Ratio & Proportion
 

  1. Question: What is the ratio whose terms differ by `40` and the measure of which is `2/7` ?

    A
    `16 : 56`

    B
    `10 : 50`

    C
    `20 : 70`

    D
    `36 : 76`

    Note: `x/(x + 40) = 2/7` `iff 7x = 2x + 80` `iff x = 16`. `:.`The ratio is `16 : 56`.
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  2. Question: If `x^2 + 4y^2 = 4xy`, then `x : y` is :

    A
    `2 : 1`

    B
    `1 : 2`

    C
    `1 : 1`

    D
    `1 : 4`

    Note: `x^2 + 4y^2 = 4xy` `iff x^2 - 4xy + 4y^2 = 0` `iff (x - 2y)^2 = 0` `iff x -2y = 0` `iff x = 2y` `iff x/y = 2/1`.
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  3. Question: If `5x^2 - 13xy + 6y^2 = 0`, then `x : y` is :

    A
    `5 : 3` or `1 : 2`

    B
    `3 : 5` or `2 : 1`

    C
    `2 : 1` only

    D
    `3 : 5` only

    Note: `5x^2 - 13xy + 6y^2 = 0` `iff 5x^2 - 10xy - 3xy + 6y^2 = 0` `iff 5x (x - 2y) - 3y (x - 2y) = 0` `iff (x - 2y)(5x - 3y) = 0` `iff x - 2y = 0` or `5x - 3y = 0` `iff x = 2y` or `5x = 3y` `iff x : y = 2 : 1` or `x : y = 3 : 5`.
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  4. Question: Two number are in the ration `3 : 5`. If each number is increased by `10`, the ratio becomes `5 : 7`. The numbers are:

    A
    `3, 5`

    B
    `12, 20`

    C
    `15, 25`

    D
    `18, 30`

    Note: Let the numbers be `3x` and `5x`. `(3x + 10)/(5x + 10) = 5/7` `iff 7(3x + 10) = 5(5x + 10)` `iff x = 5`. So, the numbers are `15` and `25`.
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  5. Question: What same number must be added to each term of the ratio `7 : 13` so that the ratio becomes `2 : 3` ?

    A
    `1`

    B
    `2`

    C
    `3`

    D
    `5`

    Note: Let `x` be added. Then, `(7 + x)/(13 + x) = 2/3` `iff 3(7 + x) = 2(13 + x)` `iff x = 5`
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  6. Question: What number should be subtracted from both the terms of the ratio `15 : 19` so as to make it as `3 : 4`?

    A
    `3`

    B
    `5`

    C
    `6`

    D
    `9`

    Note: Let `x` be subtracted. Then, `(15 - x)/(19 - x) = 3/4` `iff 4(15 - x) = 3(19 - x)` `iff x = 3`.
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  7. Question: The least whole number which when subtracted from the terms of the ratio `6 : 7` to give a ratio less than `16 : 21`, is :

    A
    `2`

    B
    `3`

    C
    `4`

    D
    `6`

    Note: Let `x` be subtracted. Then, `(6 - x)/(7 - x) < (16)/(21)` `iff 21(6 - x) < 16(7 - x)` `iff 5x > 14` `iff x > 2.8` `:.` Least such number is `3`.
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  8. Question: The ratio between two numbers is `3 : 4`. If each number is increased by `6`, the ratio becomes `4 : 5`. The difference between the numbers is :

    A
    `1`

    B
    `3`

    C
    `6`

    D
    `8`

    Note: Let the numbers be `3x` and `4x`. Then, `(3x + 6)/(4x + 6) = 4/5` `iff 5(3x + 6) = 4(4x + 6)` `iff x = 6`. So, the numbers are `18` and `24`. `:.` Difference between the numbers `=(24 - 18)` `= 6`.
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  9. Question: What number should be added to each of the numbers `8, 21, 13` and `31` so that the resulting numbers, in this order form a proportion ?

    A
    `2`

    B
    `3`

    C
    `5`

    D
    `7`

    Note: Let `(8 + x)/(21 + x) = (13 + x)/(31 + x)`. Then, ` (8 + x)(31 + x) = (13 + x)(21 + x)` or `39x + 248 = 34x + 273` or `5x = 25` or ` x = 5`.
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  10. Question: What number should be subtracted from each of the numbers `23, 30, 57` and `78`so that the remainders may be proportional ?

    A
    `4`

    B
    `5`

    C
    `6`

    D
    `7`

    Note: Let `(23 - x)/(30 - x) = (57 - x)/(78 - x)`. `:. (23 - x)(78 - x) = (57 - x)(30 - x)` or `- 101x + 1794 = 1710 - 87x` `iff 14x = 84` `iff x = 6`.
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