CAT 2019 Practice Test #1 - Discussion Thread
Please refer to the Practice Test #1 for the test questions. Please post your solutions and explanations below.
Q. N=26 * 55 * 76 * 107. How many factors of 'N' are even?
Solution - Upon complete prime factorisation, we get 213 * 512 * 76. For the number of even factors, we need a minimum of one '2', which means that while calculating the total number of factors, you must omit 20 = 1, which will not yield even factors. So, the number of even factors becomes 13*13*7 = 1183.
Alternatively, you can also calculate the total number of factors, subtract the odd number of factors and arrive at the answer, i.e., 14*13*7 = 1274, then subtract from this 13*7 = 91. You will arrive at 1274-91 = 1183.
Note - Always think of the shortest method to solve questions, such as in this case where the basics of odd and even numbers are utilised to solve a moderately complex question.
Q. 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?
Solution - Total amount of mixture is 100 litres. 1/4th, or 25% of this 25 litres, which is removed. Now, 25 litres of water is added to the remaining 75 litres of the mixture. Now, 75 litres is still in the ratio of 4:1, which means the milk is 60 litres and water is 15 litres. Now, with 25 litres of water added, this becomes 40 litres of water, with 60 litres of milk. The ratio of water to milk becomes 40:60, i.e., 2:3.
Note - For questions on Averages, Mixtures and Alligations, ensure that using formulae is a last resort. Try and solve questions through ratios, proportions and percentages.
Q. A father tells his son: "I am five times as old as you were when I was as old as you are now.'' If the sum of the ages of the father and son is 88, how old is the son now?
Solution - This question can be simplified through ratios, but can also be solved algebraically. We will take a look at the latter solution.
Let the Father's current age be 'F', and Son's current age be 'S'. Let the father's age when he was as old as the son be 'f', and the son's age then be 's'.
From the question, we identify that F=5s. Also, since the difference between their ages is always constant, F-S = f-s. Additionally, we know that f=S. Therefore, substituting the values, we get the ratio of S/s as 3:1, which implies that S=3s.
F=5s, S=3s. Multiplying with 3 and 5 respectively, we can now equate the two, which brings us to a ratio of F/S as 5:3. Since the sum of the ages is 88, we get the father's age as 55 and the son's age as 33. If you're not familiar with ratios, here's how we did it:
5k+3k = 88, ∴ k=11. Substituting the value of k, we get the ratio of ages as 55:33.
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Could you provide the solutions too, please? Thanks.
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