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 \(\frac{x}{y}\) but read as the ratio of x to y.
For example:
 The ratio of 4 to 5 is 4: 5.
 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 \(\frac{40}{80}\)= 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 \(\frac{a+b}{a}=\frac{a}{b}\)
 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.
⇒The ratio of father’s age to daughter’s age
⇒\(\frac{Father′s Age}{Daughter’s Age}=\frac{3}{1}=3:1\)
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:
 \(\frac{Weight of Joe}{Weight of James}=\frac{50}{100} = 1:2\)
 \(\frac{Number of Girls}{Number of Boys}=\frac{50}{100} = 1:2\)
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
⇒Working hours of Sheela = 2 hours
⇒The ratio of working hours of Joe to Sheela =\(\frac{8}{2}=4\).
Therefore, Joe works four times more than James.
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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
⇒Cost of 1 shirt =\(\frac{200}{2}=100\)
⇒Cost of 5 shirts = \(\left ( \frac{200}{2} \right )*5=100*5\) = 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.
⇒ 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.
⇒ \(\frac{80}{2} * \frac{40}{x}\) ⇒ 80x=80
⇒ x=1 hour
So, the man takes one hour to complete 40 km.
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