Question 1

The sum of two number is 25 and their difference is 13. Find their product.

Accepted Answer

Let the numbers be x and y. Then, x + y = 25 and x - y = 13. 4xy = (x + y)2 - (x- y)2 = (25)2 - (13)2 = (625 - 169) = 456 ∴ xy = 114.

Question 2

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:

Accepted Answer

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 ⟹ 22x2 + 26x - 140 = 0 ⟹ 11x2 + 13x - 70 = 0 ⟹ (x - 2)(11x + 35) = 0 ⟹ x = 2. Hence, required number = 11x + 2 = 24.

Question 3

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:

Accepted Answer

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 4

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:

Accepted Answer

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.

Question 5

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:

Accepted Answer

Let the ten's and unit digit be x and 8 respectively. x Then, ❨ 10x + 8 ❩ + 18 = 10 x 8 + x x x ⟹ 10x2 + 8 + 18x = 80 + x2 ⟹ 9x2 + 18x - 72 = 0 ⟹ x2 + 2x - 8 = 0 ⟹ (x + 4)(x - 2) = 0 ⟹ x = 2.

Question 6

The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is:

Accepted Answer

Let the numbers be x and y. Then, xy = 9375 and x = 15. y xy = 9375 (x/y) 15 ⟹ y2 = 625. ⟹ y = 25. ⟹ x = 15y = (15 x 25) = 375. ∴ Sum of the numbers = x + y = 375 + 25 = 400.

Question 7

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?

Accepted Answer

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.

Question 8

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?

Accepted Answer

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 9

If one-third of one-fourth of a number is 15, then three-tenth of that number is:

Accepted Answer

Let the number be x. Then, 1 of 1 of x = 15 ⟺ x = 15 x 3 x 4 = 180. 3 4 So, required number = ❨ 3 x 180 ❩ = 54. 10

Question 10

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

Accepted Answer

Let the three integers be x, x + 2 and x + 4. Then, 3x = 2(x + 4) + 3 ⟺ x = 11. ∴ Third integer = x + 4 = 15.

Question 11

The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is:

Accepted Answer

Let the numbers be x and y. Then, xy = 120 and x2 + y2 = 289. ∴ (x + y)2 = x2 + y2 + 2xy = 289 + (2 x 120) = 529 ∴ x + y = √529 = 23.

Question 12

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?

Accepted Answer

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 13

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:

Accepted Answer

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.

Question 14

Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.

Accepted Answer

Let the number be x. Then, x + 17 = 60 x ⟹ x2 + 17x - 60 = 0 ⟹ (x + 20)(x - 3) = 0 ⟹ x = 3.

Question 15

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

Accepted Answer

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 16

The largest 4 digit number exactly divisible by 88 is

Accepted Answer

To find the largest 4 digit number exactly divisible by 88, We should divide the largest possible 4 digit number by 88, and if we get any remainder than subtract it from that largest number. The largest possible 4 digit number is 9999 Now, 88) 9999 (113 88 _______ 119 88 _______ 319 264 _______ 55 _______ Therefore, the largest 4 digit number exactly divisible by 88 is given by 9999 - 55 = 9944.

Question 17

What is the place value of 6 in 64?

Accepted Answer

We know that, ● Each digit has a fixed position called its place. ● Each digit has a value depending on its place called the place value of the digit. ● The face value of a digit for any place in the given number is the value of the digit itself ● Place value of a digit = (face value of the digit) × (value of the place). Hence, the place value of 6 in 64 = 6 x 10 = 60.

Question 18

In a two-digit number, the digit in the unit's place is more than twice the digit in ten's place by 1. If the digits in the unit's place and the ten's place are interchanged,difference between the newly formed number and the oiginal number is less t…

Accepted Answer

Let the ten's digit be x. Then, unit's digit = 2x + 1. [10x + (2x + 1)] - [{10 (2x + 1) + x} - {10x + (2x + 1)}] = 1 <=> (12x + 1) - (9x + 9) = 1 <=> 3x = 9, x = 3. So, ten's digit = 3 and unit's digit = 7. Hence, original number = 37.

Question 19

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

Accepted Answer

Let the three integers be x, x + 2 and x + 4. Then, 3x = 2 (x + 4) + 3 <=> x = 11. Third integer = x + 4 = 15.

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