Problems on Numbers

Question: The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is:
[A].

20

[B].

30

[C].

40

[D].

None of these

Answer: Option A

Explanation:

Let the numbers be a, b and c.

Then, a2 + b2 + c2 = 138 and (ab + bc + ca) = 131.

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) = 138 + 2 x 131 = 400.

(a + b + c) = 400 = 20.

Video Explanation: https://youtu.be/qmJ-0X8j_xQ

Question:
A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. The number is:

[A].

18

[B].

24

[C].

42

[D].

81

Answer: Option B

Explanation:

Let the ten’s and unit digit be x and	8	respectively.
	x

Then,		10x +	8		+ 18 = 10 x	8	+ x
			x			x

10×2 + 8 + 18x = 80 + x2

9×2 + 18x – 72 = 0

x2 + 2x – 8 = 0

(x + 4)(x – 2) = 0

x = 2.

Question:
A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed. The number is:

[A].

145

[B].

253

[C].

370

[D].

352

Answer: Option B

Explanation:

Let the middle digit be x.

Then, 2x = 10 or x = 5. So, the number is either 253 or 352.

Since the number increases on reversing the digits, so the hundred’s digits is smaller than the unit’s digit.

Hence, required number = 253.

Question:
The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?

[A].

4

[B].

8

[C].

16

[D].

None of these

Answer: Option B

Explanation:

Since the number is greater than the number obtained on reversing the digits, so the ten’s digit is greater than the unit’s digit.

Let ten’s and unit’s digits be 2x and x respectively.

Then, (10 x 2x + x) – (10x + 2x) = 36

9x = 36

x = 4.

Required difference = (2x + x) – (2x – x) = 2x = 8.

Question: The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?
[A].

3

[B].

4

[C].

9

[D].

Cannot be determined

Answer: Option B

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, (10x + y) – (10y + x) = 36

9(x – y) = 36

x – y = 4.

Video Explanation: https://youtu.be/7QOJjAmGVx0

Question: A number consists of two digits. If the digits interchange places and the new number is added to the original number, then the resulting number will be divisible by:
[A].

3

[B].

5

[C].

9

[D].

11

Answer: Option D

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, number = 10x + y.

Number obtained by interchanging the digits = 10y + x.

(10x + y) + (10y + x) = 11(x + y), which is divisible by 11.

Video Explanation: https://youtu.be/lytJE8GqRvM

Question:
In a two-digit, if it is known that its unit’s digit exceeds its ten’s digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:

[A].

24

[B].

26

[C].

42

[D].

46

Answer: Option A

Explanation:

Let the ten’s digit be x.

Then, unit’s digit = x + 2.

Number = 10x + (x + 2) = 11x + 2.

Sum of digits = x + (x + 2) = 2x + 2.

(11x + 2)(2x + 2) = 144

22×2 + 26x – 140 = 0

11×2 + 13x – 70 = 0

(x – 2)(11x + 35) = 0

x = 2.

Hence, required number = 11x + 2 = 24.

Question:
The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?

[A].

69

[B].

78

[C].

96

[D].

Cannot be determined

Answer: Option D

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, x + y = 15 and x – y = 3   or   y – x = 3.

Solving x + y = 15   and   x – y = 3, we get: x = 9, y = 6.

Solving x + y = 15   and   y – x = 3, we get: x = 6, y = 9.

So, the number is either 96 or 69.

Hence, the number cannot be determined.

Question:
What is the sum of two consecutive even numbers, the difference of whose squares is 84?

[A].

34

[B].

38

[C].

42

[D].

46

Answer: Option C

Explanation:

Let the numbers be x and x + 2.

Then, (x + 2)2 – x2 = 84

4x + 4 = 84

4x = 80

x = 20.

The required sum = x + (x + 2) = 2x + 2 = 42.

Question: Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:
[A].

9

[B].

11

[C].

13

[D].

15

Answer: Option D

Explanation:

Let the three integers be x, x + 2 and x + 4.

Then, 3x = 2(x + 4) + 3      x = 11.

Third integer = x + 4 = 15.

Video Explanation: https://youtu.be/_77C9YE321Y