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Ratio gives us a relation between two quantities having similar unit. The ratio of A to B is written as A : B or $latex \frac{A}{B}$ where A is called the antecedent and B the consequent. In other words, ratio means what part one quantity is of another.  A ratio is a number, so to find the ratio of two quantities; they must be expressed  in the same units. It does not change if both of is terms are multiplied or divided by the same number.

Thus, $latex \frac{2}{3}$ = $latex \frac{4}{6}$ = $latex \frac{6}{9}$ etc.

Proportion is an expression in which two ratios are equal.

For example: $latex \frac{A}{B}$ = $latex \frac{C}{D}$  or A : B :: C : D   We have, A D = B C

Each term of the ratio  and  is called a proportional. A, B, C, and D are respectively called the first, second, third and fourth proportional. Here, A and D are known as ‘extremes’ and B and C are known as ‘means’.

Basic Formulae

  1. If four quantities are in proportion, then Product of Means = Product of Extremes. For example, in the proportion A : B :: C : D, we have B×C = A×D.  From this relation we see that if any three of the four quantities are given, the fourth can be determined.

  2. Mean Proportional: If A : x :: x : B. Here ‘x’ is called the mean or second proportional of A and B. We have, $latex \frac{A}{x}$ = $latex \frac{x}{B}$ or x2 = AB or x = √ab. Thus, the mean proportional of A and B is √ab. We can also say that A, x, B are in a continued proportion.

  3. Third Proportional: If A : B :: B : Here ‘x’ is called the third proportional of A and B. We have, $latex \frac{A}{B}$ = $latex \frac{B}{x}$  or x = $latex \frac{B^2}{A}$. Thus, the third proportional of AB is $latex \frac{B^2}{A}$.

  4. Fourth Proportional: If A : B :: C : x. Here ‘x’ is called the fourth proportional of A, B and C. We have, $latex \frac{A}{B}$ = $latex \frac{C}{x}$ or x = $latex \frac{B×C}{A}$. Thus, the fourth proportional of A, B and C is $latex \frac{B×C}{A}$.

TIPS

  1. If two numbers are in the ratio of A : B and the sum of these numbers is x, then these numbers will be $latex \frac{Ax}{A+B}$ and $latex \frac{Bx}{A+B}$ respectively.
    Question: Two numbers are in the ratio of 4: 5 and the sum of these numbers is 27. Find the two numbers.

  2. If in a mixture of x liters, two liquids A and B in the ratio of A : B, then the quantities of liquids A and B in the mixture will be $latex \frac{Ax}{A+B}$ litres and $latex \frac{Bx}{A+B}$ litres respectively.

  3. If three numbers are in the ratio A : B : C and the sum of these numbers is x, then these numbers will be $latex \frac{Ax}{A+B+C}$ , $latex \frac{Bx}{A+B+C}$ and $latex \frac{Cx}{A+B+C}$ , respectively.
    Question: Three numbers are in the ratio of 3: 4 : 8 and the sum of these numbers is 975. Find the three numbers.

  4. If two numbers are in the ratio of A : B and difference between these is x, then these numbers will be $latex \frac{Ax}{A-B}$ and $latex \frac{Bx}{A-B}$, respectively (where A > B) and $latex \frac{Ax}{B-A}$, $latex \frac{Bx}{B-A}$ respectively (where A < B). Question: Two numbers are in the ratio of 4: 5. If the difference between these numbers is 24, then find the numbers.

ANOTHER SHORT TRICK

If A: B = N1: D1 and B : C = N2: D2, then A: B : C = (N1 × N2) : (D1 × D2) : (D1 × D2). For Example:  If A: B = 3: 4 and B: C = 8: 9, find A: B: C.

Here, N1 = 3, N2 =8, D1 =4 and D2 = 9.

As we know, A: B : C = (N1 × N2) : (D1 × D2) : (D1 × D2).

A: B : C = (3 × 8): (4 × 8): (4 × 9) = 24: 32 : 36 or 6: 8: 9

If A: B = N1: D1, B: C = N2: D2, and C: D = N3: D3 then

A : B : C : D= (N1 × N2 × N3) : (D1 × N2 × N3) : (D1 × D2 × N3 ) : (D1 × D2 × D3).

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

OTHER SHORT TRICKS

Trick – I: The ratio between two numbers is A: B. If x is added to each of these numbers, the ratio becomes C : D. The two numbers are given as $latex \frac{Ax (C-D)}{AD-BC}$ and $latex \frac{Bx (C-D)}{AD-BC}$.

Question: 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 two numbers.

Trick – II: The ratio between two numbers is A : B. If x is subtracted from each of these numbers, the ratio becomes C : D. The two numbers are $latex \frac{Ax (D-C)}{AD-BC}$  and $latex \frac{Bx (D-C)}{AD-BC}$

Question: The ratio of two numbers is 5: 9. If each number is decreased by 5, the ratio becomes 5: 11. Find the numbers.

Trick – III: If the ratio of two numbers is A: B, the numbers that should be added to each of the numbers in order to make this ratio C : D, is given by $latex \frac{AD-BC}{C-D}$.

Question: Find the number that must be added to the terms of the ratio 11: 29 to make it equal to 11: 20.

Trick – IV: If the ratio of two numbers is A: B, the number that should be subtracted from each of the numbers in order to make this ratio C: D is given by $latex \frac{BC-AD}{C-D}$.

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