Question Bank  100 Arithmetic Questions From Previous CAT Papers (Solved)

Let price of 1 meter is 100
Merchant bought 1200 cm for 100 and he sells 800 cm for 80
Selling price of 1200 cm is 1200 x 80/800 = 120
Profit percentage is 20%

Q51. (CAT 1996)
The cost of diamond varies directly as the square of its weight. Once, this diamond broke into four pieces with weights in the ratio 1 : 2 : 3 : 4. When the pieces were sold, the merchant got Rs. 70,000 less. Find the original price of the diamond.
a. Rs. 1.4 lakh
b. Rs. 2 lakh
c. Rs. 1 lakh
d. Rs. 2.1 lakh

If we assume weight of diamond 10x. Well I came up with 10x because the ratio given 1 : 2 : 3 : 4 adds upto 10.
So individual weights of diamonds will be x, 2x, 3x and 4x
original price = k(10x)^2
Price for pieces = k(x^2 + 4x^2 + 9x^2 + 16x^2) = k30x^2
Thus change in price we see is 100kx^2  30kx^2 = 70kx^2 which is given as 70,000
Thus the original price 100kx^2 will be 1,00,000
will be 1,00,000

Q52. (CAT 1996)
In a mile race, Akshay can be given a start of 128 m by Bhairav. If Bhairav can give Chinmay a start of 4 m in a 100 m dash, then who out of Akshay and chinmay will win a race of one and half miles and what will be the final lead given by the winner to the loser? (One mile is 1600 m)
(a) Akshay, 1/2 mile
(b) Chinmay, 1/32 mile
(c) Akshay, 1/24 mile
(d) Chinmay, 1/16 mile

B will cover 1600m when A covers 1600 – 128 = 1472m
Let the time taken be t
Speed of B= 1600/t
Speed of A= 1472/t
Speed of B/ Speed of A= 1600 / 1472 = 25/23  (1)
Similarly
Speed of B / Speed of C = 100/96 = 25/24  (2)
From (1) and (2)
Speed of A / Speed of C = 23/24
Let speed of A = 23x, then Speed of C = 24x
Time taken by C to complete one and half miles race = (1600 x 1.5)/24x = 100/x
In same time distance covered by A = 100/x * 23x = 2300m
Therefore lead = 2400 – 2300
= 100 m
= 1/16 mile
Therefore winner is C by 1/16 mile

Q53. (CAT 1996)
Two liquids A and B are in the ratio 5 : 1 in container 1 and 1 : 3 in container 2. In what ratio should the contents of the two containers be mixed so as to obtain a mixture of A and B in the ratio 1 : 1?
a. 2 : 3
b. 4 : 3
c. 3 : 2
d. 3 : 4

Take 1 portion from container 1 and N portions from container 2.
Then we collect total 5A/6 + B/6 + N(3B/4 + A/4)
Ratio of A and B in the new mix is (5/6 + N/4)A : (3N/4 + 1/6)B
for A : B = 1 : 1, the coefficients of A and B must be equal
(5/6 + N/4) = (3N/4 + 1/6)
N = 4/3
So the required ratio is 1 : 4/3 = 3 : 4

Q54. (CAT 1996)
A man travels threefifths of a distance AB at a speed 3a, and the remaining at a speed 2b. If he goes from B to A and return at a speed 5c in the same time, then
a) 1/a + 1/b = 1/c
b) a + b = c
c) 1/a + 1/b = 2/c
d) None of these

Let distance between A and B = d
Time taken to cover 3d/5 = (3d/5)/3a = d/5a
Time taken to cover remaining distance = (2d/5)/2b = d/5b
Time taken to go from B to A and then return to B = 2d/5c
Now, d/5a + d/5b = 2d/5c
1/a + 1/b = 2/c

Q55. (CAT 1996)
A man travels from A to B at a speed x km/hr. He then rests at B for x hours. He then travels from B to C at a speed 2x km/hr and rests for 2x hours. He moves further to D at a speed twice as that between B and C. He thus reaches D in 16 hr. If distances AB, BC and CD are all equal to 12 km, the time for which he rested at B could be
a. 3 hr
b. 6 hr
c. 2 hr
d. 4 hr

Total time taken = Time taken for each stretch + rest time between stretches = 16
(12/x) + x + (12/2x) + 2x + (12/4x) = 16
12/x + x + 1/6x + 2x + 1/3x = 16
3x^2  16x + 21 = 0
x = 3 hours

Q56. (CAT 1996)
The price of a Maruti car rises by 30% while the sales of the car come down by 20%. What is the percentage change in the total revenue?
a. –4%
b. –2%
c. +4%
d. +2%

Let say 100 cars were sold before for a price of 100 rupee each.
So total revenue before = 10000
Now car price is 130 and sale is 80. Total revenue = 10400
Percentage change = + 4%

Q57. (CAT 1996)
In a watch, the minute hand crosses the hour hand for the third time exactly after every 3 hr 18 min and 15 s of watch time. What is the time gained or lost by this watch in one day?
a. 14 min 10 s lost
b. 13 min 50 s lost
c. 13 min 20 s gained
d. 14 min 40 s gained

Remember this point  In a normal watch, minute hand crosses hour hand every 1 hour 5 minutes and 27 seconds ( which is equal to 720/11 minutes or ~ 3930 seconds)
So to cross thrice, a correct watch would take  3 hours 16 minutes and 21 seconds
Here it took 3 hours 18 minutes and 15 seconds, means it is losing time. (1 minute and 54 seconds for every 3 hours 18 minutes and 15 seconds)
So the time lost in a day (24 hours) is approximately 13 minutes 50 seconds.

Q58. (CAT 1996)
I sold two watches for Rs. 300 each, one at the loss of 10% and the other at the profit of 10%. What is the percentage of loss (–) or profit (+) that resulted from the transaction?
a. (+) 10
b. (–) 1
c. (+) 1
d. (–) 10

Formula  If we sell an item at the same price, one at a profit of x% and another at a loss of x%, then the net result is LOSS of (x/10)^2
here x = 10 so it is a loss of (10/10)^2 = Loss of 1 = (  1 )

Q59. (CAT 1995)
A man invests Rs. 3,000 at the rate of 5% per annum. How much more should he invest at the rate of 8%, so that he can earn a total of 6% per annum?
a. Rs. 1,200
b. Rs. 1,300
c. Rs. 1,500
d. Rs. 2,000

Use Alligation
Present rate of interest = 5% (Cheaper quantity)
New rate of interest = 8% (Dearer Quantity)
Mean rate of interest = 6%
Ratio of Dearer Value : Cheaper Value = m  c : d  m = 6  5 : 8  6 = 1 : 2
Dearer Value = Rs.3000/2 = Rs.1500

Q60. (CAT 1995)
In a race of 200 m run, A beats S by 20 m and N by 40 m. If S and N are running a race of 100 m with exactly same speed as before, then by how many metres will S beat N?
a. 11.11 m
b. 10 m
c. 12 m
d. 25 m