Arithmetic (UPSC) Part- III

21. There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers? [2017-II]

(a) 47
(b) 49
(c) 51
(d) 45

Correct Answer: (c) 51

Solution:Since 39 is the mean of first five numbers, first five
numbers are
35, 37, 39, 41 and 43 and next 8 numbers are 45, 47, 49,
51, 53, 55, 57 and 59.

Hence, mean = (13 + 1)/2 , 7th number or middle number = 47.

22. There is a milk sample with 50% water in it. If 1/3rd of this milk is added to equal amount of pure milk, then water in the new mixture will fall down to [2017-II]

(a) 25%
(b) 30%
(c) 35%
(d) 40%

Correct Answer: (a) 25%

Solution:Let the original amount be 150 ml.
According to question.
50 ml of mixture + 50ml of pure milk
⇒ 25 ml of Milk + 25ml of water + 50 ml of pure milk.

Hence, % of water in new mixture = (25/100) × 100 = 25%.

23. There are 4 horizontal and 4 vertical lines, parallel and equidistant to one another on a board. What is the maximum number of rectangles and squares that can be formed? [2017-II]

(a) 16
(b) 24
(c) 36
(d) 42

Correct Answer: (c) 36

Solution:

Number of squares and rectangles in a 3 × 3 grid
= 1³ + 2³ + 3³ = 36.

24. Certain 3-digit numbers have the following characteristics: [2017-II]

All the three digits are different.
The number is divisible by 7.
The number on reversing the digits is also divisible by 7.

How many such 3-digit numbers are there?

(a) 2
(b) 4
(c) 6
(d) 3

Correct Answer: (b) 4

Solution:Let the numbers are of the form abc.
So, According to question,
100 a + 10b + c = 7K ....(i)
100c + 10b + a = 7m ....(ii)

From, (i) – (ii)
99a – 99c = 7 (k–m)
99 (a – c) = 7n
a – c = 7
a = 9, c = 2 a = 8, c = 1
Hence, 4 numbers, 259, 952, 168 and 861.

25. How many numbers are there between 99 and 1000 such that the digit 8 occupies the units place? [2017-II]

(a) 64
(b) 80
(c) 90
(d) 104

Correct Answer: (c) 90

Solution:—— 8
There are 9 values (1 to 9) for hundreds place digit. While 10 values (0 to 10) for ten’s place digit.
Hence, 9 × 10 = 90 Numbers.

26. If for a sample data [2017-II]

Mean < Median < Mode
then the distribution is

(a) symmetric
(b) skewed to the right
(c) neither symmetric nor skewed
(d) skewed to the left

Correct Answer: (d) skewed to the left

Solution:Skewed to the left.

27. The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again? [2017-II]

(a) 42
(b) 38
(c) 25
(d) 16

Correct Answer: (b) 38

Solution:Mr. x’s present age 26, because 25 was a perfect square
and the next year would be a perfect cube.
Next cube number → 64
Hence, minimum years required = 64 – 26 = 38 years.

28. P works thrice as fast as Q, whereas P and Q together can work four times as fast as R. If P, Q and R together work on a job, in what ratio should they share the earnings? [2017-II]

(a) 3 : 1 : 1
(b) 3 : 2 : 4
(c) 4 : 3 : 4
(d) 3 : 1 : 4

Correct Answer: (a) 3 : 1 : 1

Solution:Let the work done by Q in 1 day = x units.
So the work done by P in 1 day = 3x units

Work done by R in 1 day = (3x + x)/4 = x units

Hence, the ratio of earnings =

P Q R
3x : x : x

= 3 : 1 : 1.

29. There are certain 2-digit numbers. The difference between the number and the one obtained on reversing it is always 27. How many such maximum 2-digit numbers are there? [2017-II]

(a) 3
(b) 4
(c) 5
(d) None of the above

Correct Answer: (d) None of the above

Solution:Let the two digit numbers are of the form ab.
So, 10a + b – (10b + a) = 27
⇒ 9a – 9b = 27
⇒ a – b = 3
(a, b) ⇒ (9, 6), (8, 5), (7, 4) (6, 3) (4,1), (5, 2)
Hence, 96 , 85 , 74 , 63 , 41 , 52 and 25, 14, 36, 47, 58, 69
are the required numbers.

30. A 2-digit number is reversed. The larger of the two numbers is divided by the smaller one. What is the largest possible remainder? [2017-II]

(a) 9
(b) 27
(c) 36
(d) 45

Correct Answer: (d) 45

Solution:94 divided by 49 leaves the largest remainder 45.