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Lesson 7 of 7

Proportional Reasoning-2 · Lesson 7 of 7

Solving Proportion and Work-Rate Problems

“Apply direct proportion, inverse proportion and combined work rates to workers, pumps, supplies, schedules and other real situations.”

Objectives
  • Solve inverse-proportion problems involving workers, machines, pumps and time.
  • State the assumptions that make an inverse model reasonable.
  • Distinguish mixed direct- and inverse-proportion situations.
  • Combine individual work rates to find the time taken when people or machines work together.
  • Check answers by interpreting how the quantities should change.

Using the Constant Product in Real Problems

Many inverse-proportion problems have the same hidden structure: the total job or resource is fixed, while the number of workers, machines, pumps or users changes. Under suitable assumptions, multiplying the two changing quantities gives a constant.

Example — Workers and days

Problem
Twenty workers can lay a fixed length of road in 4 days. How many days would 10 equally productive workers need?

  1. 1.The total road work is fixed, so fewer workers require more days.
  2. 2.Use workers × days = constant: 20 × 4 = 10 × d.
  3. 3.80 = 10d.
  4. 4.d = 8 days.
  5. 5.The workforce halved from 20 to 10, so the time doubled from 4 to 8 days.
Example — Pumps filling one tank

Problem
Two identical pumps fill a tank in 18 hours. How long will four identical pumps take?

  1. 1.The same tank is being filled, and the pumps are assumed to work at equal constant rates.
  2. 2.Use pumps × time = constant: 2 × 18 = 4 × t.
  3. 3.36 = 4t.
  4. 4.t = 9 hours.
  5. 5.Doubling the number of pumps halves the time.
Example — Food provisions

Problem
Food is enough for 80 students for 15 days. If 20 more students join and everyone consumes the same amount each day, how long will the food last?

  1. 1.The total amount of food is fixed.
  2. 2.The number of students rises from 80 to 100, so the number of days must fall.
  3. 3.Use students × days = constant: 80 × 15 = 100 × d.
  4. 4.1200 = 100d, so d = 12 days.
  5. 5.The answer is smaller than 15 days, which fits the situation.

Assumptions Matter

Proportion models simplify the real world. Workers may have different speeds, pumps may not have identical flow rates, and families may use different amounts of water. Before using a proportional equation, identify the condition that is assumed to stay constant.

Useful assumptions

Typical assumptions include equal worker productivity, identical pumps or machines, a constant price per item, a constant fuel efficiency, equal daily consumption per person or family, and no interruptions or changing conditions.

Working Together — Add Rates, Not Times

When two people or machines work at the same time, their completion times cannot simply be added or averaged. Instead, convert each completion time into a work rate: the fraction of one whole job completed per unit of time. Then add the rates.

Combined work rateLaTeX
If one worker completes a job in T hours, that worker's rate is 1/T job per hour.
Example — Two people cutting vegetables

Problem
Ram can finish a fixed quantity of vegetables in 1 hour and Shyam can finish the same quantity in 1.5 hours. How long will they take together?

  1. 1.Treat the whole task as 1 unit of work.
  2. 2.Ram's rate is 1/1 = 1 unit per hour.
  3. 3.Shyam's rate is 1/1.5 = 2/3 unit per hour.
  4. 4.Combined rate = 1 + 2/3 = 5/3 units per hour.
  5. 5.Time for 1 unit = 1 ÷ (5/3) = 3/5 hour.
  6. 6.3/5 hour = 36 minutes.
Example — Two pumps together

Problem
A small pump fills a tank in 3 hours and a large pump fills it in 2 hours. How long do they take together?

  1. 1.Small-pump rate = 1/3 tank per hour.
  2. 2.Large-pump rate = 1/2 tank per hour.
  3. 3.Combined rate = 1/3 + 1/2 = 5/6 tank per hour.
  4. 4.Time for one tank = 1 ÷ (5/6) = 6/5 hour.
  5. 5.6/5 hour = 1.2 hours = 1 hour 12 minutes.
Common mistake

If one pump takes 3 hours and another takes 2 hours, their joint time is not 5 hours and not 2.5 hours. Convert each time into a rate first, add the rates, and then invert the combined rate to get the time.

