Quantitative Lecture-01

$Age Problems-image$

Quantitative Lesson-01

(Age Problem)

If you ask which topic is the easiest in the whole quantitative section, the answer is ‘Age Problem’. And you’ll be witness after covering this topic here using only one method and solving every question of any difficulty level that come from ‘Age’ topic of word problems. Before begin, let’s know some basic sentences and learn their meaning and know how to convert those basic sentences in mathematical equation.

“Mr A is twice as old as Mr B”

Is that statement means A = 2B or B = 2A? First think and do by your own, then go below for answer and explanation.

Remember that you just need to multiply twice with the variable (i.e. person), which come after the second as. For instance, have a look at the sentence: “Mr A is twice as old as Mr B”. Notice that Mr B comes after the second as, so we should multiply twice (i.e. 2) with B and equate this to A. i.e. A = 2B

The obscure but very important point that you need to know is below:

There are two different ways to translate this statement (i.e. “Mr A is twice as old as Mr B”) into math:

Theoretically, when no equality sign is there, if we suppose age of B as x, then age of A becomes 2x (because age of A is twice as much as age of B).

Notice that, when equality sign is there, twice multiply with B. But when no equality sign is there, twice multiply with x (i.e. supposed age of B), and that gives 2x (i.e. supposed age of A).

Many students ask why we didn’t write 2x as age of B, if we multiply 2 with B when equality sign is there? That’s the only point where few students confuse. Let me elucidate this point:

You will never commit a mistake in Age Problem if you remember that when writing the statement in equation (i.e. when equality sign is there), simply extract the statement as below:

On other hand, when writing statement without equation (i.e. when no equality sign is there), suppose age of B as x and hence age of A will become 2x (because A is twice of B by age).
A B

2x x

Also, you must know that if age of A is twice age of B, then it means age of B is actually half the age of A. So, if we suppose age of A as x, then age of B will need to be ^x⁄₂ (because A is twice the age of B, so B is half of A) as follows:
A B

x ^x⁄₂

Bottom-line of the whole discussion is that if it is given that A is twice as old as B, and age of B is supposed as x, then age of A will be 2x (i.e. twice of B). On other hand, if age of A is supposed as x, age of B will be ^x⁄₂
(i.e. half of A).

Similarly, let’s know another statement and learn how to convert it into equation and non-equation mathematical form:

“Age of Mr A is five less than twice the age of Mr B”

There are two different ways to translate this statement (i.e. “Age of Mr A is five less than twice the age of Mr B”):

Similarly,

“Mr A is five years older than Mr B”

There are two different ways to translate this statement (i.e. “Mr A is five years older than Mr B”):

Similarly,

“Mr A is five times as old as Mr B” or “Mr A is five times the age of Mr B”

There are two different ways to translate this statement (i.e. “Mr A is five times as old as Mr B”):

Let’s move ahead to next stage and learn a basic question scenario that comes in Age Problem topic:

“15 years ago, age of Mr A was x years, what’s the present age of Mr A?”

It’s just as simple as you think, and do not require any paper work. Those who are good in math quickly answer x + 15. For instance, if your age was y, 2 years ago, then at present your age must increase by 2 (i.e. y + 2). But we want you to use following method because this method is used to solve any type of age problem, whether easy or hard:

Era (i.e. Time) Mr A
–15) x (This relation states 15 years ago, age of A was x)

0) x + 15 (When we move from 15 years ago to present, age of A must increase by 15)

Thus, the present age of Mr A is x + 15.

Similarly,

“15 years ago, age of Mr A was x years, how old Mr A will be in 4 years?”

Instead of asking present age, now question asks age of A, 4 years from now. The question statement says “how old A will be in 4 years?”, which means we need to find future age of A, 4 years from now. For that purpose, we should always first find the present age of Mr A as we did before, and then add 4 to present age so we’ll get future age in 4 years as follows:

Era Mr A
–15) x (This relation states 15 years ago, age of A was x)

0) x + 15 (When we move from 15 years ago to present, age of A must increase by 15)

+4) (x + 15) + 4 = x + 19 (When we move from present age to future age 4 years, 4 will added to present age)

Thus, 4 years from now, age of A will be x + 19.

