# Ratio and Proportion Questions Answers Formulas, Tips and Tricks

• #### 1. What is Ratio

The ratio of two quantities a and b in the same units, is the fraction \begin{aligned} \frac{a}{b} \end{aligned} and we write it as a : b.

In the ratio a : b, we call a as the first term which is also know as antecedent and b, the second term which is also called consequent.

For example:
\begin{aligned}
4:5 = \frac{4}{5}
\end{aligned}
Here 4 is antecedent and 5 is consequent

Important Rule :
The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

For example: 2:3 = 4:6 = 6:9

• #### 2. What is Proprotion

Equality of two ratios is called 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 <=> (b x c) = (a x d).

• #### 3. Fourth, third and mean proportional

i). Fourth Proportional:

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

ii). Third Proportional:

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

iii). Mean Proportional:

Mean proportional between a and b is \begin{aligned} \sqrt{ab} \end{aligned}

• #### 4. Comparison of Ratios and Compounded Ratio

i). Comparison of Ratios:
When we say that a:b > c:d, then it means
\begin{aligned}
\frac{a}{b} > \frac{c}{d}
\end{aligned}

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

Please note it is ratio of first ratio term of every ratio and second ratio term of every ratio

• #### 5. Important results of Ratio

i). Duplicate ratio of (a:b) is
\begin{aligned}
(a^2:b^2)
\end{aligned}

ii). Sub-duplicate ratio of (a : b) is
\begin{aligned}
(\sqrt{a}:\sqrt{b})
\end{aligned}

iii). Triplicate ratio of (a : b) is
\begin{aligned}
({a}^3:{b}^3)
\end{aligned}

iv). Sub Triplicate ratio of (a : b) is
\begin{aligned}
({a}^\frac{1}{3}:{b}^\frac{1}{3})
\end{aligned}

v).
\begin{aligned} if& \frac{a}{b} = \frac{c}{d}, \\
then \frac{a+b}{a-b} = \frac{c+d}{c-d}

\end{aligned}

• #### 6. Ratio Variation

i). We say that x is directly proportional to y, if x = ky for some constant k and we write,
\begin{aligned}
x \propto y
\end{aligned}

ii). We say that x is inversely proportional to y, if xy = k for some constant k and
we write

\begin{aligned}
x \propto \frac{1}{y}
\end{aligned}