Problems on Numbers Practice Test Questions – Aptitude -- Free Practice

Question 1

The sum of two numbers is 75 and their difference is 20. Find the difference of their squares.

A

1500✓✗

B

1600✓✗

C

1550✓✗

D

None of these✓✗

Explanation

Difference of squares = (sum) × (difference) = 75 × 20 = 1500
(Using the identity a² - b² = (a + b)(a - b))

Question 2

A number consists of two digits. The digit in the tens place is 3 times the digit in the unit's place. If 54 is subtracted, the digits are reversed. The number is:

A

39✓✗

B

92✓✗

C

93✓✗

D

94✓✗

Explanation

Let the units digit = u, so the tens digit = 3u.
Number = 30u + u = 31u. Reversed number = 10u + 3u = 13u.
31u - 54 = 13u
18u = 54
u = 3
Number = 31 × 3 = 93

Question 3

In a two-digit number, the digit in the ten's place is twice the digit in the unit's place. If 18 is subtracted from the number, the digits are reversed. Find the number.

A

63✓✗

B

21✓✗

C

42✓✗

D

None of these✓✗

Explanation

Let the unit's digit = u, so the ten's digit = 2u.
Number = 20u + u = 21u. Reversed number = 10u + 2u = 12u.
21u - 18 = 12u
9u = 18
u = 2
Number = 21 × 2 = 42

Question 4

The sum of two numbers is 62 and their product is 960. The sum of their reciprocals is

A

31480✓✗

B

29480✓✗

C

61960✓✗

D

41960✓✗

Explanation

Sum of reciprocals = 1a + 1b = a + bab = 62960 = 31480

Question 5

A number consists of two digits. The digit in the ten's place is 3 times the digit in the unit's place. If 54 is subtracted from the number, then the digits are reversed. The number is:

A

39✓✗

B

62✓✗

C

93✓✗

D

31✓✗

Explanation

Let the unit's digit = u, so the ten's digit = 3u.
Number = 30u + u = 31u. Reversed number = 10u + 3u = 13u.
31u - 54 = 13u
18u = 54
u = 3
Number = 31 × 3 = 93

Question 6

The product of two consecutive odd numbers is 4623. Which is the smaller of the two numbers?

A

71✓✗

B

73✓✗

C

67✓✗

D

57✓✗

Explanation

Let the two consecutive odd numbers be n and n + 2. Then, n(n + 2) = 4623, i.e., n² + 2n - 4623 = 0. Discriminant = 4 + 4 × 4623 = 18496; √18496 = 136. So, n = -2 + 1362 = 67. (Check: 67 × 69 = 4623.) The smaller number is 67.

Question 7

The denominator of a fraction is 2 more than numerator. If the numerator as well as denominator is increased by 4, the fraction becomes 810. Find the original fraction.

A

23✓✗

B

13✓✗

C

47✓✗

D

25✓✗

Explanation

Let the numerator be n, so the denominator is n + 2. n + 4n + 6 = 810 = 45. Cross-multiplying: 5(n + 4) = 4(n + 6), so 5n + 20 = 4n + 24, giving n = 4. The denominator is 6, so the original fraction is 46 = 23.

Question 8

Find the sum of the second largest and the smallest number, if the sum of five consecutive odd numbers is 595.

A

236✓✗

B

351✓✗

C

263✓✗

D

298✓✗

Explanation

Let the five consecutive odd numbers be n, n+2, n+4, n+6, n+8. Sum = 5n + 20 = 595, so n = 115. The numbers are 115, 117, 119, 121, 123. Second largest = 121, smallest = 115. Required sum = 121 + 115 = 236.

Question 9

A positive integer, which when added to 100, gives a sum which is greater than when it is multiplied by 100. What is the number?

A

5✓✗

B

2✓✗

C

3✓✗

D

1✓✗

Explanation

n+100 > 100n means n < 10099 ≈ 1.01. The only positive integer satisfying this is n=1.

Question 10

The sum of two numbers is 73 and their difference is 28. Find the difference between their squares.

A

101✓✗

B

45✓✗

C

2044✓✗

D

Cannot be determined✓✗

Explanation

Difference of squares = (sum) × (difference) = 73 × 28 = 2044.

Question 11

67 of a certain number is 96. Find a quarter of that number.

A

112✓✗

B

32✓✗

C

56✓✗

D

28✓✗

Explanation

Number = 96 × 76 = 112. Quarter = 1124 = 28.

Question 12

A train starts with X passengers. At the first station, one third of the passengers get down and 280 new passengers board the train. At the second station, half of the new total passengers get down and 12 new passengers board the train. On reaching the final destination, the train has 248 passengers. How many passengers did the train start with?

