Mathematical numbers used in comparing two things which are similar to each other in terms of units are ratios. A ratio can be written in three different ways viz, x to y, x: y and

For example:

1. The ratio of 4 to 5 is 4: 5.
2. Ram’s weight is 40 kgs and Ali’s weight is 80 kgs. To find out the ratio of Ram’s weight to Ali’s weight we need to divide Ram’s weight to Ali’s weight. Therefore, the ratio between Ram’s and Ali’s weight is
   \(\begin{array}{l}\frac{40}{80}\end{array} \)
   = 1:2

Comparing things similar to each other is the concept of ratio. And when two ratios are the same, they are said to be in proportion to each other. It is represented by the symbol ‘::’ or ‘=’.

Introduction to Ratio and Proportion

Golden Ratio

Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to larger of the two quantities.

• • If two numbers a and b are in golden ratio, then 
    \(\begin{array}{l}\frac{a+b}{a}=\frac{a}{b}\end{array} \)
  • It is approximately equal to 1.618.

Ratio

• The ratio is the comparison of a quantity with respect to another quantity.
• It is denoted by “:“.
• Two quantities can be compared only if they are in the same unit.

Example: Father’s age is 75 years and the daughter’s age is 25 years.

$\Rightarrow$The ratio of father’s age to daughter’s age

$\Rightarrow$

Difference between Fractions and Ratios

• A fraction describes a part of a whole and its denominator represents the total number of parts.
  Example: 13 means one part out of 3 parts.
• A ratio is a comparison of two different quantities.
• Example: In a society, 10 people like driving, 20 people like swimming and the total number of people in society is 30.
• The ratio of the number of people liking driving to the total number of people = 10:30.
• The ratio of the number of people liking swimming to the number of people liking driving is 20:10.

For more information on Difference between Fractions and Ratios, watch the below video

Same Ratio in Different Situations

• Ratios can remain same in different situations.
• Example:

1. \(\begin{array}{l}\frac{Weight of Joe}{Weight of James}=\frac{50}{100} = 1:2\end{array} \)
2. \(\begin{array}{l}\frac{Number of Girls}{Number of Boys}=\frac{50}{100} = 1:2\end{array} \)

Here, both the above ratios are equal.

Comparing Quantities Using Ratios

• Quantities can be compared using ratios.
• Example: Joe worked for 8 hours and James worked for 2 hours. How many times Joe’s working hours
  is of James’ working hours?
  Solution: Working hours of Joe = 8 hours
  $\Rightarrow$Working hours of Sheela = 2 hours
  $\Rightarrow$The ratio of working hours of Joe to Sheela $=\backslash (\backslash begin\{array\}\{l\}\backslash frac\{8\}\{2\}=4\backslash end\{array\}\; \backslash ).$

Therefore, Joe works four times more than James.

To know more about Comparison of Ratios, visit here.

Equivalent Ratios

When the given ratios are equal, then these ratios are called as equivalent ratios.

• Equivalent ratios can be obtained by multiplying and dividing the numerator and denominator with the same number.
• Example: Ratios $10:30(=1:3)$ and $11:33(=1:3)$ are equivalent ratios.

Unitary Method

The method in which first we find the value of one unit and then the value of required number of units is known as Unitary Method.

• Example: Cost of two shirts in a shop is Rs.200. What will be the cost of 5 shirts in the shop?
  Solution : Cost of 2 shirts $=Rs.200$
  $\Rightarrow$Cost of 1 shirt $=\backslash (\backslash begin\{array\}\{l\}\backslash frac\{200\}\{2\}=100\backslash end\{array\}\; \backslash )$
  $\Rightarrow$Cost of 5 shirts $=\; \backslash (\backslash begin\{array\}\{l\}\backslash left\; (\; \backslash frac\{200\}\{2\}\; \backslash right\; )*5=100*5\backslash end\{array\}\; \backslash )\; =Rs.500$

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Proportions

If two ratios are equal, then they are said to be in proportion.

• Symbol “::” or “=” is used to equate the two ratios.
• Example: Ratios 2:3 and 6:9 are proportional.
  $\Rightarrow$ 2:3 :: 6:9 or 2:3 = 6:9

Uses of Ratios and Proportions

• Example: Suppose a man travelled 80 km in 2 hours, how much time will he take to travel 40 km?
  Solution: If x is the required time, then the proportion is
  $80:2::40:x$.
  $\Rightarrow$ 
  \(\begin{array}{l}\frac{80}{2} * \frac{40}{x}\end{array} \)
  $\Rightarrow$  $80x=80$
  $\Rightarrow$ $x=1hour$
  So, the man takes one hour to complete 40 km.

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