MathOratory - Quant Notes - Arithmetic - Part 1


  • 99.9+ Quant %ile in CAT, XAT | IIM Lucknow | Managing Partner - MathOratory


    We will primarily solve questions from SNAP papers as today we are focusing on the basics of arithmetic. We will start with some basic concepts before we solve problems.

    Successive Percentage Change

    This is a very important concept in percentages and we can use it when the percentage values are good.

    The formula goes like this: If a certain value is first changed by a%, then by b%; then the net percentage change is given by
    (a + b + ab/100)%

    3 important things to NOTE here:

    1. The order of the changes does NOT matter. So, if it was b% change followed by a% change, even then the net change would have been the same.

    2. Note that I wrote percentage change, not percentage increase/decrease. It works good with both. Just take the appropriate signs. For increase, use a (+) value and for decrease a (-) value. If the final result is (+) or (-), that will denote whether there is a net increase or decrease

    3. Remember that the final result is also in percentage terms, not the value.

    Examples:
    20% increase followed by 30% decrease
    20 - 30 - 20*30/100 = -16%

    20% increase, followed by 50% increase, followed by 30% decrease
    Use the formula on any two of them first
    20+50+20 * 50/100 = 80%
    Then use the result with the 3rd value
    80 - 30 - 80 * 30/100 = 26%

    Componendo-Dividendo

    Now will see another interesting observation. Most of you by now must be knowing that reapplying componendo-dividendo on the componendo-dividendo form will give back the original form
    That is if we have,
    (x+y)/(x-y) = 5/2
    Reapplying comp-div on the left side, we will get x/y
    But if we apply on one side, we should be applying on both sides, to get:
    x/y = (5+2)/(5-2) = 7/3
    Even if you know this, do you know why this happens?
    The reason is that the coefficient of 'x' in the numerator and that of 'y' in the denominator adds to 0
    +1-1 = 0
    So, if we reapply the same process we get back the original form.
    This is a very helpful understanding, example:

    (2x+3y)/(2x-3y) = 5/4 ... this is a com-div form over 2x and 3y
    => 2x/3y = (5+4)/(5-4) = 9
    =>x/y = 27/2

    Let us extend this to something a bit different
    (2x+3y)/(3x-2y) = 5/4
    It's not a comp-div form, but observe that the coefficient of 'x' in the numerator (+2) and that of 'y' in the denominator (-2), adds to 0
    thus, if we reapply the whole process we should get back x/y
    But what is the process?
    (2 * numerator + 3 * denominator)/(3 * numerator - 2 * denominator)
    so, x/y = (2 * 5 + 3 * 4)/(3 * 5 - 2 * 4) = 22/7

    Multiplication factor

    This is a very important understanding of percentages from calculations point of view. What this basically does is: it converts a percentage question to a ratio question. And dealing with ratios are always easier than to deal with percentages, as ratios are easier to calculate.

    So, if a value increases by 20%, which is 1/5th: the multiplication factor must be (1+1/5) = 6/5, as the value is increasing
    If the value reduces by 30%, which is 3/10: the multiplication factor must be (1 - 3/10) = 7/10, as the value is reducing

    Try to use multiplication factors in the basic questions to ease up your calculations

    10 years ago the average age of all 25 teachers in a school was 45 years. 4 years ago one teacher retired from her post at the age of 58. After one year a new teacher whose age was 54 was recruited from outside. The present average age of all the teachers is ?

    A teacher retired at the age of 58 and a year later a teacher joined at the age of 54. The year on which the teacher joined, if that year the first teacher would have retired, then the first teacher's age at that point would have been 59

    So there's a net reduction in total of 5
    This net reduction will be carried forward throughout all the years
    If nobody would've retired or joined the average of the 25 people would've been 45+10=55, as we are moving 10 years ahead
    But remember there's a net reduction of 5 years in total
    As there is no change in the number of teachers, the net reduction in average would be 5/25 = 1/5
    So the current average should be 55 - 1/5 = 54 4/5

    A, B, C and D purchased a hotel for 56 lakhs. Contribution of B, C and D together is 460% of the contribution of A alone. Contribution of A, C and D is 366.66% that of the B's contribution and contribution of C is 40% of A, B and D together. Contribution of D is ?

