AVERAGES and ALLIGATIONS

In this chapter, we will study two powerful concepts—Averages and Alligations.
Let us first study the topic of Averages.

Averages are a very important concept in our daily life. From the marks of a student to the performance of a sports player, everything is measured in averages. If a cricketer scores a century (100 runs) in one innings and gets out for a duck (0 runs) in the next innings, then we say that the average score of the cricketer is 100+0 divided by 2 = 50 (we divide by 2 as he played two innings)

Averages are also called as arithmetic mean in Statistics. Before we move to some complex questions, we shall first solve two simple questions as a warm-up.

Q1) A lady purchased 5 dresses at an average of 1500 each, 4 bags at an average of1200 each and 3 wallets at the average of `500 each. Find the average cost of her total purchases.

Q2) There are four members in a family (Father, Mother, Son, Daughter). Their average age is 30 years. The son gets married and after the arrival of the wife, the average age of the family becomes 29. Find the age of the wife.

Ans: Total age of the full family of 4 members = 30 × 4 = 120 Total age of the full family of 5 members (after arrival of wife) 29 × 5 = 145 So, the age of the new entrant (wife) is 145 – 120 = 25 years. Now that we have refreshed our basic concepts, let us move towards some advanced examples.

Q3) In a concert, there are 30 musicians whose average age is decreased by 2 months when a musician aged 45 years is replaced by another musician. Find the age of the new musician.

Ans: As you can see this question involves some kind of ‘replacement’ (new musician coming in place of old musician). Such replacement questions can be easily solved by the secret short-cut approach that I have discussed here.
As we can see, the average age of 30 musicians is decreased by 2 months (on replacement).

So the net effect of the replacement is 30 × (-2) = -60 months.

Now we have to remove 60 months from the person who is being replaced.
The person being replaced is of 45 years.

Thus when we remove 60 months (or 5 years) from a person of 45 years, we get 40 years. Hence the age of the new musician is 40 years.

This gives us a kind of a formula, 40 years = 45 years – (30 × 2 months)
Therefore Age of new person = Age of removed person – (Number of people × Decrease in age)

Alternatively, when there is an increase in age instead of decrease in age, we can say that Age of new person = Age of removed person + (Number of people × Increase in age) You can directly use these formulas to solve faster!

Q4) The average salary payout of 5 people in a company was 10,000 per month. One person whose salary was15,000 resigned and he was replaced by another person. After this replacement, the average salary of the entire office became `12,000. Find the salary of the new entrant.

Ans: As it can be seen from the formula given above

Salary of New Person = Salary of Removed Person

(Number or People × Increase)
= 15,000 + (5 × 2000)
= `25,000

I will now introduce you to a very powerful concept of Cross-Ratios. It is also called Rule of Alligation or the Rule of Mixtures.

Q5) The average salary of the entire staff in an office is 5000. Out of this, the average salary of the men is 4600 and the salary of women is `5200. Find the number of men if the number of women are 40.

Ans: Whenever we have to find the average mixture, average ratio or proportion we use the concept of cross-ratios as given below.

On the bottom left-hand side we will write the combined average salary minus average salary of women (5000- 5200) and on the bottom right-hand side we will write combined average salary minus average salary of men.

The bottom two figures will actually give us the ratio of the men and women in the organization! It may sound unbelievable at first sight but it is true. The bottom two figures will indeed give us the ratio between men and women. In fact, as we will see in the examples that follow later, in this method of cross-ratios, the bottom two figures will give us the ratio of almost any quantity that we require to deduce.

So we can see that the ratio of men: women is 200:400 (which is same as 1:2). In other words, for every 2 women there is 1 man.

We know from the question that there are 40 women and hence there are 20 men in the office. Our final answer is 20 men.

So we see that this rule of Cross-Ratios (also known as Method of Alligation) can be quite useful in solving such kinds of questions.

So now we will move on to the second part of this chapter. Let me further delve in to the concept of Alligations and Mixtures. You will often find questions on these in competitive exams.

Q6) At what rate should wheat at 14 per kilo be mixed with wheat at20 per kilo so as to get a bag of wheat with an average rate of `18 per kilo?

Ans: We have two quantities of wheat here. The expensive wheat is 20 per kilo and the cheaper wheat is14 per kilo. By the method of cross-ratios that we just studied, this is how the working will look like:

The bottom two figures give us the ratio of the mixture. Hence the ratio between expensive wheat and cheaper wheat should be 4:2. Our final answer is that for every 4 kilos of expensive wheat (20 per kilo) there must be 2 kilos of cheaper wheat (14 per kilo) in the mixture.

Q7) For a class of 70 students, the school declared a bonus prize of 10,500. Find out how many boys and girls are there in the class if each boy got110 and each girl got `180.

Ans: This is a classical problem on cross-ratios. Let us solve it by the method that we have just learnt. The total prize money 10,500 was divided between 70 students 10,500/70 = 150. Hence the average amount each student got was150. We also know that each boy was given 110 and each girl was given180. This is how our working will look like:

Our final answer is there are 30 boys and 40 girls in the class. (Additionally, if time permits, we can double check the answer. We know that each boy got 110 and each girl got180. The total amount distributed was `10,500.

We multiply 110 per boy × 30 boys =3300 We multiply 180 per girl × 40 girls =7200 The total amount (3300 + 7200) is `10,500 and hence our calculation is correct!)

So, in this chapter we learnt two powerful concepts of averages and alligations. The cross-ratios method explained in the latter half will be used again in the chapters that follow. This method is very useful in solving questions fast.

In the following chapters, we will study some amazing secret techniques for various topics that you will encounter in competitive exams. Keep reading.