Ratio

This is an expression used to compare two or more
 quantities relative to each other.

The quantities are separated by a colon (:)

Example

There are two cats and one dog.

ccd

The ratio of cats to dogs is  2:1

 

cats : dogs
2 : 1

 

The ratio of dogs to cats is  1:2

dogs : cats
1 : 2

 

This means there are twice as many cats as dogs,
or half as many dogs as cats.

Equivalence

There are six cats and three dogs.
There are still twice as many cats as dogs.

 

ccd

ccd

ccd

 

Dividing through by the HCF
reduces the ratio to itssimplest form.

Here, the HCF of 6 and 3 is 3,
so divide both sides by 3.

  cats : dogs  
3 6 : 3 3
2 : 1

The  ratio  6:3  reduces to the  ratio  2:1
 when expressed in simplest form.

Multiplying the simplest form will give equivalent ratios

cats : dogs
2 : 1
4 : 2
6 : 3
8 : 4

etc.

The ratio
8:4 is equivalent to 
2:1 , 4:2,    200:100  etc.

So how can this be used ?

Given the ratio of boys to girls as  2:3
 how many girls will there  be if there are 10 boys ?

bbggg

 

boys : girls
2 : 3
10 : *

Notice that the original number of boys has
been multiplied by 5.

This means that the original number of girls must
also be multiplied by 5

  boys : girls  
5 2 : 3 5
10 : 15

So there are 15 girls.

Example

Given the ratio of girls to boys as  3:2
 how many boys will there  be if there are 12 girls?
 


girls
: boys
3 : 2
12 : *

 

  girls : boys  
4 3 : 2 4
12 : 8

There are 8 boys

Notice that the term which has both known
quantities goes on the left.

Other methods 

These are sometimes written as ratio equations
e.g.    3:2 :: 12:x


      In Vedic maths,
the product of the means
equals the product of the extremes

Multiplying the outer two numbers will have the same answer as multiplying the inner two numbers.
     a:b :: c:d
gives  ad =bc

 

Example
Find the value of x
3:2 :: 12:x

1

Find the value of x
5:x :: 35:9

2

Proportion

There are two cats and one dog.

ccd

cats : dogs Total animals
2 : 1 3

The proportion of  cats is  2/3
The proportion of  dogs is 1/3

Two thirds of the animals are cats.
One third of the animals are dogs.

Sharing in proportion

Example

Divide 45 Kg in the ratio  3: 2

 

3

The total number of parts is 5
each part has a value of 45 ÷ 5  = 9Kg
 9 x 3 = 27 Kg     9 x 2 = 18 kg
so 3 parts is 27kg, 2 parts is 18 Kg

( check: 27 + 18 = 45,  the original number)

Example

A box of 84 chocolates is shared amongst
Tom, Ryan  and Ashleigh in the ratio
3:4:5 respectively.
How many chocolates does each person
receive ?

Tom : Ryan : Ashleigh Total
3 : 4 : 5 12

4

5

6

 

 

© Alexander Forrest