Chain Rule and Direct and Inverse Variation: Formulas, Tricks and Solved Examples for SSC CGL, CHSL and MTS

The chain rule is the base of many arithmetic topics such as time and work, pipes, wages, food supply and speed. It tells you how a quantity changes when other quantities change. This guide on chain rule and direct and inverse variation for SSC explains the two types of variation, the man-day formula and the fastest solving method, with solved examples, tricks, practice questions and FAQs.

What Is Variation?

When two quantities are connected, a change in one causes a change in the other. There are two types.

The Chain Rule Formula

When many quantities are involved, write the “cause” side and the “effect” side:

(M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂

Here M is the number of men (or machines), D is days, H is hours per day, and W is the work done. This is the man-day-hour method.

Step-by-Step Method

1. Write all the quantities as two rows, one for the first case and one for the second.
2. Decide for each quantity whether it is direct or inverse with the unknown.
3. For direct variation, multiply by the ratio as it is. For inverse variation, multiply by the ratio inverted.
4. Cancel the common factors and get the answer.

Direct and Inverse Pairs to Remember

Solved Examples

Example 1. If 5 pens cost 60 rupees, find the cost of 12 pens.
Direct: 60 × 12/5 = 144 rupees.

Example 2. 8 men can build a wall in 15 days. In how many days can 12 men build it?
Inverse: 15 × 8/12 = 10 days.

Example 3. 15 men working 8 hours a day finish a work in 24 days. In how many days will 20 men working 6 hours a day finish it?
15 × 24 × 8 = 2,880 man-hours. 20 × 6 = 120 per day. 2880/120 = 24 days.

Example 4. A camp of 500 men has food for 30 days. After 10 days, 100 men leave. For how many more days will the food last?
Food left is for 20 days for 500 men = 10,000 man-days. Now 400 men: 10000/400 = 25 days.

Example 5. x varies directly as y. If x = 12 when y = 4, find x when y = 7.
k = 12/4 = 3. x = 3 × 7 = 21.

Example 6. 12 men produce 1,080 units in 15 days. How many units will 18 men produce in 12 days?
1080 × (18 × 12)/(12 × 15) = 1080 × 216/180 = 1,296 units.

Example 7. 12 pipes fill a tank in 20 minutes. In how many minutes will 8 pipes fill it?
20 × 12/8 = 30 minutes.

Common Types of Chain Rule Questions

Tricks for Speed

• Always write the unknown at the end and place the ratio carefully; do not try to solve in your head.
• Keep the same side for a cause (men, hours) and the other for effect (work), and cross-check by common sense: more men should mean fewer days.
• Use man-days as a common unit for food and work problems.
• If the work to be done is the same, the product of men, days and hours is constant.
• In wages problems, first find the wage of one man for one day.

Quick Tips

• Decide direct or inverse for each pair before you calculate.
• Write man-days and man-hours as the common unit.
• Check your answer with common sense: more workers means fewer days.
• Practise the camp and food type, which is asked often.

Quick Revision Points

• Direct variation: x/y = constant; inverse variation: xy = constant.
• Men and days are inversely related for the same work.
• M1D1H1/W1 = M2D2H2/W2.
• Food problems: total man-days of food stay constant.
• Wages are directly proportional to man-days.

Practice Questions With Answers

1. 9 notebooks cost 135 rupees. Find the cost of 14 notebooks.
   Answer: 210 rupees
2. 20 men finish a job in 18 days. How many days will 30 men take?
   Answer: 12 days
3. 12 men working 8 hours a day finish a job in 15 days. How many days will 10 men working 9 hours a day take?
   Answer: 16 days
4. Food for 60 men lasts 40 days. For how many days will it last for 40 men?
   Answer: 60 days
5. A car at 60 km/h takes 5 hours. How long at 75 km/h?
   Answer: 4 hours
6. y varies inversely as x. If y = 6 when x = 8, find y when x = 12.
   Answer: 4
7. 4 men earn 1,000 rupees in 5 days. How much will 6 men earn in 8 days?
   Answer: 2,400 rupees
8. 15 workers dig 120 m of trench in 12 days. How many metres will 24 workers dig in 10 days?
   Answer: 160 m
9. A hostel has food for 200 students for 45 days. If 50 more join, how long will the food last?
   Answer: 36 days
10. 5 machines make 400 items in 8 hours. How many items will 8 machines make in 5 hours?
    Answer: 400 items

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Quantitative Aptitude notes page and the next topic Logarithms and Exponents.

Frequently Asked Questions

What is the chain rule in maths?

It is a method to find an unknown quantity when many connected quantities change, using the ratio of each pair.

What is direct variation?

When one quantity increases, the other increases in the same ratio, for example cost and number of articles.

What is inverse variation?

When one quantity increases, the other decreases in the same ratio, for example men and days.

What is the man-day-hour formula?

M1 × D1 × H1 / W1 = M2 × D2 × H2 / W2.

How do I solve food problems in the chain rule?

Convert the food into man-days and divide by the new number of persons.

Are men and days directly or inversely proportional?

Inversely, for the same work.

Are wages directly proportional to days?

Yes. More days worked means proportionally more wages.

Is chain rule asked in SSC exams?

Yes. It is asked directly and also as part of time and work, pipes and wages problems.

How much time should one question take?

About 40 seconds.

What is the difference between chain rule and unitary method?

The unitary method finds the value of one unit first, while the chain rule solves in one step using ratios.

What is the constant of variation?

It is the fixed value k in x = ky or x = k/y.

What are common mistakes in chain rule?

Mixing up direct and inverse pairs, and forgetting hours per day.