# Ch.17 :- Arithmetical Reasoning : Introduction & Formula

**Example 1**

**Ex. 1. The number of boys in a class are three times the number of girls. Which one of the following number can’t represent the total number of children in the class?**

(a) 48

(b) 44

(c) 42

(d) 40

**Sol.** Let the number of girls be x, then from the question it is clear that number of bays are 3x.

Therefore, total no. of students

= Number of boys + Number of girls

= 3x + x

= 4x

Now, the total number of children in the class must be a multiple of 4. Out of the four options given (c) does not qualify this condition. Therefore, 42 does not represent the total number of children in the class. Hence, the correct answer is (c).

**Example 2**

**Ex. 2. The sum of the ages of a son and father is 56 years. After four years, the age of the father will be three times that of the son. Their ages respectively are:**

(a) 12 years, 44 years

(b) 16 years, 42 years

(c) 16 years, 48 years

(d) 18 years, 36 years

**Sol.** Let the age of the father be x, then the age of the son would be (56 – x). After four years, the age of father would be (x + 4) and that of son would be (56 -x + 4) years. Now, from the information given in the question,we have

(x + 4) = 3 (56 -x + 4)

=> x + 4 = 168 – 3x + 12

=> 4x = 168 + 12 – 4 = 176

=> x = 44 years

Therefore, the age of father and son is 44 years and 12 years, respectively. Hence, the correct answer is (a).

**Example 3**

**Ex. 3. In 10 years, A will be twice as old as B was 10 years ago. If at present A is 9 years older than B, the present age of B is:**

(a) 19 years

(b) 29 years

(c) 39 years

(d) 49 years

**Sol.** Let the present age of B be x years. Then, the present age of A would be (x + 9) years. After 10 years, the age of A would be

(x + 9 + 10)

= (x + 19) years

and before ten years, the age of B was (x – 10) years.

Now, from the information given in the question,

(x + 19) = 2 (x – 10)

or x + 19 = 2 x – 20

or x = 19 + 20 = 39 years

Therefore, the present age of B is 39 years.Hence, the correct answer is (c).

**Example 4**

**Ex. 4. The 1st bunch of bananas has 1/4 excess to as many as bananas in 2nd bunch. If the 2nd bunch has 3 bananas less than the 1st bunch, then the number of bananas in 1st bunch is :**

(a) 9

(b) 10

(c) 12

(d) 15

**Sol.** Let the number of bananas in 2nd bunch be x.

Then, the number of bananas in 1st bunch = x + ^{x}⁄_{4} = ^{5x}⁄_{4}

Therefore,

^{5x}⁄_{4} – x = 3 => 5x – 4x = 12 => x = 12.

Then, the number of bananas in 1st bunch

=^{5}⁄_{4} x 12 = 15

Hence, the correct answer is (d).

**Example 5**

**Ex. 5. In a town. 65% people watch the news on television, 40% read a newspaper and 25% read a newspaper and watch the news on television also. What percentage of the people neither watch the news on television nor read a newspaper?**

(a) 5%

(b) 10%

(c) 15%

**Sol.** Let the total number of people Let be 100.

Let circle A represents people

who watched television and B represents people who read newspaper.

Then, x + y = 65, Y + z = 40,

We get, x = 40, y = 25,

Then, the number of people who television nor read newspaper

= 100 – (x + y + z)

= 100- (40 + 25 + 15)

= 100-80 = 20

Therefore, the required percentage is 20%. Hence, the correct answer is (d).

**Example 6**

**Ex. 6. In a group of 15 people, 7 read French, 8 read English while 3 of them read none of these two. How many of them read French and English both?**

(a) 0

(b) 3

(c) 4

(d) 5

**Sol.** Let circles x and y represent people who read French and English, respectively.

Area A shows the people who read French only.

Area C shows the people who read English only.

Area B shows the people who read French and English both.

Now,

(A + B + C) + 3 = 15 or A + B + C = 12 … (i)

A + B = 7, B + C = 8

Adding these two, we get A + 2B + C = 15 … (ii)

Subtracting (i) from (ii), we get

B = 15 – 12 = 3

Therefore, number of people who read French and English both is 3.

Hence, the correct answer is (b).

**Example 7**

**Ex. 7. In a chess tournament each of six players will play with all other players exactly once. How many matches will be played during the tournament?**

(a) 12

(b) 15

(c) 30

(d) 36

**Sol.** The situation in which the matches will be played would be as follows:

(i) 1st player will play matches with other 5 players.

(ii) 2nd player will play matches with 4 players other than the 1st player.

(iii) 3rd player will play matches with 3 players other than 1st and 2nd players.

(iv) 4th player will play matches with 2 players other than 1st, 2nd and 3rd players.

(v) 5th player will play match with 6th player only.

Therefore, the number of matches played during the tournament is

= 5 + 4 + 3 + 2 + 1 = 15.

**Example 8**

**Ex.8. Consider the diagram given below:**

**Five hundred candidates appeared in the examination conducted for the tests in English, Hindi and Mathematics. The diagram gives the number of candidates who failed in different tests. What is the percentage of candidates who failed in at least two subjects ?**

(a) 0.078%

(b) 1.0%

(c) 6.8%

(d) 7.8%

**Sol.** From the diagram, it is clear that number of candidates who failed in at least two subjects

= number of candidates who failed in two or more subjects.

= (10 + 12 + 12 + 5)

= 39.

Therefore, the required percentage

= (^{39}⁄_{500} x 100)%

= 7.8%

Therefore, the option (d) is the correct answer.

**Example 9**

**Ex. 9. In an examination, 42% students failed in Hindi and 52% failed in English. If 17% failed in both the subjects, the percentage of those who passed in both the subjects, is :**

(a) 23%

(b) 27%

(c) 34%

(d) 40%

**Sol.** Let the total number of students who appeared for the examination be 100. Circles X and Y represent the students who failed in Hindi and English, respectively .

Now, number of students who failed in Hindi only

= (42-17)

= 25%

Number of students who failed in English only

(52 -17) = 35%

Total number of students failed

= students failed in Hindi only + students failed in English only

= 25 + 35 = 60%.

Number of students passed = 100 – 60 = 40%

Hence, option (d) is the correct answer.

**Example 10**

**Ex.10. A shepherd had 17 sheep. All but nine died. How many was he left with ?**

(a) Nil

(b) 8

(c) 9

(d) 17

**Sol.** “All but nine died” means “All except nine died”. It means that nine sheep remained alive and other died. Hence, option (c) is the correct answer.