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Ratio is a comparison of two or more numbers that indicate their quantities in relation to each other whereas a Proportion is an equation that defines the two given ratios are equal to each other. It helps in understanding the relationship between two ratios. Ratio and Proportion are commonly used concepts in mathematics and other allied subjects to compare the quantities and establish a relationship between them.
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Ratio & Proportion
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Ratio of two different quantities is represented by using the symbol (:) between the two numbers. For example, if we want to represent the comparison between the number of boys and girls in a class, it can be done in the form of 30:45 as the ratio between boys and girls. The number being divided (at left) is called antecedent, and the divisor (at right) is called a consequent.
Proportion is an equation that defines the two given ratios as equal to each other. It helps in understanding the relationship between two ratios. When two given ratios are equal in respect of value, they can be said to be in proportion. When we denote proportions between two ratios, it can be done by using the double dividend(::) or equal to (=) sign.
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Properties of Ratio
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A ratio will remain the same if both its antecedent and consequent are multiplied or divided with the same non-zero number.
- a/b = pa/pb = qa/qb where p, q ≠ 0
- a/b = (a/p)/(b/p) = (a/q)/(b/q) where p, q ≠ 0
If any two ratios a/b and c/d are equal
- a/b = c/d ? b/a = d/c
- a/b = c/d ? a/c = b/d
- a/b = c/d ? (a+b)/b = (c+d)/d
- a/b = c/d ? (a-b)/b = (c-d)/d
We can compare any two ratios in their fraction notation just as we compare real numbers:
- a/b = p/q ? aq = bp
- a/b > p/q ? aq > bp
- a/b < p/q ? aq < bp
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Types of Ratio
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There are various types of ratios. Some are as follows.
- Profit and Loss Ratios - When both the figures are derived from the statement of profit and loss account then it will be known as profit and loss ratio. It can also be called income statement ratio or revenue statement ratio.
- Balance sheet Ratios - If both the figures are derived from balanced sheets, then it will be called as balance sheet ratio. When this type of ratio expresses a relation between two accounts of the balance sheet, then it will be called financial ratios.
- Composite Ratios - It is a type of combined ratio which compares two variables from two different accounts. One will be taken from the profit and loss account and the other one from the balance sheet.
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Properties of Proportion
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If a:b = c:d are in a proportion, then
- Product of extremes = product of means i.e. ad = bc.
- a:b = b:c then b is known as mean proportional and b2 = ac.
- a, b, c, d… are in continuous proportion i.e. a:b = b:c = c:d.
- The third proportional of two numbers a and b is c such that a:b = b:c.
- d is defined as the fourth proportional to numbers a, b, c if the ratio a:b = c:d.
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Types of Proportion
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There are two basic types of proportion. These are:
- Direct Proportion - This proportion describes the direct relation between any two quantities. If one quantity increases, then the opposite quantity also increases and vice versa simultaneously.
- Inverse Proportion - This proportion describes the indirect relation between any two quantities. If one quantity increases, then the opposite quantity will gradually decrease and vice versa simultaneously.
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Things to Remember
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- If we multiply and divide each term of the ratio by the same number, the ratio won’t be affected.
- For any three quantities, if the ratio between the first and second number will be equal to the ratio between second and third number, then it is said to be a continued proportion.
- In the case of any four quantities in a continued proportion, the ratio between the first and second number will be equal to the ratio between the third and fourth number.
- In the case of ratio & proportion, we can write in two ways either by using equal sign as a/b = c/d or using colon, a:b = c:d.
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Sample Questions
Ques. In a mixture/compound of 60 liters, the ratio of water and milk is given in the ratio of 2:1. If this ratio will be changed to 1:2, then what will be the amount of water added? (2 Marks)
Ans. Quantity of milk: (60*2/3) = 40 liters
Quantity of water: (60-40) = 20 liters
The ratio given is 1:2,
Let the quantity of water be x liters
Accordingly, if milk:water = (40/20+x) = 1/2,
then 20+x = 80
x = 60 liters
Ques. An amount of money is to be distributed among four candidates A, B, C, D which are in the proportion of 5:2:4:3. If C will get Rs 1000 more than D, then calculate the share of B? (2 Marks)
Ans. Let the shares of A, B, C, D be 5x, 2x, 4x, 3x respectively.
