Ratio and Proportion

Ratio and Proportion

Ratio is a relation between two quantities or numbers. It is a relation that one quantity bears to another with respect to magnitude. A ratio of a and is denoted by a : b and is read as a is to b. In a ratio, the first part (a) is called antecedent and second part (b) is called consequent.

For example, the ratio 3:5 represents 3/5 with antecedent 3 and consequent 5.

A ratio is a number, so to find the ratio of two quantities, they must be expressed in the same units. A ratio does not change if both of its terms are multiplied or divided by the same number.

Types of Ratios

  1. Duplicate Ratio: The ratio of the squares of two numbers is called the duplicate ratio of the two numbers.
  2. Triplicate Ratio: The ratio of the cubes of two numbers is called the triplicate ratio of the two numbers.
  3. Sub-duplicate Ratio: The ratio of the square roots of two numbers is called the sub-duplicate ratio of two numbers.
  4. Sub-triplicate Ratio: The ratio of the cube roots of two numbers is called the sub-triplicate ratio of two numbers.
  5. Inverse Ratio or Reciprocal Ratio: If the antecedent and consequent of a ratio interchange their places, the new ratio is called the inverse ratio of the first.
  6. Compound Ratio: The ratio of the product of the antecedents to that of the consequents of two or more given ratios is called the compound ratio. Thus, if a:b and c : d are two given ratios, then ac : bd is the compound ratio of the given ratios.

Proportion

The equality of two ratios is called proportion. When two ratios are equal, the four terms involved, taken in order are called proportional, and they are said to be in proportion. If a/b = c/d, then a, b, c and d are said to be in proportion and written as a:b::c:d. This is read as a is to b as c is to d. Here a, d are known as extremes and b, c are known as means.

If four quantities are in proportion, then

Product of means = Product of extremes

For example, in the proportion a:b::c:d, bc = ad. From this relation, if any three of the four quantities are given, the fourth can be determined.

Fourth Proportional: If a:b::c:x, x is called the fourth proportional of a, b, c.

Continued Proportion: Three quantities are said to be in continued proportion, if the ratio of the first to the second is same as the ratio of the second to the third. If a/b = b/c; b is called mean proportion.

b2 = ac or b = √(ac)

Invertendo: If a : b :: c : d then b : a :: d : c

Alternendo: If a : b :: c : d then a : c :: b : d

Componendo: If a : b :: c : d then (a+b) : b :: (c+d) : d

Dividendo: If a : b :: c : d then (a - b) : b :: (c - d) : d

Componendo and Dividendo: If a : b :: c : d then (a+b) : (a - b) :: (c+d) : (c - d)


Examples

1. If two numbers are in the ratio of a:b and the sum of these numbers is x, then these numbers are ax/(a+b) and bx/(a+b) respectively.

Let the three numbers in the ratio a:b:c be A, B and C.

A = ka, B = kb, C = kc

A + B + C = ka + kb + kc = x

k(a + b + c) = x

k = x/(a + b + c)

Example 1: Two numbers are in the ratio of 4:5 and the sum of these numbers is 27. Find the two numbers.

Here a = 4, b = 5 and x = 27

x/(a+b) = 27/9 = 3

The first number = 4 × 3 = 12

The second number = 5 × 3 = 15

Example 2: Three numbers are in the ratio of 3:4:8 and the sum of these numbers is 975. Find the three numbers.

Here a = 3, b = 4, c = 8 and x = 975

x/(a+b+c) = 975/15 = 65

The first number = 3 × 65 = 195

The second number = 4 × 65 = 260

The third number = 8 × 65 = 520

2. If two numbers are in the ratio of a:b and difference between these numbers is x, then these numbers are ax/(a-b) and bx/(a-b), respectively.

Let the two numbers be ak and bk.

Let a > b.

Given ak – bk = x

⇒ (a – b)k = x

k = x/(a – b)

Example 3: Two numbers are in the ratio of 4:5. If the difference between these numbers is 24, then find the numbers.

Here a = 4, b = 5 and x = 24

x/(b - a) = 24

The first number = 4 × 24 = 96

The second number = 5 × 24 = 120

Example 4: If A:B = 3:4 and B:C = 8:9, find A:B:C

As B is common, make it same in both the given ratios.

A:B = 6:8 and B:C = 8:9

So, A:B:C = 6:8:9

Example 5: If A:B = 2:3, B:C = 4:5 and C:D = 6:7, find A:D.

A:B = 8:12 and B:C = 12:15

So, A:C = 8:15 and C:D = 6:7

Or, A:C = 16:30 and C:D = 30:35

So, A:D = 16:35

Example 6: Given two numbers which are in the ratio of 3:4. If 8 is added to each of them, their ratio is changed to 5:6. Find the two numbers.

Let two numbers be 3k and 4k.

(3k + 8) : (4k + 8) = 5 : 6

6(3k + 8) = 5(4k + 8)

18k + 48 = 20k + 40

k = 4

Numbers are 3k = 12 and 4k = 16.

Example 7: Find the number that must be subtracted from the terms of the ratio 5:6 to make it equal to 2:3.

Let the required number be k.

(5 - k) : (6 - k) = 2 : 3

3(5 - k) = 2(6 - k)

15 - 3k = 12 - 2k

k = 3

Example 8: Find the number subtracted from each of the numbers 54, 71, 75 and 99 leaves the remainders which are proportional.

Let k be subtracted from each of the numbers.

The remainders are 54 – k, 71 – k, 75 – k and 99 – k

These are in proportional. So,

(54 - k)(99 - k) = (71 - k)(75 - k)

(54 × 99) - 54k - 99k + k2 = (71 × 75) - 71k - 75k + k2

k = 3

Example 9: Annual income of A and B is in the ratio of 5:4 and their annual expenses bear a ratio of 4:3. If each of them saves Rs.500 at the end of the year, then find their
annual income.

Let their incomes be 5k and 4k, respectively.

Expenditure of first person = 5k - 500

Expenditure of second person = 4k - 500

(5k - 500) : (4k - 500) = 4 : 3

3(5k - 500) = 4(4k - 500)

15k - 1500 = 16k - 2000

k = 500

Incomes are 5k = 2500 and 4k = 2000.

Example 10: A mixture contains alcohol and water in the ratio of 6:1. On adding 8 litres of water, the ratio of alcohol to water becomes 6:5. Find the quantity of water in the mixture.

Let mixture contains 6k litres of alcohol and k litres of water. On adding 8 litres of water,

6k : (k + 8) = 6 : 5

30k = 6(k + 8)

30k = 6k + 48

k = 2

Example 11: In what ratio the two kinds of tea must be mixed together into one at Rs. 9 per kg and another at Rs. 15 per kg, so that mixture may cost Rs. 10.2 per kg?

Let the ratio of two kinds of tea be a : b. So,

9a + 15b = 10.2(a + b)

1.2a = 4.8b

a/b = 4/1

Example 12: Two alloys contain silver and copper in the ratio 3:1 and 5:3. In what ratio the two alloys should be added together to get a new alloy having silver and copper in the ratio of 2:1?

Let the ratio of two alloys to be added together be a : b.

Amount of silver = 3a/4 + 5b/8

Amount of copper = 1a/4 + 3b/8

3a/4 + 5b/8 : 1a/4 + 3b/8 = 2 : 1

3a/4 + 5b/8 = a/2 + 3b/4

a/4 = b/8

a/b = 1/2

Hence, the two alloys should be mixed in the ratio 1:2.