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Ratio and Proportions

Ratio

Ratio

Ratio definition: The ratio of two quantities of the same kind in the same units is the fraction that one quantity is of the other. Thus the ratio 'a is to b' is the fraction written as a : b, here   b ≠ 0. In the ratio "a : b", 'a' is called as the 'first term' or 'antecedent' and 'b' is called as the second term or 'consequent'.

Important property: The value of a ratio remains unchanged, if both of its terms are multiplied or divided by the same non-zero quantity.

Ratio in simplest form: A ratio is said to be in simplest form, if antecedent (first term) and consequent (second term) have no common factors other than one. [It means their HCF is "1"].

Compound ratio: If a/b, c/d are two ratios then the compound ratio of a/b and c/d is a/b : c/d = ac/bd = ac : bd. In other words, compound ratio is the ratio of the product of antecedents to the product of consequents of the given ratios.

Comparison of ratios: If a : b = a/b and c : d = c/d are two ratios, then we say that:

                                 

Continued ratio: If P : Q and Q : R be two ratios given in such a way that the second term of the first ratio is equal to the first term of the second ratio then we say that P, Q, R are in continued ratio that is P : Q : R.

Ex 1: If , then Q : P is ?
Sol:
21P = 20Q
Ex 2: If a : b = 5 : 6 and b : c = 12 : 25, then a : c is ?
Sol:
a : b b : c
5 : 6 12 : 25
× 2 × 1
10 : 12 12 : 25
a : b : c = 10 : 12 : 25
    a : c = 10 : 25
      = 2 : 5
Alternate Method:
a : b = 5 : 6
b : c = 12 : 25
By multiplying (1) and (2), we get:
a : c = 2 : 5
Proportions

Proportions are a method of relating one quantity to another. The equality of two ratios is called proportion. The symbol for proportion is : : . Thus, if a : b = c : d then we say that a, b, c, d are in proportion and a : b :: c : d. Here a, b, c, d are called first, second, third, fourth terms of the proportion respectively. Here ‘a’ and ‘d’ are called extremes, ‘b’ and ‘c’ are called means.

a, b, c, d are in a proportion

Geometrical mean of a, b, c: Geometrical mean of a, b, c is or b2 = ac. In this case ‘b’ is called mean proportional or geometrical mean of a, b, c.
Direct proportion: If two quantities are so related to each other that an increase (or decrease) in the first causes an increase (or decrease) in the second, then the two quantities are said to vary directly.

Indirect proportion: If two quantities are so related to each other that an increase (or decrease) in the first causes a decrease (or increase) in the second, then the two quantities are said to vary indirectly or inversely.

Efficiency is indirectly proportional to number of days D taken to complete the work. Then mathematically, or , where k is a constant.

Examples on ratio definitions:

  • The ratio between Rs.50 and Rs.10 is 50 : 10 = 50/10 = 5/1 = 5 : 1
  • The ratio between 5 kg and 15 kg is 5 : 15 = 5/15 = 1/3 = 1 : 3
  • The ratio between Rs.1 and 5 kg is not possible, since they are of different units.

Examples on important property:
        5/15 = (5 * 8)/(15 * 8) shows that 5 : 15 is equal to 40 : 120
        3/7 = [3/5]/[7/5] shows that 3 : 7 is equal to 3/5   :   7/5.

Example on ratio in simplest form:
Find out the ratio of cherries to the grapes and what is the simplest form of that ratio ?

Example on compound ratio
Find the compound ratio of following figures ?

Example on compound ratio:
        If P : Q = 8 : 6 and Q : R = 6 : 5 then P : Q : R = 8 : 6 : 5.

Example on direct proportion: If the cost of one book is Rs.6 then the cost of 10 books is Rs.60 and 15 books cost is Rs.90....
So, if number of books are increased automatically cost also increases i.e, the number of books is in direct proportion with the cost of books.
(Ratio of number of books) :: (Ratio of cost of books)
Example on indirect proportion: Time taken to finish a work is inversely proportional to number of persons at work (More persons, less is the time taken to finish a work).
(Ratio of men) :: (Inverse ratio of time taken).

Rate & Speed

Rate
Ratio compares two quantities of the same kind. A rate is a special kind of ratio that compares two quantities having different units of measure. In general, rate is determined per unit time or per unit of length or mass or another quantity.

Ex: Joy sells bananas at Rs.60 per half dozen whereas Ian sells bananas of the same size at Rs.108 per dozen. From whom should we buy?
Sol: To find the answer to the problem, we have to find the price for an equal number of bananas, such as one banana, in each case. (Note that 1 dozen = 12 pieces).

Joy's selling price for each banana = .
Ian's selling price for each banana = .

From above, we conclude that the price of one banana sold by Ian is less than the price of Joy. Therefore, we should buy the bananas from Ian. Here Rs.10 per banana and Rs.9 per banana are known as rates.

Average rate: The average rate of a quantity is defined as the total quantity divided by the total number of quantities.

Ex: A school charges Rs.5.40 per kilometer to ferry 15 boys and 12 girls for a picnic by bus. Find the average amount that each student has to pay if the total distance travelled for the trip is 400 km.
Sol:
Amount to be paid per kilometer = Rs.5.40
Total distance travelled = 400 km
Total amount to be paid = Rs.(400 × 5.40)
  = Rs.2160
Total number of students = 15 + 12 = 27
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