2IIM Quant Notes - Percentage


  • Director, 2IIM Online CAT Preparation | IIT Madras | IIM Bangalore | CAT 100th percentile - CAT 2011, 2012 and 2014.


    Author: Vimal Gopinath - 2IIM (Bengaluru). .

    Percentage is a fairly intuitive idea for most of us. As we know, a percentage is essentially a comparison with 100. But test-makers do not always make it easy by giving us a round sum of 100 to work with. So, the first strategy is to understand how the problem works IF the sum were 100, and then extrapolate it to fit any scenario. In other words, work out the problem assuming a total of 100, and then scale it up or down to fit your problem. Let us try a few questions based on this idea:

    Question 1. 60% of the people in a class speak French. Of the remaining, 60% speak spanish. If 32 people speak neither french nor spanish, how many people are there in the class?

    60% Speak French - this means 40% of the class remains. Of this, 60% speak Spanish, so 40% of the remaining does not speak Spanish.
    so 40% of 40% do not speak either French or Spanish. 2/5*2/5 of the class does not speak either language, or 4/25 of the class counts for 32 people.
    32/4 * 25 = 200.
    There are 200 people in the class.

    Question 2. A and B are in the ratio 11: 14. By what percentage is A less than the sum of A and B?

    A and B need not be 11 and 14, but they COULD be 11 and 14, in which case their sum is 25. If we "scale this up" to 100, then A gets scaled up to 44, which is 56 less than the sum

    Question 3. If the length of a rectangle increases by 25%, then by what percentage should its width reduce so that its area remains the same?

    One of the ways of solving this question is to assume an area of 100 (lets say length = 10 and width = 10). Now, if the length increases by 25%, it becomes 12.5. So for the area to remain the same, the width should be 100/12.5 = 8. If the width is 8, its a 20% decrease

    Here's a slight variation on the previous question:

    Question 4. If the length of a rectancle decreases by 8%, by what percentage should the width increase so that its area increases by 15% ?

    Let original area be A. Desired area is 1.15A (an increase of 15%)
    Now the original product was L*B
    New product so far is 0.92L*B.
    We want to get an area that is 1.15 times the original. In essence, B has to be multiplied by some 'k', so that the product of 'k' and '0.92' is 1.15.
    So 0.92k = 1.15,
    k = 115/92 = 1.25
    Since B becomes 1.25B, it must increase by 25%

    The second strategy I want to talk about today concerns percentage changes. Measuring percentage changes the traditional way requires us to do the following calculation: (New – Old)*100 / Old. Now, with a bit of manipulation, this can be rearranged into: {(New/Old) – 1 }* 100. This just requires us to find out the New/Old ratio. If the ratio is equal to 1, it means that there is no change. Therefore, the percentage calculation gets scaled down to a comparison against 1, with a New/Old ratio of “1” indicating zero change. So, if the New / Old ratio is 1.1, it indicates a 10% increase, and if the New / Old ratio is 0.9, it indicates a 10% decrease, and so on.

    Question 5. If the New / Old ratio is 4.5, what is the percentage increase in this scenario?

    The formula is (New/Old - 1)*100, which gives us 350%

    Question 6. By selling an item at half its selling price, a trader makes a 25% loss. What would have been the profit percentage if he had sold it at the selling price?

    A loss of 25% means a SP/CP ratio of 0.75. But this happens only when we take the SP as half of the original SP. So, (0.5*SP)/CP = 0.75, which means SP/CP = 1.5 - > 50% profit.  As you can see, the New/Old method saves you a few precious seconds. the old school method will probably take you a couple of additional steps. there are no "step marks" on the CAT! :)

    Question 7. A milk merchant mixes milk and water in the ratio 4:1, and sells the mixture at the cost price of milk. What is his profit percentage?

    4:1 means that for every 800 ml of milk (for which he is paying), he is adding 200 ml water. So, he is getting paid for 1 litre (1000ml) of adultrated milk, but he is incurring the cost of only 800 ml of that. So, the SP/CP = 1000/800 = 1.25, which is a 25% profit.

    Here's a slightly modified version of the previous question:

    Question 8: An absent-minded rice merchant expects to make an illegal profit of 25% by using adding extra weight in his weighing balance scale. However, instead of adding the illegal weight to the weighing pan containing rice, by mistake he adds it to the weighing pan containing the weights and ends up making a loss. What was his loss percentage, if he claims to sell rice at the cost price?

    Let's break this down. Let's say the merchant wants to sell 1000 g of rice and make a profit of 25%, he has to pay for only 800 g of it. His cost is 800, and his selling price is 1000. So, he has to use an extra weight of 200 g on the weighing pan containing the rice. Now lets say he puts the 200 g on the other pan (containing the weight of 1000g), he ends up selling 1200 g of rice for the price of 1000. So, the SP / CP ratio is 1000/1200, or 5/6, which indicates a loss of 1/6. Or, 16.66% loss.

    Question 9. A vendor buys 16 items but he lost 4 items in a fire, and he can only sell 12 items. By what least percentage by which he should he mark up the 12 items so that he doesn't make a loss?

    He wants to break even on his sale. So he has to sell 12 items at the same cost of 16 items originally. so he has to sell his new items at 16/12, or 4/3 the original cost.
    He has to mark up by 33% to prevent loss

    Question 10: A two digit number ab is 60% of x. The two-digit number formed by reversing the digits of ab is 60% more than x. Find x.

    10a + b = 0.6x
    10b + a = 1.6x
    Subtracting one from the other, we have
    9b - 9a = x or x = 9 (b - a)
    x should be a multiple of 9.
    10a + b = this implies that x should also be a multiple of 5.Or, x should be a multiple of 45
    x should be equal to 45, ab = 27, ba = 72


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