Two-person scenario:

This scenario of Age Problem is usually tested in GAT (General) test. So, let’s learn this with an example scenario using same method as we have done before.

“Mr. A is 10 years older than Mr. B at present. In 5 years, Mr. A will be twice as old as Mr. B. How old Mr. A is at present?”

Solution:

Always follow this way to solve age problems:

Suppose that present age of Mr. B = x

While teaching, few students, especially from medical background, have asked me why we suppose x? Well, x has to be supposed, because we have not given age of B that can help us to find age of A accordingly. So, we have to suppose something for Mr B in order to solve the question.

As A is 10 years older than B, so age of A will become x + 10. And in second part of the questions, which says “In 5 years, A will be twice as old as B”, the word ‘will’ shows something about future; i.e. in 5 years when A will become (x + 5), while B will become {(x – 10) + 5, which is equal to x – 5}. Methodologically,

Thus, you always need to follow the following steps:

Notice that, in 5 years, age of A will be x + 15, while that of B will become x + 5. Now, question states that “In 5 years, A will be twice as old as B”, and at this stage, we always make an equation as follows:

➩ x + 15 = 2(x + 5)
➩ x + 15 = 2x + 10
➩ 15 – 10 = 2x – x
➩ x = 5

Now, the question asks about present age of Mr A. In terms of x, you can see that present age of A is (x + 10), so if we put value of x here, we’ll get the required answer i.e.:

Present age of A = x + 10 = 5 + 10 = 15 Answer!

Notice that, at beginning, we need to write the first relationship of two persons (i.e. A is 10 years older than B, at present) in terms of x without form of equality sign. And then the second relationship (i.e. In 5 years, A will be twice as old as B) is drawn in terms of x with equality sign form. Finally solve for value of x, and put accordingly where required according to the question.

Also, you can do so by supposing present age of A as x, and solve to find its value (i.e. present age of A) as follows:

Given that, at present, A is 10 years older than B, so B is 10 years younger than A. So, if present age of B is x, then age of A become x

Now, at this age the second part of the question provides relationship between the two ages i.e. “In 5 years A will be twice as old as B“. You have learned that ‘as many …. as’ or ‘as much …. as’ precedes whichever variable (as here Mr. B came after these words) will always be multiplied by the word ‘twice’ or ‘three times’ etc. So we can say:

(After 5 years): A = 2B
➩ x + 5 = 2(x – 5)
➩ x + 5 = 2x – 10
➩ 5 + 10 = 2x – x
➩ x = 15

As, present age of A is x, thus present age of A is 15.

Note that, in both ways (i.e. whether we suppose present age of A as x or present age of B as x), we will get same result (i.e. present age of A = 15)

Many people use equation method, rather than this method as discussed. But problem with the equation method is that the chance of getting incorrect answer is maximum, as the change of era makes different calculations

Similarly, lets try another question with a bit different wording of question:

“7 years ago, A was three times as old as B. How old A would be in 5 years, when he will be 20 years older than B?”

First, try this by your own, then go ahead for answer explanation.

Solution:

Suppose 7 years ago, B was x years old; As it is given that in same era (i.e. 7 years ago) A was three times as old as B, so A must be 3 times the age of B. i.e. (3 times of x = 3x)

After this always come to present, so you can conveniently go to future age without doing any mistake.

➩ At present, age of B must be = x + 7

And, present, age of A must be = 3x + 7

Methodologically,

Now, second part of question (where equational relation is required), statues:

➩ 3x + 12 = (x + 12) + 20
➩ 3x + 12 = x + 32
➩ 3x – x = 32 – 12
➩ 2x = 20
➩ x = 10

Now, question asks about the age of Mr A in 5 years, i.e. after 5 years from present. In terms of x, age of A in 5 years is 3x + 12, and that’s what we need to find. So, let’s put value of x in it:

Age of Mr A in 5 years = 3x + 12 = 3(10) + 12
= 30 + 12 = 42 Answer!

