About Chain Rule
The chain rule is proportion done in one pass. If one quantity rises while another falls, or two quantities rise together, the answer is found by scaling one factor at a time, and the only real question is which way each factor moves the answer.
What you need to understand
- More men doing the same work means less time. That relation is inverse.
- More hours worked each day means fewer days are needed. That relation is inverse too.
- More machines working for more days produces more output. Those relations are direct.
- The product of the quantities that together make up the whole stays constant: men multiplied by days is the total man-days.
- Scaling from a known case is safer than memorising a formula, because each step can be checked.
- Reducing everything to a single rate, such as output per machine per day, makes the final multiplication obvious.
Formulas to remember
How to work through these questions
- Decide for each factor whether it pushes the answer up or down, and write that down before calculating.
- Reduce the problem to a rate for a single unit, such as one man for one day, which makes the final step obvious.
- Scale one factor at a time, checking after each step that the direction of the change is sensible.
- Check the answer lies on the correct side of the original figure: more men must mean fewer days.
- Keep the units in view throughout, because the unit of the answer often gives away the correct relation.
Mistakes that cost marks
- Using a direct proportion where the relation is inverse, which puts the answer on the wrong side of the original value.
- Scaling only one of the factors when two of them change.
- Dividing where the proportion calls for multiplying.
- Forgetting that the question asks for days rather than for men.
- Repeating the quantity that was already given instead of computing a new one.
Worked example
If 12 men can do a piece of work in 24 days, how many days will 16 men take?
- Total work = 12 x 24 = 288 man-days.
- With 16 men that work is shared between more people, so the time is 288 / 16.
- 288 / 16 = 18 days.
- Check: more men must mean less time, and 18 days is indeed less than 24 days.
Answer: 18 days
Practice questions with answers
A few Chain Rule questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
18 men finish a job in 4 days. How long will 12 men take for the same job?
Answer: Option B — with explanation
Total work = 18 x 4 = 72 man-days.
With 12 men the time is 72 / 12 = 6 days.
More men means less time, so the answer must be smaller than 4.
Common mistakes
- gave the number of men in place of the number of days: 12
- repeated the days already given: 4
- inverted the proportion: 2.67
Question 2
4 machines produce 312 units in 13 days. How many units will 8 machines produce in 12 days?
Answer: Option B — with explanation
Output per machine per day = 312 / (4 x 13) = 6 units.
So 8 machines in 12 days make 8 x 12 x 6 = 576 units.
Common mistakes
- repeated the output already given: 312
- allowed for the machines but not for the change in the number of days: 624
- allowed for the days but not for the change in the number of machines: 288
Question 3
13 articles are bought for Rs. 507. Find the cost of 17 such articles.
-
A
663
-
B
507
-
C
387.71
-
D
39
Answer: Option A — with explanation
Price of one article = 507 / 13 = Rs. 39.
So 17 articles cost 17 x 39 = Rs. 663.
Common mistakes
- multiplied where the proportion called for dividing: 387.71
- gave the price of a single article: 39
- repeated the cost of the first lot: 507
Question 4
If 17 men can do a piece of work in 45 days, how many days will 15 men take to do the same work?
Answer: Option C — with explanation
Total work = 17 x 45 = 765 man-days.
With 15 men the time is 765 / 15 = 51 days.
More men means less time, so the answer must be smaller than 45.
Common mistakes
- gave the number of men in place of the number of days: 15
- repeated the days already given: 45
- inverted the proportion: 39.71
Question 5
11 machines produce 1320 units in 15 days. How many units will 4 machines produce in 4 days?
Answer: Option D — with explanation
Output per machine per day = 1320 / (11 x 15) = 8 units.
So 4 machines in 4 days make 4 x 4 x 8 = 128 units.
Common mistakes
- repeated the output already given: 1320
- allowed for the machines but not for the change in the number of days: 480
- allowed for the days but not for the change in the number of machines: 352
Frequently asked questions
Why do more men take less time?
Because the work is fixed. If the total comes to a certain number of man-days, spreading that total over more men reduces the number of days needed.
How do I handle a question where both the men and the hours change?
Reduce everything to a rate for one man for one hour, then build the answer back up from that rate by multiplying in each factor in turn.
What does the name chain rule actually mean?
It means chaining the proportions together, scaling the known case by each factor in turn until it matches the case being asked about.
How do I know whether a relation is direct or inverse?
Ask what happens to the answer when the factor increases. More machines give more output, which is direct. More men give fewer days, which is inverse.
