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Points to Remember

  • The order of terms in a ratio is very important.

    Example: 5:6 is not the same as 6:5.


  • Ratio between two quantities should be in the most simplified form obtained by cancelling out all common factors of the two terms.

    Example: If there is a ratio 30:24, it has to be written in its most simplified form which is 5:4.


  • Ratio is a comparison of two or more quantities of the same kind only. By “same kind”, we mean that the units of the quantities being compared must be same.

    Example: Comparison of a girl’s height to a boy’s height is possible. But, comparison of a girl’s height to a boy’s age or weight is not meaningful.


  • Ratio is a comparison of two or more quantities of the same kind expressed in the same unit only. The quantities compared must have same unit.

    Example: Comparison of 10 kg rice to 200 grams of sugar is not possible, because rice is expressed in kg but sugar is expressed in grams.



    (By converting kg in terms of grams, the comparison is made possible and is written as 10,000:200.)


  • If the ratio of two quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable.
    Otherwise, they are said to be incommensurable.

    Example: Description: 5499.pngcannot be expressed as the ratio of two integers; therefore, Description: 5492.pngare incommensurable quantities, whereas the ratio 3:2 is a ratio of commensurable quantities.


  • Both terms of a ratio can be multiplied or divided by the same number (except zero).

    Example: The ratio 2:3 is same as 4:6 because 2:3 is multiplied by 2 to get 4:6.


    Description: 5486.png

  • Continued ratio is the relation between the magnitudes of more than two quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a:b:c.

    Example: Let the weight of three boys be 35 kg, 39 kg and 41 kg.



    Then, the continued ratio of the weight of the three boys is 35:39:41.

  • Two or more ratios can be bridged in order to have a continued ratio between them.
    When bridging two or more ratios to form a continued ratio, we have to make sure that the consequent of a ratio and the antecedent of the succeeding ratio are equal.
    If they are not equal, then LCM of the two numbers is considered and they are made equal to the LCM by applying appropriate mathematical operations. 
a:b = 1:2, b:c = 3:5, c:d = 6:8. Find a:b and c:d.
b:c = 3:5, c:d = 6:8
We have to make sure that the consequent of the firsi ratio is equal to the antecedent of the second ratio.
LCM of 5 and 6 is 5 × 6 = 30.
Multiplying first ratio by 6 and second ratio by 5, we get
b:c = 18:30 and c:d = 30:40
Now a:b = 1:2 and b:c = 18:30
LCM of 2 and 18 is 18.
Multiplying first ratio by 9 and second ratio by 1, we get
a:b = 9:18, b:c = 18:30 and c:d = 30:40
a:b:c:d = 9:18:30:40
  • Whenever a ratio a:b is given between two magnitudes, we always express the actual magnitudes as ka and kb, where k is a constant.
Let two quantities be in the ratio 2:3. If the first quantity is 20, find the second quantity.
Given, ratio between two quantities = 2:3
The two quantities can be assumed as 2k and 3k.
2k = 20 ⇒ k = 10
3k = 3 × 10 = 30
Hence, the second quantity is 30.
  • To compare two ratios, convert them into equivalent like fractions. 
To find which ratio is greater between Description: 5480.png
Description: 5474.png 
Description: 5468.png 
So, Description: 5462.png(applying LCM to 7/10 and 3/4)
And Description: 5454.png
As 15 > 14,Description: 5448.png
Hence, 3.6: 4.8 is greater ratio.
  • In the ratio a:b, if a > b, then it is known to be of greater inequality.


    Example: The ratios 6:5, 7:3, 18:5, etc., are all ratios of greater inequalities.

  • In the ratio a:b, if a < b, then it is known to be of lesser inequality.

    Example: The ratios 3:4, 7:10, 15:25, etc., are all ratios of lesser inequalities.

  • If a:b and c:d are two ratios, then ac:bd is their compound ratio.
    Compound ratio of the given ratios is the ratio of the product of their antecedents to product of their consequents.

    Example: Let 5:6 and 7:8 be two ratios, then the compound ratio of these two ratios will be 5 × 7:6 × 8 = 35:48.

  • One ratio is the inverse of another if their product is 1. Thus, a:b is the inverse of b:a and vice-versa. Also, inverse ratio of a:b:c is bc:ac:ab.

    Example: Inverse ratio of 5:4 is 4:5.

  • a2:b2 is the duplicate ratio of a:b.

    Example: Let 4:5 be a ratio, then its duplicate ratio is 42:52 = 16:25.


    Note: Duplicate ratio is obtained by compounding a ratio with itself.

  • a3:b3 is the triplicate ratio of a:b.

    Example: Let 4:5 be a ratio, then its triplicate ratio is 43:53 = 64:125.


    Note: Triplicate ratio is obtained by compounding a ratio with itself twice.

  • Description: 5328.png

    Example: Let 25:36 be a ratio, then its sub-duplicate ratio isDescription: 5322.png


    Note: When a sub-duplicate ratio is compounded with itself, we get the original ratio. 

  • Description: 5258.png.

    Example: Let 27: 8 be a ratio, then its sub-triplicate ratio isDescription: 5252.png.


