Ratio & Proportion

  • IIM Lucknow | MathOratory

    This article will be targeted to simplify the process/calculation related to Ratios & Proportion. Before we start, understand a simple fact about competitive exams. Solving a particular question is not sufficient unless we solve it in MINIMAL time.

    I'll start with a very basic question.

    What is the easiest way of solving (x+18 )/(x-12) = x/(x-20)

    Firstly try to understand a simple fact about ratio
    If I say 1/3 = 2/6, that doesn't imply that 1=2 and 3=6
    But, if I notice the difference of the parts 1/3 (difference of parts is 2). And, 2/6 (difference of parts is 4)
    If, we try to equate the differences, we should be doubling the difference of the first one or halving that of the second one. In either scenario, notice that the numerator as well as the denominator becomes equal.

    Let's apply this here. We have (x+18 )/(x-12) = x/(x-20)
    The difference of parts on the left side is 30, and that of the right side is 20
    In order to equate them, we should be multiplying the left side by 2 and the right side by 3. But, that in turn should also equate the numerators.
    So, we can imply that: 2(x+18 ) = 3x > > x = 36

    If (3x+2y)/(3x-2y) = 5/2, Find (5x+4y)/(7x-2y)

    Apply Componendo Dividendo, to get
    3x/2y = 7/3
    or, x/y = 14/9
    In order to find (5x+4y)/(7x-2y), simply substitute 'x' with 14 and 'y' with 9
    So the reqd answer is:
    (5 * 14 + 4 * 9)/(7 * 14 - 2 * 9) = 106/80 = 53/40

    If a shopkeeper wrongly calculates his profit on SP and gets 30% Loss, Find the actual Loss percentage.

    Play with fractions.
    We have L/SP = 3/10
    Hence, L/CP = 3/13 = 23.1 % (approx)

    There are three discount types that a shop offers:
    Type 1: Buy 3 get 1 free
    Type 2: Buy 2 get 50% off on 2nd one
    Type 3: 30% flat off
    Type 4: Buy 5 get 2 free
    Which option offers the highest discount ?

    If you buy 'a' items and get 'b' for free, the discount is b/(a+b)
    So, the fractions that we are dealing with are:
    Type 1: 1/4
    Type 2: 0.5/2
    Type 3: 30% (flat mentioned)
    Type 4: 2/7
    Every other type has less than 30%. Answer is type 3

    Given(2x+3y)/(5x-2y) = 5/7, Find: x/y

    Many people might have known that reapplying componendo-dividendo on the comp-div form, gives back the original form.
    So, if we have (a+b)/(a-b) = 5/3
    we can re-apply comp-div to get a/b = (5+3)/(5-2) = 4

    Or, even in cases like:
    (2x+3y)/(2x-3y) = 5/3
    implies, 2x/3y = (5+3)/(5-3)

    I'll extend this one step forward:
    If, we have something like (2x+3y)/(5x-2y) where the coeff of 'x' in the numerator and coefficient of 'y' in the denominator adds up to 0 (which in this case it does), we can reapply the process to get back x/y

    So, in this case, we operate
    (2 times the num + 3 times the den)/(5 times the num - 2 times the den)

    In the given question we had:
    (2x+3y)/(5x-2y) = 5/7
    x/y = (2 * 5 + 3 * 7)/(5 * 5 - 2 * 7)
    reapplying the process, x/y = 31/11

    A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?

    Focus on three things

    1. Relationship ... The ratio given
    2. The data provided ... Difference C-D
    3. The data required .... B's share

    Let us relate the two relationships
    The Ratio of A:B:C:D = 5:2:4:3
    C-D = 4-3 = 1 part
    B = 2 parts
    So, B's share has to be double the difference
    The difference being 1000, B's share must be 2000

    Due to an increase of 30% in the price of eggs, 3 eggs less are available for rs. 7.80. The present rate of eggs per dozen is:
    (a)Rs. 8.64
    (b) Rs. 8.88
    (c) Rs. 9.36
    (d) Rs. 10.40

    Let's look at it in this way
    The price gone up by 30%, implies that the new price is to the old price is in the ratio = 13/10
    So, in the same budget, the new quantity is to the old quantity = 10/13
    Notice the difference of parts (10:13) is 3, which correspond to the data give. So, we can infer that initially at Rs 7.80, we could have bought 13 eggs and now we can purchase only 10. So the price of 12 eggs would be 7.8 *12/10 = 7.8 + 1.56 = 9.36

    PS: If the question would've said 6 less eggs can be bought ... then we should have scaled up the entire data by a factor of 2. That is, initially 26 eggs could've been bought, and now 20

    Four years ago, the ratio of ages of me and my brother was 13:9. Eight years hence, it would be 4:3. Find our current ages.