    B,C,D is 460% of A, so A,B,C,D must be 560% of A
    As A,B,C,D together is 56 lakhs, and that corresponds to 560% of A
    100% of A corresponds to 10 lakhs (that's A's share)

    A,C,D is 366.66%, which is 3 + 2/3 = 11/3 of B
    So, A,B,C,D must be 1+11/3 = 14/3 of B

    14 corresponds to 56 lakhs, so 3 must correspond to 12 lakhs (B's share)

    C is 2/5 of A,B,D
    so, A,B,C,D must be 1 + 2/5 = 7/5 of A,B,D
    7 corresponds to 56 lakhs
    5 corresponds to 40 lakhs (that's sum of A,B, D share)

    out of which A = 10
    B = 12
    So, D = 40 - (10+12) = 18

    Weekly income of two persons are in the ration 7:2 and their expenses are in the ratio 5:2. If each of them save 300 rupees per week then the weekly income of the first person is ?

    Observe few things first: If we have two numbers. Both are increased/decreased by the same value. The difference of the two values should remain constant.

    Now, let us apply it here. But before we do that, understand that we can apply it here as both person saves equal amount. And Income - Savings = Expenditure

    So, the numerator and the denominator of the first ratio MUST reduce by a constant value to get the second ratio. If that happens the difference between the numerator and denominator must remain constant.

    Now, we have the first ratio 7/3 and the second one 5/2
    In the first ratio, the difference between numerator and denominator is 4 and the in the second one is 3

    So multiply both parts in the first ratio by 3, and both parts in the second ratio by 4, to make the difference equal.

    we now have 21/9 and 20/8

    Now, if we observe the numerator reduce by 1 and so does the denominator. but it should actually reduce by 300. So, we need to multiply by 300, effectively everything here by 300

    As, we are looking for the income of the first person, that is 21 parts: it would be 21 * 300 = 6300

    The ratio between present age of A and B is 3:4. The present age of B is 10 years more than the age of A 5 years ago. Find out the present age of B

    If B's age is 10 years more than A's age 5 years back
    Then B's age should be 5 years more than A's present age
    Now, A's and B's present ages are in the ratio 3:4
    and if the gap is 5 years, which corresponds to 1 part in the ratio
    B's age must be 20 years, as it is 4 parts in the ratio

    If 60% of (x-y) = 40% of (x + y), then what percent of x is y?

    (x+y)/(x-y) = 6/4 = 3/2
    Apply componendo-dividendo
    x/y = (3+2)/(3-2) = 5/1
    thus, y/x = 1/5 = 20%

    The MRP of a chocolate is Rs 65. A customer purchased the chocolate for 56 Rupees 16 Paise. He got two successive discounts of which one is 10%. Find the other discount of the discount scheme that was given by the seller.

    Observe that both 65 and 56.16 are multiples of 13
    dividing both by 13, we have 5 and 4.32
    reduce 10% of 5 => you get 4.5
    4.5 to 4.32 is a reduction of .18
    which is 4%

    If all the sides of a cuboid increases by 8%, its volume increases by x%
    If all the sides of a different cuboid reduces by 8%, its volume reduces by y%
    Which of the following is true?
    A ) x = y
    B ) x > y
    C ) x < y
    D ) We need the initial side ratio of the two cuboids that we begin with

    Understand this LOGICALLY
    when we have successive change over +a and +a, the net change will be more than 2a
    when we have successive change over -a and -a, the net change will be less than 2a
    Think over the successive change formula, you'd understand
    So, the x% must be greater than y%

    A grocer sells a soap markerd Rs 30 at 15% discount and gives a shampoo sachet costin Rs 1.5 free with each soap. He then makes 20% profit. His cost price per soap is

    After the 15% discount over 30, the new SP would have been 25.5.
    But the shopkeeper returns a sachet worth rs. 1.5 to the customer, so the net SP = 24
    CP = 24/1.2 = 20

    A trader makes a profit equal to the selling price of 75 articles when he sold 100 of the articles. What % profit did he make in the transaction?

