Introduction
Ratio is a term used to compare two quantities. A Ratio of two numbers tells how many times one quantity is to another quantity.
Ratio and its example
When two quantities, which are of the same kind and have the same units of measurement, are compared by dividing
one quantity to another quantity, it is called ratio.
∴ ratio of two quantities a and
b can be written as ab , where b ≠ 0
It is denoted by : symbol
Ratio of and b is written as a : b and read as a ratio b
a and b are called terms of ratio.
In ratio a : b, the first term a is called antecedent.
The later term b is called consequent.
What is the ratio of pears in two bags with weights 15 kg and 20 kg?
The ratio of their weight = 1520
which is further = 34
(∵ the ratio must be expressed in its lowest terms)
So, the ratio is 3:4
Proportion and its example
When two ratios are equal, that implies the two ratios are in proportion also.
It means the equality of two ratios is called proportion.
The symbol of proportion is ::
To understand it, consider four quantities a, b, c and d.
If the ratio of first two quantities a and b is equal to ratio of last two quantities c and d , then four
quantities a, b, c and d are said to be in proportion.
It is written as ab =
cd
∴ the proportion of above ratios can be written as a : b :: c : d
In a : b :: c : d, the first term a and fourth term d are called extremes.
The second term b and third term c are called means.
If four terms are in proportion then:
Product of extremes = Product of means
Is this a correct 15 : 45 :: 40 : 120 proportion?
Here, 15 and 120 are called extremes and 45 and 40 are called means.
The product of extremes = 15 × 120 = 1800
The product of means = 45 × 40 = 1800
i.e. The product of extremes and means both are equal to 1800
the numbers 15, 45, 40 and 120 in proportion.
What is Continued Proportion?
Let there are three quantities a, b and c. If the ratio between first and second quantity is equal to ratio between second and third quantity. It implies that the three quantities are in continued proportion.
It is written as following:
a : b = b : c
Second quantity, here b, is called mean proportional between first quantity a and third
quantity c.
Third quantity, here c, is called third proportional to first quantity a and second
quantity b.
Are the numbers 4,6 and 9 are in continued proportion?
Reduce 46, which can be written as
23
Reduce 69, which can be written as
23
∴ both ratios 46 and
69 are equal to
23
∴ 46 =
69
or 4 : 6 = 6 : 9
Hence, 4, 6 and 9 are in continued proportion
Also, 6 is mean proportional between 4 and 9.
9 is third proportional to 4 and 6.
Unitary method in ratios and proportions
This method is used to find the value of any number of units of a quantity when the value of more
than one unit of same quantity is known. With this method first the value of a single unit is calculated and
then that value is multiplied by the number of units for which the value is to be calculated.
This method can solve ratios and proportions problems such as find the price of 8 apples
when the price of 6 apples is known.
This method is called as Unitary method because it involves an intermediate step to find the value of a
single unit.
Steps of unitary method
Step 1: Find the value of single unit by dividing the given total value by the total number of units.
Step 2: Multiply the number obtained in step 1 with the total number of units to know their value.
Example 1: If the cost of 20 toy cars is $220. Find the cost of 5 such toy cars.
Step 1: Calculate the cost of single unit i.e. one toy car by dividing the cost of 20 toy cars i.e. $220
by the total number of toy cars i.e. 20
Cost of 20 toy cars = $220
Cost of 1 toy car = 22020 = 11
Step 2: Multiply 11 obtained in step 1 with the 5 toy cars.
∴ Cost of 5 toy cars = 11 × 11 = $55