Have you noticed that we just used unique method and followed unique sets of steps to answer Age Problems? If not, remember the following steps that we have used so far to answer Age Problem:

Step 1: At first, we suppose age of one person as x based on given Era in first part of the question (e.g. given that: 7 years ago, A was three times as old as B, so suppose age of B is x years old, 7 years ago).

Step 2: Then find age of other person accordingly (e.g. 7 years ago, age of A three times that of B, so 7 years ago, age of A became 3x).

Step 3: Write the relationship of the two persons in terms of x without equation form; e.g.:
Era Mr. A Mr. B
–7) 3x x

Step 4: Come to present Era, so we can go to future era accordingly if required. Here it is extremely important to change the ages of A and B accordingly (i.e. add 7 years to age of each person to come from Era of “7 years ago” to present Era), e.g.:

Step 5: After coming to present Era, it’s now easy to go to future Era accordingly as the question states in second part. Again, never forget to change the ages of A and B accordingly (i.e. add 5 years to the present ages of each person to go from present Era to 5 years in future), e.g.:
Era Mr. A Mr. B
–7) 3x x
0) 3x + 7 x + 7
+5) (3x + 7) + 5 (x + 7) + 5
= 3x + 12 = x + 12

Step 6: Make equation this time according to the statement in the second part of the question. (i.e. In 5 years, when A will be 20 years older than B) that means 3x + 12 = x + 12 + 20. After equation made, then solve it for value of x. And we have found x = 10.

Step 7: Plugin value of x where required according to the question. (e.g. according to the last question, we were asked to find age of A in 5 years, and that is 3x + 12. Plugin and solve to get answer. We plugin value of x, and got 42 years age of A, 5 years from now.

That’s it, we are damn sure on this answer!

Let’s do a hard difficulty level question scenario:
“Faiza is three years younger than Bilal. When they got married 30 years ago, the difference between thrice Bilal’s age and twice Faiza’s age was 25. How old will Bilal be when he celebrate his 50^th anniversary of his marriage?”

Solution:

Era F B
0) x – 3 x (At present: 0 years)
–30) (x – 3) – 30 ‘ x – 30 (30 years ago: –30 years)
= x – 33

Notice that, we move from present to 30 years back in the past, so 30 will be subtracted from ages of both persons, as shown above. Now given that, 30 years ago, difference between thrice Bilal’s age and twice Faiza’s age was 25 years, i.e. 3(age of Bilal 30 years ago) – 2(age of Faiza 30 years ago)

➩ 3(x – 30) – 2(x – 33) = 25 (Don’t forget to place parenthesis i.e. brackets)
➩   3x – 90 – 2x + 66 = 25 {As we know, – 2(x – 33) = – 2x + 66}
➩   x – 24 = 25
➩   x = 49

49 is the present age of Bilal, as we have initially supposed present age of Bilal as x. Given that, Bilal got married 30 years ago (i.e. when he had an age of 49 – 30 = 19 years), so after 50 years from his marriage with Faiza, he’ll have his 50th anniversary of marriage. Thus,
Age of Bilal on his 50th anniversary of marriage = 19 + 50 = 69 years Answer!

So far, we have learned how to solve age problem (with two-person scenario), when a person’s age is required at some given time period (e.g. present age, 5 years from now or 3 years ago etc.). Now is the best time to learn how to solve such problems when the time period is required to find if the age of a person is given. For instance,

let’s discuss one-person scenario where instead of age of person, the question asks to find Era. We’ll use same methodology as discussed earlier:
“x years ago, Mr A was 15 years old. In 2x years, he will become 39 years old. What is x?”

Solution:
Era       Mr A
–x)       15
0)    15 + x
+2x) (15+x) + 2x
= 15 + 3x
So, we have found that 2x years from now, age of A will be 15 + 3x. Also, it is given in the question that, 2x years from now, age of A will be 39 years. Thus,