    Note: When a sub-triplicate ratio is compounded with itself twice, we get the original ratio.

  • If a quantity is increased or decreased in the ratio a:b, it means,
    Description: 5189.png
A company employs 10 employees. The salary of each employee is 6000 per month. If the number of employees is reduced in the ratio of 5:4 and the salary of each employee is increased in the ratio of 3:4, then find out what will be the number of employees and what will be their salary?
First, let us find out the number of employees after reducing their number in the ratio of 5:4.
Given: Old number of employees = 10, a:b = 5:4
Description: 5183.png 
Now, let us find the new salary of the employees after increasing their salary in the ratio of 3:4.
Given: Old salary = 6000, a:b = 3:4
Description: 5176.png 

Note: In the above point, if a > b, then the quantity is reduced in the ratio a:b; and if a < b, then the quantity is increased in the ratio a:b.

  • If a given number N has to be divided into two parts, A and B, which are in the ratio a:b, then Description: 5113.png.
    Divide 1458 into two parts in the ratio of 2:7.
    N = 1458, a = 2, b = 7
    Description: 5107.png 
  • If a given number N has to be divided into three parts A, B and C, which are in the ratio a:b:c, thenDescription: 5101.png
    A pencil of length 12 cm. is broken down into three parts in the ratio 1:2:3. What are the lengths of the three parts?
    N = 12, a = 1, b = 2, c = 3
    Description: 5095.png
    Find three numbers in the ratio 2:3:4, so that the sum of their cubes is equal to 792.
    Let the numbers be 2x, 3x and 4x. Then, we have
    (2x)3 + (3x)3 + (4x)3 = 792
     99x 3 = 792  x 3 = 8  x = 2
    Hence, the required numbers are 2x = 4, 3x = 6, 4x = 8

    A:B = 1:2, B:C = 3:4, C:D = 8:12 and D:E = 12:16. Find A:B:C:D:E.
    A:B = 1:2 = 3:6
    B:C = 3:4 = 6:8
    C:D = 8:12
    D:E = 12:16
    (Bring it to a form where the second term of one ratio will be equal to the first term of the succeeding ratio.)
    Combining, we get A:B:C:D:E = 3:6:8:12:16.

    The ratio of money with Rahul and Sachin is 7:15 and that with Sachin and Laxman is 7:16. If Rahul has ₹980, how much money does Laxman have?

    Two different ratios are given, we should first express them as a continued ratio to get the ratio of money that the three boys have.

    The ratio of money with Rahul, Sachin and Laxman is 49:105:240

    Description: 5089.png 


    Hence, Laxman has ₹4800.

    ₹850 is divided among 8 men, 10 women and 12 boys such that their respective shares are in the ratio of 9:8:4. What is the share of a boy?
    The ratio of shares of groups of 8 men, 10 women and 12 boys:
    9 × 8:8 × 10:4 × 12 = 72:80:48
    Now, we know how to divide a quantity into three parts in a given ratio. Using that, we can find out the share of 12 boys as Description: 5083.png
    So, the share of 1 boy Description: 5077.png 
    Divide ₹1540 among ABC so that A will receive Description: 5069.png as much as B and C together and B will receive Description: 5062.png of what A and C together do.
    A’s share : (B + C)’s share = 2:9 ..................... (1)
    B’s share : (A + C)’s share = 3:11 ..................... (2)
    Now, dividing ₹1540 in the ratio of 2:9 and 3:11, we get
    A’s share Description: 5056.png ₹280
    B’s share Description: 5050.png = ₹330
     C’s share = ₹1540 – (₹280 + ₹330) = ₹930.
    How many one rupee coins, fifty paise coins and twenty five paise coins of which the numbers are proportional to 2.5, 3 and 4, together worth ₹210?
    Ratio of number of 1 rupee, 50 paise and 25 paise coins = 2.5:3:4 = 5:6:8.
    Ratio of values of the coins = 5 × 100:6 × 50:8 × 25 = 500:300:200 = 5:3:2
    From the above ratio,
    value of 1 rupee coins in ₹210 = Description: 5044.png
    value of 50 paise coins in ₹210 = Description: 5038.png
    value of 25 paise coins in ₹210 = Description: 5031.png
    Hence, number of 1 rupee coins = 1 × 105 = 105.
    Number of 50 paise coins = 2 × 63 = 126.
    Number of 25 paise coins = 4 × 42 = 168.
    If a carton containing 25 mirrors is dropped, which of the following cannot be the ratio of broken mirrors to unbroken mirrors?
    (a) 4:1 (b) 3:2 (c) 2:3 (d) 1:1
    There are 25 mirrors in the carton.
    So, the sum of terms in the ratio must divide 25 exactly.
    We see that 4 + 1 = 5 divides 25 exactly.
    3 + 2 = 5 also divides 25 exactly.
    2 + 3 = 5 also divides 25 exactly.
    1 + 1 = 2 does not divide 25 exactly.
    Thus, our answer is (d).
    A 48 litre mixture contains water: milk in the ratio 3 : 5. How much water should be added to this mixture to reverse this ratio
    Now, Description: 5009.png 

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