# Ratio and Proportion Formulas, Questions and Study Material

## Ratio and Proportion Formulas

Ratio: The ratio of two quantities a and b in the same units, is the fraction $\frac{a}{b}$ and we write it as a : b.
In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.

Example: The ratio 5 : 9 represents $\frac{5}{9}$ with antecedent = 5, consequent = 9.

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.

Proportion: The equality of two ratios is called proportion.
If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.
Thus, a : b :: c : d $\Leftrightarrow$ (b x c) = (a x d).

Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c.

Third Proportional: a : b = c : d, then c is called the third proportion to a and b.

Mean Proportional: Mean proportional between a and b is ab.

Comparison of Ratios: We say that (a : b) > (c : d) $\Leftrightarrow$ $\frac{a}{b}>\frac{c}{d}$

Compounded Ratio: The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate Ratios: Duplicate ratio of (a : b) is ($a^{2}$ : $b^{2}$).

Sub-duplicate ratio of (a : b) is (a : b).
Triplicate ratio of (a : b) is $a^{3}$ : $b^{3}$)..
Sub-triplicate ratio of (a : b) is $a^{\frac{1}{3}}\, :\, b^{\frac{1}{3}}$
If $\frac{a}{b}\, =\frac{c}{d}$, then $\frac{a\,+\,b}{a\,-\,b}\, =\,\frac{c\,+\,d}{c\,-\,d}$ [componendo and dividendo]

Variations: We say that x is directly proportional to y, if x = ky for some constant k and we write, x ∞ y.
We say that x is inversely proportional to y, if xy = k for some constant k and we write, x ∞ $\frac{1}{y}$

## Ratio and Proportion Important Examples

Example: 1. A and B together have Rs. 1210. If $\frac{4}{15}$ of A's amount is equal to $\frac{2}{5}$ f B's amount, how much amount does B have?
Explanation: $\frac{14}{5}A =\frac{2}{5}B$
$\Rightarrow A = \left ( \frac{2}{5}\times \frac{15}{4} \right ) B$
$\Rightarrow A = \frac{3}{2}B$
$\Rightarrow \frac{A}{B}B = \frac{3}{2}B$
$\Rightarrow A : B = 3 : 2.$
$\therefore$ B's share = Rs.$\left ( 1210x^{2} \right )= 484$

Example 2: A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?
Explanation: Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.
Then, 4x - 3x = 1000
$\Rightarrow$ x = 1000.
$\therefore$ B's share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.

Example 3: Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?
Answer: 2 : 3 : 4
Explanation: Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.
Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).
$\left ( \frac{140}{100}\times 5x \right )\, ,\left ( \frac{150}{100}\times 7x \right )\, ,\left ( \frac{175}{100}\times 8x \right )$
$7x,\,\frac{21x}{2}\,and\,14x.$
$\therefore$ The required ratio = $7x\,:\,\frac{21x}{2}\,:\,14x.$
$\Rightarrow$ 14x : 21x : 28x
$\Rightarrow$ 2 : 3 : 4

Example 4: In a mixture 60 litres, the ratio of milk and water 2 : 1. If the this ratio is to be 1 : 2, then the quantity of water to be further added is:
Explanation: Quantity of milk = $\left ( 60\times\frac{2}{3}\right )$ litres = 40 litres.
Quantity of water in it = (60- 40) litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added further be x litres.
Then, milk : water = $\frac{40}{20 + x}\, =\, \frac{1}{2}$
$\Rightarrow$ 20 + x = 80
$\Rightarrow$ x = 60.
$\therefore$ Quantity of water to be added = 60 litres.

## Ratio and Proportion Questions from Previous Year Exams

IBPS Questions Related to Ratio and proportion.

## Ratio and Proportion Video Lecture

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