"Chain rule problems" is one of the topics in quantitative aptitude. For some students who prepare quantitative aptitude to get prepared for competitive exams, it is a bit difficult topic to understand.

The reason for why it is bit difficult for them to understand is, they do not know the basic stuff about solving chain rule problems.

Even though many websites give example problems on chain rule, some students are not able to understand how the problems have been solved and the steps involved in solving.

To solve chain-rule problems, we have to understand the two important stuff.

They are,

1. Direct variation 2. Inverse variation

**Direct Variation: **

We have direct variation in the following situations.

(i) When one quantity increases, other quantity also increases.

(ii) When one quantity decreases, other quantity also decreases.

**That is, both the quantities are traveling on the same path. (increase - increase or decrease - decrease) **

**Inverse Variation: **

We have inverse variation in the following situations.

(i) When one quantity increases, other quantity decreases.

(ii) When one quantity decreases, other quantity increases.

**That is, the given two quantities are traveling on the opposite directions.(increase-decrease or decrease-increase) **

**Formula:**

**No. of man days**

**= No. of men X No.of days**** **

**No. of man hours**

**= No. of men X No.of hours per day ****X** No. of days

**No. of machine hours**

**=** No. of machines **X** No.of hours per day **X** No. of days

Let us see, how to solve word problems on chain rule.

**Problem 1 :**

If the wages of 6 men for 15 days be $2100, then find the wages of 9 men for 12 days.

**Solution** :

**Step 1 :**

No. of man days (6 men /15 days)

= 6X15 = 90 days

No. of man days (9 men / 12 days)

= 9X12 = 108 days

**Step 2 :**

**Given :** For 90 man days, the wages is $2100

**Target :** To find the wages for 108 man days (9 men/12 days)

**Step 3 :**

More man days ---> more wages = Increase ----> Increase

So, it is direct variation.

Let "A" be the wages for 108 man days ( 9 men /12 days)

**Step 4 :**

**Step 5 :**

**Hence the wages of 9 men for 12 days is $ 2520**

**Problem 2 :**

If 9 engines consumes 24 metric tonnes of coal, when each engine is working 8 hours a day, how much coal will be required for 8 engines, each running 13 hours a day, it is being given that 3 engines of former type consume as much as 4 engines of later type ?

**Solution** :

**Step 1 :**

The most important thing that we have to notice in this problem is, the engine in the first type and engine in the second type are not same in consumption of coal.

First we have to make it to be same and proceed the problem.

**Step 2 :**

In coal consumption, we have

3 engines of type-I = 4 engines of type-II

**Step 3 :**

In the second type, we have 8 engines.

8 engines of type-II = 2x 4 engines of type-II

8 engines of type-II = 2x 3 engines of type-I

8 engines of type-II = 6 engines of type-I

**Step 4 :**

From step 3, one point is very clear.

That is, in type-II, instead of taking 8 engines, if we take 6 engines, it will become type-I. Coal consumption for both the types will be same**. **

**Step 5 :**

No. of engine hours for **type-I **

= No. of engines X No. pf hours per day

= 9 x 8

= **72 hours**

No. of engine hours for **type-II**

= No. of engines X No. pf hours per day

= 6 x 13

= **78 hours**

**Step 6 :**

More engine hours ----> more coal consumption

= Increase ----> Increase

So, it is direct variation.

Let "A" be the coal required for 8 engines of type-II (6 engines of type-I)

**Step 7 :**

**Step 8 :**

**Hence, 26 metric tonnes of coal will be required for 8 engines, each running 13 hours a day. **

Chain rule problems explained above will give a clear idea to students on solving word problems on chain rule .

And also we hope that chain rule problems explained above would be much useful for the students who struggle to solve word problems on chain rule.

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