Choosing the Right Relationship

Before calculating, ask what should happen when one quantity increases. If the other should increase by the same factor, the relation is direct. If the other should decrease by the reciprocal factor while a product stays fixed, it is inverse. Some problems, such as combined work, are easiest when expressed through rates rather than a single proportion.

Pair of quantitiesLikely relationCondition
Number of taps and filling timeInverseSame tank; equal tap flow
Number of painters and daysInverseSame wall; equal productivity
Distance travelled and petrol usedDirectConstant fuel efficiency
Speed and travel timeInverseSame route distance
Length of cloth and priceDirectFixed price per metre
Pages read and reading timeDirectFixed reading speed
Example — Rearranging chairs

Problem
Chairs are arranged in 25 rows with 12 chairs in each row. If the same chairs are rearranged with 20 chairs per row, how many rows are needed?

  1. 1.The total number of chairs is fixed: 25 × 12 = 300.
  2. 2.Rows and chairs per row are inversely related for the same total.
  3. 3.Rows needed = 300 ÷ 20 = 15.
  4. 4.Check: 15 × 20 = 300 chairs.
Example — Changing the number of periods

Problem
A school day has 8 periods of 45 minutes each. If the same teaching time is divided into 9 periods, how long is each period?

  1. 1.Total teaching time = 8 × 45 = 360 minutes.
  2. 2.With the same total time, number of periods and length of each period are inversely related.
  3. 3.New period length = 360 ÷ 9 = 40 minutes.

Quiz

Quick check

Three workers finish a job in 4 days. Under equal productivity, how long will four workers take?

Quick check

A machine completes one job in 5 hours. What is its work rate?

Quick check

Which assumption is needed when treating number of workers and completion time as inversely proportional?

Quick check

A pump fills one tank in 6 hours. At the same constant rate, how long will the same pump take to fill 5 identical tanks one after another?

Quick check

A car covers a fixed route in 2 hours at 60 km/h. What time is expected at 80 km/h?

Practice Problems

Practice Problems
  1. Classify each pair as direct proportion, inverse proportion or neither under the stated condition: (a) number of equal taps filling one tank and filling time, (b) painters and days for one fixed wall, (c) distance driven and petrol used at fixed fuel efficiency, (d) speed and time for one fixed route, (e) cloth length and price at a fixed rate per metre, (f) pages read and time at fixed reading speed.
  2. If 24 pencils cost ₹120, how much will 20 pencils cost at the same price per pencil?
  3. A tank has enough water for 20 families for 6 days. If 10 more families move in, how long will the water last, assuming equal daily water use per family? State the assumption.
  4. For each sleep time 15, 2.5, 20, 8, 3.5, 13, 10.5 and 18 hours in a 24-hour day, calculate the angle that would represent sleep in a circular day chart.
  5. A transport pie chart has Bus 120°, Walk 90°, Cycle 60°, Two-wheeler 60° and Car 30°. Identify the most common mode, find the fraction travelling by car, and if 18 travel by car find the total number surveyed and the number using two-wheelers.
  6. Three workers can paint a fence in 4 days. If a fourth equally productive worker joins, how many days will the job take? State the assumption.
  7. One pump takes 6 hours to fill 2 identical tanks one after another. How long will it take to fill 5 such tanks at the same constant rate?
  8. A set of chairs forms 25 rows of 12 chairs. If the same chairs are rearranged with 20 chairs in each row, how many rows are formed?
  9. A school has 8 periods of 45 minutes. If the same total time is split into 9 periods, how long is each period?
  10. A small pump fills a tank in 3 hours and a large pump fills it in 2 hours. Find the time taken if both pumps work together.
  11. A factory needs 42 identical machines to complete a production target in 63 days. How many machines are needed to complete the same target in 54 days?
  12. A car takes 2 hours to cover a fixed route at 60 km/h. Find the time taken at 80 km/h.

Key Takeaways

Key Takeaways

Inverse-proportion problems often keep a total job, distance, resource or amount fixed. For suitable inverse pairs, the product of corresponding values remains constant. Always state the assumptions that make the proportional model reasonable. Direct and inverse proportion must be distinguished before choosing an equation. When people or pumps work together, add their work rates rather than their completion times. A result should make sense directionally: more equally productive workers should reduce time, while more items at a fixed price should increase total cost.

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Understanding Inverse Proportion

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