    Let us simplify the problem here
    4 years ago 13/9 (difference of parts = 4)
    8 years hence 4/3 (difference of parts = 1)
    We can agree to the fact that the difference between the ages have not changed. Hence, we can say for sure that we need to equate the difference of parts first. So, let's multiply the parts of the 2nd ratio by 4. Now we have,
    4 years ago: 13/9
    8 years hence: 16/12
    If we check now, the numerator and denominator have both increased by 3, which should have actually increased by 12 (4 years ago to 8 years hence)
    So, in order to make 3 to 12, we need to multiply 4: So, multiply everything by 4
    4 years ago: 52/36
    8 years hence: 64/48
    Now, all the data matches. So, these should be the actual ages...
    We can now find the current ages, as 52+4 = 56 and 36 + 4 = 40

    There are four natural numbers which are in continuous proportion. If the ratio of the smallest and the next in the series is 3:5, find the minimum possible sum of the four numbers.

    I hope everyone understand continuous proportion.
    In this case, it simply means that
    a:b = b:c = c:d, taking a,b,c,d as the four numbers
    Further, we know that these are each equal to 3:5
    After this, it is a straightforward, simple combining of ratios problem.
    a:b = b:c = 3:5
    Combining, we have a : b : c = 9 : 15 : 25
    Also, we have c:d = 3:5
    Combining these two, we have a : b : c : d = 27 : 45 : 75 : 125
    As, this is the reduced ratio and a,b,c,d are NATURAL NUMBERS, in order to minimize the sum, we need to minimize the values of each one of them, which cannot be reduce further from the above ratio
    So, the minimum sum = 27+45+75+125 = 272

    Two natural numbers are taken, such that their mean proportional is 12 and their third proportional is 324. Find the numbers.

    Please understand that if few numbers are in continuous proportion, they MUST be in GP series as well.
    The important part of the question is to understand that there are two continuous proportions
    A, 12, B & A, B, 324
    We can then treat these two as GP series as well
    Let us say the common ratio of the first series is 'r' ...
    implying B = A * r^2
    And let us say the common ratio of the second series be 'R' ...
    implying B = A * R
    equating the B,
    we have R = r^2
    Now, obviously we cannot have the same value of r and R, as they are definitely different series ... otherwise B has to be both 12 and 324 simultaneously (which is funny)
    Let us take few more values that can satisfy: If r = 2, R= 4
    Then B, according to the first series = 12 * 2 = 24, and in the 2nd series B = 324/4 which is not 24
    Taking, r = 3, R = 9
    first series, B = 12 * 3 = 36
    second series, B =324/9 = 36 ... BANG, we have our solution
    So, A = 12/r = 12/3 = 4
    Thus, the two numbers are 4 and 36

    The sides of a right angled triangle are a, a+17x, a+18x. If 'a' and 'x' are both positive, find a/x.

    Everyone is aware of Pythagoras theorem. The question is: How many Pythagorean triplets are you aware of?
    Please remember the following five:
    3,4,5; 5,12,13; 7,24,25; 8,15,17; and 9,40,41
    If we consider the triplet in our scenario:
    a, a+17x, a+18x: we find that the difference between the first two is 17x and difference between the last two is x, where the differences are in the ratio 17:1, which i satisfied by the triplet 7,24,25
    So, if we take our 'x' to be 1, our 'a' must be 7, implying a/x = 7

    A , B and C , three friends are enjoying a bonfire. A is contributing 5 wood logs , B is contributing 4 wood logs for the bonfire. C is not contributing any wood logs. Hence he is giving Rs 27 to A and B . What amount of money should be taken by A and B respectively? ( all the wood logs are identical)

    First thing that we need to understand is what would C pay for ?
    C is not going to pay for the woods A and B provided to the group. Actually C is going to pay for the woods A and B have provided to C
    So, let's figure that out A gives 5 wood, B gives 4. So total contribution is 9. But, that is equally shared among the three.
    So, each one of them consumes 3 in return
    So, A gives 5, but consumes 3 himself, providing 5-3 = 2 to C
    And, B gives 4, but consumes 3 himself, providing 4-3 = 1 to C
    So, A and B gives wood to C in the ratio 2:1, their payment should also be in the ratio 2:1
    the Rs. 27 distributed in the ratio 2:1, would be 18 and 9 ... These should be the amounts

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