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### Formulas of Problems on Ratio and Proportion - Aptitude Questions and Answers.

TIPS FOR SOLVING QUESTIONS RELATED TO RATIO AND PROPORTION:

Ratio: The ratio of two quantities of the same kind and in the same unit is a comparison by division of the measure of two quantities.
It determines how many times one quantity is greater or lesser than the other quantity.
Thus, the ratio of two quantities 'a' and 'b' in the same units, is the fraction \begin{aligned} \frac{a}{b} \end{aligned} and is generally expressed as a : b.
Here, 'a' is called the antecedent and 'b' is called the consequent.

Proportion: The equality of two ratios is called proportion.
For example, 5 : 7 = 10 : 14 i.e. 5 : 7 :: 10 : 14.
Thus, If a : b = c : d, then a : b :: c : d and a, b, c, d are said to be in proportion. Here first and fourth terms i.e. a and d are called extremes, while second and third terms i.e. b and c are called mean terms.

1. If four quantities are in proportion, then

Product of means = Product of extremes

Thus, if a, b, c and d are in proportion i.e. a : b :: c : d, then

b x c = a x d

2. The mean proportion between any two numbers is equal to the square root of their product.
For example, if a : x :: x : b, then

x2 = ab

x =  ab

3. If a : b and c : d are two ratios, and
\begin{aligned}
\frac{a}{b} > \frac{c}{d}
\end{aligned}
\begin{aligned}
\frac{a}{b} < \frac{c}{d}
\end{aligned}
\begin{aligned}
\frac{a}{b} = \frac{c}{d}
\end{aligned}

4. Duplicate ratio of a : b = a2 : b2

5. Sub-duplicate ratio of a : b =  a:  b

4. Triplicate ratio of a : b = 3 : b3

5. Sub-triplicate ratio of a : b = \begin{aligned}
({a}^\frac{1}{3}:{b}^\frac{1}{3})
\end{aligned}

6. If a, b, c and d are four quantities such that a : b :: c : d, then
\begin{aligned} \frac{a+b}{a-b} = \frac{c+d}{c-d}

\end{aligned}

7. Important formula-
\begin{aligned} \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{a + c + e}{b + d + f}
\end{aligned}

8. If a quantity 'k' has to be divided in the ratio of a : b : c, then the proportional parts are
\begin{aligned} \frac{ka}{a + b + c}, \frac{kb}{a + b + c}, \frac{kc}{a + b + c}
\end{aligned}