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About Direct and Inverse Proportion — Class 8 Olympiad

Identify direct and inverse variation; solve time-work, speed-distance, and mixture problems. This topic is part of the Olympiad Class 8 mathematics syllabus (chapter: Module 9). On this page you can practice 47 questions across three difficulty levels — 20 easy, 17 medium, and 10 hard — each with a visual step-by-step solution, plus a timed 24-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Direct and Inverse Proportion — solved examples for Class 8 Olympiad

Example 1easy

If quantity P is directly proportional to quantity Q, and quantity Q is directly proportional to quantity R, which of the following statements is true?
  1. A)P is inversely proportional to R.
  2. B)P is directly proportional to R.
  3. C)P is directly proportional to R².
  4. D)P is proportional to (1/R).

Step-by-step solution

  1. Given that P is directly proportional to Q, we can write P = kQ for some constant k.
  2. Given that Q is directly proportional to R, we can write Q = mR for some constant m.
  3. Substitute Q = mR into the first equation: P = k(mR).
  4. Since k and m are constants, their product (km) is also a constant. Let km = C. So, P = CR. This shows P is directly proportional to R.

Answer: P is directly proportional to R.

Example 2medium

A specialized machine can produce 'P' units of a product in 'H' hours, working at a constant rate. If 2 such machines work together for 'D' days (where 1 day = 24 hours), how many units will they produce? (Assume 'P' units in 'H' hours for one machine.)
  1. A)(P × D × 24 × 2) / H
  2. B)(P × H × 2) / (D × 24)
  3. C)(P × D × 2) / (H × 24)
  4. D)(H × D × 24 × 2) / P

Step-by-step solution

  1. Rate of one machine = P units / H hours.
  2. Total hours for D days = D × 24 hours.
  3. Production by one machine in D days = (P/H) × (D × 24) units.
  4. Production by two machines in D days = 2 × (P/H) × (D × 24) = (P × D × 24 × 2) / H units.

Answer: (P × D × 24 × 2) / H

Example 3hard

A construction project requires 180 workers to be completed in 90 days. After 30 days, it is observed that only 1/5 of the total work is completed. To ensure the project is completed exactly on schedule, how many *additional* workers must be employed?
  1. A)36 workers
  2. B)72 workers
  3. C)120 workers
  4. D)180 workers

Step-by-step solution

  1. Total work planned = 180 workers × 90 days = 16200 worker-days.
  2. Work completed in the first 30 days = 1/5 of total work = 1/5 × 16200 = 3240 worker-days.
  3. Remaining work = 16200 - 3240 = 12960 worker-days.
  4. Time remaining to complete the project = 90 - 30 = 60 days.
  5. Let 'X' be the total number of workers required to complete the remaining work in 60 days. Using the formula Workers × Days = Work, we have X × 60 = 12960.
  6. X = 12960 / 60 = 216 workers.
  7. Since 180 workers are already on the job, the additional workers needed = 216 - 180 = 36 workers.
  8. Correction: The initial calculation of 'work completed' might be interpreted as actual work done. If 1/5 of the work is done in 30 days by 180 workers, then the work completed is 180 workers × 30 days = 5400 worker-days. If this is 1/5 of the total work, then total work = 5 × 5400 = 27000 worker-days.
  9. Remaining work = 27000 - 5400 = 21600 worker-days.
  10. Remaining time = 60 days.
  11. Workers needed for remaining work = 21600 / 60 = 360 workers.
  12. Additional workers = 360 - 180 = 180 workers.

Answer: 180 workers

Practice questions on Direct and Inverse Proportion

  1. Q1.easy

    A car travels a fixed distance at a certain speed. If the car increases its speed by 25%, by what percentage will the time taken to cover the same distance decrease?
    1. A)20%
    2. B)25%
    3. C)30%
    4. D)15%
    Show answer

    Answer: 20%

    Hint: Remember that for a fixed distance, speed and time are inversely proportional. Express the new speed as a fraction of the original speed.