    Profit of 100 = SP of 75
    Thus, P/SP = 75/100
    If P = 75, SP = 100, CP = 25
    Thus, P/CP = 75/25, or 300%

    The fees of a school increased 9.09%, then by 8.33%, and then by 7.69% over 3 years. If the current school fees is Rs. 3,360, then find the approximate school fees 3 years back.

    The multiplication factors are:
    9.09% increase = 12/11
    8.33% increase = 13/12
    7.69% increase = 14/13
    So, the value becomes
    (12/11) * (13/12) * (14/13) = 14/11
    14 is the new part so that corresponds to 3360 = 14 * 240
    11 should then correspond to 2640 (=11 * 240)

    A student scored 43% in an exam and failed by 12 marks. Another student scored 48% marks in the same exam and passed by 18 marks. Find the passing percentage in the exam.

    The percentage points difference is 5%, which corresponds to 30 marks
    So each percentage point should be 6 marks. Now, someone who failed by 12 marks, must have failed 2 percentage points. As, he gets 43%, the passing percentage must be 45%

    In September 2009, the sales of a product were 2/3rd of that in July 2009. In November 2009, the sales of the product were higher by 5% as compared to September 2009. How much is the percentage of increase in sales in November 2009 with respect to the base figure in July 2009.

    5% increase = 21/20
    So, effectively = (2/3) * (21/20) = 7/10, which implies a reduction of 3/10 = 30% decrease

    The length, breadth and height of a room are in ration 3:2:1. The breadth and height of the room are halved and length of the room is doubled. Then area of the four walls of the room will
    a) Decrease by 13.64%
    b) Decrease by 15%
    c) Decrease by 18.75%
    d) Decrease by 30%

    I don't like unnecessary fractions. As we are about to half the breadth and height, so let us keep these two values EVEN number to begin with
    Length = 6
    Breadth = 4
    Height = 2
    Area of the four walls, initially = (6+4) * 2 * 2 = 40
    New values
    L = 12
    B = 2
    H = 1
    Area of four walls, new = (12+2) * 1 * 2 = 28
    Reduction = 12/40 = 30%

    The difference between the value of a number increased by 25% and the value of the original number decreased by 30% is 22. What is the original number?

    Increased by 25%, multiplication factor of 5/4
    Reduced by 30%, multiplication factor of 7/10
    As, we are dealing with the same number, let us find the difference between these two fractions
    5/4 - 7/10 = 25/20 - 14/20 = 11/20
    11 corresponds to 22
    Thus, 20 corresponds to 40
    I have purposely added few extra steps to enable you to understand how i calculate these mentally

    Wheat is now being sold at Rs 27 per kg. During last month its cost was rs 24 per kg. Find by how much percent a family reduces its consumption so as to keep the expenditure fixed.

    Just reverse the multiplying factor here.
    AS the price becomes 27/24 = 9/8
    The consumption must become 8/9 to keep the expenditure constant
    So, the reduction must be 1/9 = 11.11%
    You can also treat this as a proportionality question, where price and consumption are inversely proportional.

    Sonali invest 15% of her monthly salary in insurance policies. She spends 55% of her monthly salary in shopping and household expenses. She saves the remaining amount of Rs 12750. What is Sonali’s monthly income?

    After taking off 15+55 = 70%, we are left with 30%
    30% is 12750
    100% is 42500

    A bakery opened with its daily supply of 40 dozen rolls. Half of the rolls were sold by noon, and 60% of the remaining rolls were sold between noon and closing time. How many dozen rolls were left unsold?

    50% + 60% of 50% are gone = 80% gone
    left with 20% of 40 = 8


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