A

248✓✗

B

288✓✗

C

280✓✗

D

None of these✓✗

Explanation

After station 1: 2X/3+280. After station 2 (half get down, 12 board): (2X/3+280)/2+12 = X/3+152. Setting this equal to 248 gives X/3=96, so X=288.

Question 13

A man earns Rs 20 on day 1 and spends Rs 15 on the next day. On day 3 he again earns Rs. 20 and spends Rs 15 on the next day. If this continues, when will he have Rs 60 in hand?

A

On 17th day✓✗

B

On 26th day✓✗

C

On 27th day✓✗

D

On 30th day✓✗

Explanation

Tracking day by day: he has 5 more Rs every 2 days after the first earn-spend cycle (20,5,25,10,30,15,...). Following this pattern, he reaches Rs 60 exactly on day 17.

Question 14

The sum of a number and its square is 182. What is the number?

A

14✓✗

B

13✓✗

C

26✓✗

D

28✓✗

Explanation

x+x²=182 gives x²+x-182=0. Discriminant=729=27². x=(-1+27)/2=13.

Question 15

One-third of one-fourth of a number is 15. Find three-tenths of that number.

A

35✓✗

B

36✓✗

C

54✓✗

D

74✓✗

Explanation

(13)(14)x=15 gives x=180. Three-tenths of 180 = 54.

Question 16

A, B and C distribute Rs. 1,000 among themselves. A and C together have Rs. 400, B and C together have Rs. 700. How much does C have?

A

Rs 100✓✗

B

Rs 50✓✗

C

Rs 200✓✗

D

Rs 300✓✗

Explanation

(A+C)+(B+C)-(A+B+C) = C, so C = 400+700-1000 = Rs.100.

Question 17

A 40 feet rope is cut into 2 pieces. One piece is 18 feet longer than the other. What is the length of the shorter piece?

A

9✓✗

B

11✓✗

C

18✓✗

D

22✓✗

Explanation

Let shorter=x, longer=x+18. 2x+18=40, so x=11 feet.

Question 18

A boy was asked to multiply a number by 78. Instead he divided the number by 78 and got a result 1514 more than what he should have got if he had multiplied the number by 78. The number is:

A

4✓✗

B

7✓✗

C

8✓✗

D

9✓✗

Explanation

8n/7 - 7n/8 = 1514 simplifies to 15n/56=1514, giving n=4.

Question 19

A student, instead of finding the value of 78 of a number, found the value of 718 of the number. If his answer differed from the actual one by 770, find the number.

A

1584✓✗

B

2520✓✗

C

1728✓✗

D

1656✓✗

Explanation

(78-718) × N = 770. Since 18-118=572, 7N × 5/72 = 770 gives N=1584.

Question 20

P and Q are two integers such that PQ = 64. Which of the following cannot be the value of P + Q?

A

20✓✗

B

65✓✗

C

16✓✗

D

35✓✗

Explanation

Integer factor pairs of 64 give sums 65, 34, 20, 16 (and their negatives), but never 35.

Question 21

Seema weighs 59 of her weight plus 20 kg. What is the actual weight of Seema?

A

36 kg✓✗

B

72 kg✓✗

C

27 kg✓✗

D

45 kg✓✗

Explanation

W = 5W/9+20 gives 4W/9=20, so W=45 kg.

Question 22

How many natural numbers n, where n < 50, exist satisfying the condition that (n-1)! is not a multiple of n?

A

15✓✗

B

17✓✗

C

14✓✗

D

16✓✗

Explanation

By Wilson's theorem, (n-1)! is not a multiple of n exactly when n is prime, plus the special case n=4. There are 15 primes below 50, plus n=4, giving 16.

Question 23

I bought 2 varieties of pens, the first variety for Rs 12 each and the second variety costing Rs 17 each, spending Rs 157 in total. In how many different combinations could I have purchased the pens?

A

2✓✗

B

5✓✗

C

1✓✗

D

4✓✗

Explanation

Solving 12x+17y=157 in non-negative integers gives only one solution: x=6, y=5.

Question 24

Find the number of integers satisfying the inequality (x²+x-6)(x²-x-20) < 0.

A

4✓✗

B

2✓✗

C

0✓✗

D

infinite✓✗

Explanation

Factoring gives (x+3)(x-2)(x-5)(x+4)<0, satisfied on (-4,-3) union (2,5). Only integers strictly inside these intervals are 3 and 4, i.e. 2 integers.

Question 25

There are rabbits and pigeons. There are 200 heads and 580 legs. How many pigeons are there?

A

100✓✗

B

120✓✗

C

110✓✗

D

130✓✗

Explanation

Let r=rabbits(4 legs), p=pigeons(2 legs). r+p=200 and 4r+2p=580. Solving gives r=90, p=110.