  2. Q2.easy

    12 men can complete a construction project in 20 days. After 8 days, 4 men leave the project. How many more days will the remaining men take to complete the remaining work?
    1. A)16 days
    2. B)18 days
    3. C)20 days
    4. D)24 days
    Show answer

    Answer: 18 days

    Hint: First, calculate the amount of work completed in the first 8 days. Then, determine the remaining work and the number of men available to complete it.

  3. Q3.easy

    5 identical machines can produce 100 toys in 10 hours. How many toys can 8 such machines produce in 15 hours, assuming they work at the same rate?
    1. A)180 toys
    2. B)200 toys
    3. C)240 toys
    4. D)280 toys
    Show answer

    Answer: 240 toys

    Hint: The number of toys produced is directly proportional to both the number of machines and the number of hours they work.

  4. Q4.medium

    A group of 'N' workers can complete a project in 'M' days. If 'K' workers leave after 'X' days, how many more days will it take for the remaining workers to complete the project? (Assume K < N and X < M).
    1. A)(N × M - N × X) / (N - K)
    2. B)(N × M - N × X) / K
    3. C)(N × M - K × X) / (N - K)
    4. D)(N × M - K × X) / N
    Show answer

    Answer: (N × M - N × X) / (N - K)

    Hint: Calculate the total work units needed and the work already completed. The remaining work must be done by the reduced number of workers.

  5. Q5.medium

    A car travels from city A to city B at an average speed of 60 km/h and returns from B to A at an average speed of 40 km/h. What is the average speed of the car for the entire journey?
    1. A)50 km/h
    2. B)48 km/h
    3. C)45 km/h
    4. D)42 km/h
    Show answer

    Answer: 48 km/h

    Hint: Assume a convenient distance for the one-way journey to calculate total time and total distance. Remember, average speed is total distance divided by total time.

  6. Q6.medium

    If 'X' men working 'Y' hours a day can complete a task in 'Z' days, then how many men working 'P' hours a day would be required to complete a task twice as large in 'Q' days?
    1. A)(2 × X × Y × Z) / (P × Q)
    2. B)(X × Y × Z) / (2 × P × Q)
    3. C)(X × Y × P) / (2 × Z × Q)
    4. D)(2 × X × P × Q) / (Y × Z)
    Show answer

    Answer: (2 × X × Y × Z) / (P × Q)

    Hint: Establish a relationship between men, hours, days, and work (M1D1H1/W1 = M2D2H2/W2). The new task is twice the original work.

  7. Q7.hard

    A tank has three pipes: P1, P2, and P3. P1 can fill the tank in 6 hours, P2 in 8 hours, and P3 can empty it in 12 hours. If P1 is opened at 9:00 AM, P2 at 10:00 AM, and P3 at 11:00 AM, at what time will the tank be exactly half-full?
    1. A)11:00 AM
    2. B)11:12 AM
    3. C)11:24 AM
    4. D)11:36 AM
    Show answer

    Answer: 11:12 AM

    Hint: Determine the individual rates of each pipe and calculate the net work done in different time intervals. The goal is to reach half the tank's capacity.

  8. Q8.hard

    Three gears A, B, and C are meshed in series such that A drives B, and B drives C. Gear A has 30 teeth, Gear B has 45 teeth, and Gear C has 20 teeth. If Gear A makes 120 rotations per minute (RPM), what is the *difference* in the number of rotations made by Gear B and Gear C in 10 minutes?
    1. A)1200 rotations
    2. B)1800 rotations
    3. C)2000 rotations
    4. D)2400 rotations
    Show answer

    Answer: 2400 rotations

    Hint: The number of teeth and rotations are inversely proportional for meshed gears. Calculate the RPM for each gear, then total rotations over 10 minutes.

  9. Q9.hard

    The cost of a diamond varies directly as the square of its weight. If a diamond weighing 10 grams costs ₹10,000, what would be the loss incurred if this diamond breaks into two pieces whose weights are in the ratio 2:3?
    1. A)₹4800
    2. B)₹5000
    3. C)₹5200
    4. D)₹6000
    Show answer

    Answer: ₹5200

    Hint: First, establish the constant of proportionality. Then, calculate the value of the original diamond and the values of the two broken pieces separately.

These are 9 of the 47 questions available for Direct and Inverse Proportion. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.