Then, C - D = 1000
4x-3x = 1000
x = 1000
Therefore, the share of B will be 2x which is 2*1000 = 2000.
Ques. The ratio of A’s salary to B’s was 4:5. A’s salary is increased by 10% and B’s by 20%. What will be the ratio of their salaries now? (2 Marks)
Ans. Current ratio = 4:5
If A’s salary is increased by 10% and B’s salary by 20%.
New ratio of salaries will be 4*1.1:5*1.2 = 11:15.
Ques. If x:y = 1:2, find the value of (2x+3y):(x+4y). (3 Marks)
Ans. x:y = 1:2 that depicts x/y = 1/2
Now, (2x+3y):(x+4y) = (2x+3y)/(x+4y)
Dividing numerator and denominator with y
(2x+3y):(x+4y) = [(2x+3y)/y] / [(x+4y)/2] = [2(x/y)+3] / [(x/y)+4]
Put x/y = 1/2
We get,
(2x+3y):(x+4y) = [(2(1/2) + 3) / (1/2+4)]
(2x+3y):(x+4y) = (1+3) / [(1+8)/2]
(2x+3y):(x+4y) = 4/(9/2)
(2x+3y):(x+4y) = 4/1*2/9
(2x+3y):(x+4y) = 8/9
Ques. State the uses of Ratio & Proportions in daily life. (3 Marks)
Ans. Ratios are useful in transforming values from one unit to another, expressing quantities in a mix (like number of boys/girls in a crowd or quantity of sugar in water), and expressing probability/chance.
Proportions are useful in calculating material for recipes/formulas which includes multiple elements in different quantities, calculating profit/expenses for business, calculating money required for fuel according to trip distance and for many other different types of computations.
Ques. There are a certain number of Rs 10, Rs 20, Rs 50 notes available in a box. The ratio of the number of notes of Rs 10, Rs 20, Rs 50 is 3:4:6. The total amount available in a box is Rs 2460. Calculate the amount of Rs 10 and Rs 50 in the box available. (2 Marks)
Ans. Let the number of Rs 10, Rs 20, Rs 50 be 3a, 4a, 6a.
Given,
10*3a + 20*4a + 50*6a = 2460
410a = 2460
a = 6
Number of notes of Rs 10 = 3*6 = 18
Number of notes of Rs 20 = 4*6 = 24
Number of notes of Rs 50 = 6*6 = 36
Final amount = 10*18 + 50*36 = Rs 1980
Ques. If 18:13.5::16:x and (x+y):y::18:10, then calculate the value of y. (2 Marks)
Ans. 18:13.5::16:x*x = (16*13.5)/18x = 12
Now,
(x+y):y::18:10
(12+y):y::9:5 5(12+y) = 9y
60 + 5y = 9y
4y = 60
y = 15
Ques. State the differences between Ratio and Proportion. (5 Marks)
Ans.
| Basis of Comparison | Ratio | Proportion |
|---|---|---|
| Meaning | Ratio refers to the comparison of two values of the same unit. | Proportion is a condition in which two ratios are set equal to each other. |
| Represents | Quantitative relationship between two categories. | Quantitative relationship of a category and the total. |
| Sign | Colon (:) sign | Double colon (::) or equal to (=) sign |
| Keyword | The keyword used for ratio is "to every". | The keyword used for proportion is 'Out of'. |
| Nature | Ratio as a certain number of parts, for example, three parts to one part. | Proportion as the same value of increase or decrease, for example, doubles, half. |
| Denoted by | Parts of total quantity | An equal part of a different quantity. |
| What is it | Ratio is an expression. | Ratio is an equation. |
Ques. Out of 30 students in a class, 6 like football, 12 like cricket and the remaining like tennis. Find the ratio of
(a)Number of students who like football to the number of students who like tennis.
(b)Number of students who like cricket to the total number of students. (3 Marks)
Ans. Number of students in the class=30
Number of students who likes football = 6
Number of students who likes cricket = 12
Number of students who likes tennis = 30-(6+12) = 30-18 = 12
- Ratio of number of students liking football to the number of students liking tennis
= Number of students liking football / Number of students liking tennis
= 6/12 = 1/2 or 1:2
- Ratio of number of students liking cricket to the total number of students
= Number of students liking cricket / Total number of students
= 12/30 = 2/5 = 2:5
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