{11}{2} \) hours. How much of it can be filled in \( \frac{3}{4} \) hours?

- Analysis: The amount of water filled is directly proportional to the time the tap is open. As time increases, the amount of water filled also increases.
- Answer: Direct variation.

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

- Analysis: Cost is directly proportional to the length of the cloth. As the length increases, the cost also increases.
- Answer: Direct variation.

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#### Question 2: Paper Mass Calculation
If the mass of 35 sheets of paper is 280 g, how many sheets will weigh 7.2 kg?

- Given:
- 35 sheets = 280 g
- 7.2 kg = 7200 g (since 1 kg = 1000 g)

- Step 1: Find the mass of 1 sheet.
\[
\text{Mass of 1 sheet} = \frac{280 \text{ g}}{35} = 8 \text{ g}
\]

- Step 2: Find the number of sheets that weigh 7200 g.
\[
\text{Number of sheets} = \frac{7200 \text{ g}}{8 \text{ g/sheet}} = 900 \text{ sheets}
\]

- Answer: \( \boxed{900} \)

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#### Question 3: Family Expenses
A family’s expenses for 4 weeks 5 days amount to Rs. 1980. What is their expense at the same rate for 6 weeks 2 days?

- Step 1: Convert weeks and days into total days.
- 4 weeks 5 days = \( 4 \times 7 + 5 = 33 \) days
- 6 weeks 2 days = \( 6 \times 7 + 2 = 44 \) days

- Step 2: Find the daily expense.
\[
\text{Daily expense} = \frac{1980 \text{ Rs}}{33 \text{ days}} = 60 \text{ Rs/day}
\]

- Step 3: Calculate the expense for 44 days.
\[
\text{Expense for 44 days} = 60 \text{ Rs/day} \times 44 \text{ days} = 2640 \text{ Rs}
\]

- Answer: \( \boxed{2640} \)

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#### Question 4: Workers and Days
If 25 workers can finish a job in 40 days, how many workers will complete the same job in 25 days?

- Analysis: This is an inverse proportion problem. The product of workers and days is constant.
\[
\text{Workers} \times \text{Days} = \text{Constant}
\]

- Step 1: Calculate the constant.
\[
\text{Constant} = 25 \text{ workers} \times 40 \text{ days} = 1000
\]

- Step 2: Find the number of workers needed for 25 days.
\[
\text{Workers} \times 25 \text{ days} = 1000
\]
\[
\text{Workers} = \frac{1000}{25} = 40
\]

- Answer: \( \boxed{40} \)

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#### Question 5: Rice Consumption
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?

- Step 1: Calculate the total consumption rate per day.
- Total people = 8 persons
- Rice consumed per day = \( \frac{40 \text{ kg}}{30 \text{ days}} = \frac{4}{3} \text{ kg/day} \)

- Step 2: New total people when 2 guests join.
- New total people = 8 + 2 = 10 persons

- Step 3: Assume the consumption per person remains constant. Calculate the new daily consumption.
- Daily consumption per person = \( \frac{\frac{4}{3} \text{ kg/day}}{8 \text{ persons}} = \frac{4}{3 \times 8} = \frac{1}{6} \text{ kg/person/day} \)
- New daily consumption = \( 10 \text{ persons} \times \frac{1}{6} \text{ kg/person/day} = \frac{10}{6} = \frac{5}{3} \text{ kg/day} \)

- Step 4: Calculate the number of days 40 kg of rice will last.
\[
\text{Days} = \frac{40 \text{ kg}}{\frac{5}{3} \text{ kg/day}} = 40 \times \frac{3}{5} = 24 \text{ days}
\]

- Answer: \( \boxed{24} \)

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#### Question 6: Work Hours
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?

- Analysis: This is an inverse proportion problem. The product of hours per day and days is constant.
\[
\text{Hours per day} \times \text{Days} = \text{Constant}
\]

- Step 1: Calculate the constant.
\[
\text{Constant} = 10 \text{ hours/day} \times 28 \text{ days} = 280
\]

- Step 2: Find the number of hours per day needed for 20 days.
\[
\text{Hours per day} \times 20 \text{ days} = 280
\]
\[
\text{Hours per day} = \frac{280}{20} = 14
\]

- Answer: \( \boxed{14} \)

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#### Question 7: Notebook Cost
The cost of 18 notebooks is Rs. 423. Find the cost of 20 notebooks.

- Step 1: Find the cost of 1 notebook.
\[
\text{Cost of 1 notebook} = \frac{423 \text{ Rs}}{18} = 23.5 \text{ Rs}
\]

- Step 2: Find the cost of 20 notebooks.
\[
\text{Cost of 20 notebooks} = 23.5 \text{ Rs/notebook} \times 20 = 470 \text{ Rs}
\]

- Answer: \( \boxed{470} \)

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Final Answers:

1. (b), (c), (d)
2. \( \boxed{900} \)
3. \( \boxed{2640} \)
4. \( \boxed{40} \)
5. \( \boxed{24} \)
6. \( \boxed{14} \)
7. \( \boxed{470} \)