Arithmetic

Dear students,
Arithmetic constitutes a very important part of the Quantitative Ability section in CAT. Here is a set of 5 questions from previous year CAT. This will help you to adopt the correct strategies while solving the question and improve your accuracy.

  1. Ten years ago, the ages of the members of a joint family of eight people added up to 231 years. Three years later, one member died at the age of 60 years and a child was born during the same year. After another three years, one more member died, again at 60, and a child was born during the same year. The current average age of this eight-member joint family is nearest to
    (1) 23 years
    (2) 22 years
    (3) 21 years
    (4) 25 years
    (5) 24 years
  2. A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja. giving her rupees instead of paise and paise instead of rupees. After buying a toffee for 50 paise, Shailaja noticed that she was left with exactly three times as much as the amount on the cheque. Which of the following is a valid statement about the cheque amount?
    (1) Over Rupees 13 but less than Rupees 14
    (2) Over Rupees 7 but less than Rupees 8
    (3) Over Rupees 22 but less than Rupees 23
    (4) Over Rupees 18 but less than Rupees 19
    (5) Over Rupees 4 but less than Rupees 5
  3. Arun, Barun and Kiranmala start from the same place and travel in the same direction at speeds of 30, 40 and 60 km per hour respectively. Barun starts two hours after Arun. If Barun and Kiranmala overtake Arun at the same instant, how many hours after Arun did Kiranmala start?
    (1) 3
    (2) 3.5
    (3) 4
    (4) 4.5
    (5) 5
  4. A milkman mixes 20 litres of water with 80 litres of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he has sold. What is the current proportion of water to milk?
    (1) 2:3
    (2) 1:2
    (3) 1:3
    (4) 3:4
  5. Karan and Arjun run a 100-metre race, where Karan beats Arjun by 10 metres. To do a favour to Arjun, Karan starts 10 metres behind the starting line in a second 100-metre race. They both run at their earlier speeds. Which of the following is true in connection with the second race?
    (1) Karan and Arjun reach the finishing line simultaneously.
    (2) Arjun beats Karan by 1 metre.
    (3) Arjun beats Karan by 11 metres.
    (4) Karan beats Arjun by 1 metre.

Solution :-

  1. The sum of the ages of the family ten years ago = 231
    The sum of the ages of the family seven years ago = 231 + (3 × 8) – 60 = 195
    The sum of the ages of the family four years ago = 195 + (3 × 8) – 60 = 159
    The sum of the ages of the members of the family now = 159 + (4 × 8) = 191
    So ,average = 191/8 = 23.875 ≈ 24 Hence, option 5.
  2. Let the amount on Shailaja’s cheque be Rs. x and paise y = (100x + y) paise (x and y are positive integers)
    The teller gives her (100y + x) paise.
    Now, 100y + x – 50 = 3(100x + y)
    97y – 299x = 50
    y = (50 + 299x)/97 = (50 + 8x)/97 + 3
    Now as y is an integer, (50 + 8x) has to be a multiple of 97 with x, y ≤ 99
    50 + 8x = 97k (k is an integer) x = 12k – 6 + (k – 2)/8 k = 2, 10, 18… x = 18, 115, 212… x = 18 is the only possible value.
    This implies that y = 5
    The amount on Shailaja’s cheque is over Rs. 18 but less than Rs. 19.
    Hence, option 4.
  3. Arun has travelled 60 km when Barun starts.
    Barun overtakes Arun in 60/(40 – 30) = 6 hrs
    In this time, Barun travels 6 × 40 = 240 km from the starting point.
    Kiranmala overtakes Arun at the same point.
    Kiranmala takes 240/60 = 4 hrs to reach there.
    Arun takes 240/30 = 8 hrs to reach there.
    Kiranmala starts 8 – 4 = 4 hrs after Arun.
    Hence, option 3.
  4. Total mixture = 100
    Milk = 80 litres, water = 20 litres
    ¼ of the mixture = 25 litres, which is sold
    Sold quantity of milk = 25 × 4/5 = 20 litres
    Sold quantity of water = 25 × 1/5 = 5 litres Remaining quantity of milk =80-20=40 litres Remaining quantity of water =20-5=15 litres After adding 25 litres of water, quantity of water =15+25 =40 litres Now ratio of water to milk =40:60 =2:3
  5. Karan = 100
    Arjun = 90
    Now, Karan has to run 110 metre.
    Arjun will run = 9900/100 = 99 metre.
    Karan beats Arjun by 1 metre.

Set analysis:-
Level of difficulty:- Moderate 
Expected time to solve:- 10 to 12 minutes.

Hope you learnt!
See you with the next one tomorrow!
Happy learning!
Gajendra Sir,
CAT Coach

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