RATIO & PROPORTION
☛ A fraction a/ b can be expressed as a:b
☛ Generally, ratio is used as a tool to compare two quantities.
Let us say, in a jar there are 5L milk and 4 L water. We say the ratio of milk to water in the Jar is 5:4.
☛ The ratio remains unaffected when it is multiplied by any natural number.
eg. 5:4 =5 x 4: 4 x 4 = 20:16
=5 x 16: 4 x 16 = 80:64
=5 x 17: 4 x 17= 85:68
☛ Proportion :
When two ratios a:b and c:d are equal, we say that they are in proportion.
They are denoted as a:b::c:d
a/b = c/d
ad = bc
The third term c in the proportion a:b::c:d is known as ‘Third Proportional‘.
The fourth term d in the proportion a:b::c:d is known as ‘Fourth Proportional‘.
☛ If a person’s age is x years, then after n years, his age will be x+n years
☛ Age difference two persons always remains constant.
Eg. Ram is 16 years today and his mother is 38 years old. What will the age difference between then after 10 years?
Ram is 16 years and his mother is 38 years old today. Difference in their ages is 22 years.
It means Ram’s mother was 22 years when he was born.
The difference will be always present.
10 years hence, Ram will be 26 years and his mother will be 48 years. Difference in ages = 48-26= 22 years
Ratio of ages
☛ The ratio of ages between two persons keeps changing every year.
☛ All the age problems can be solved by converting the problem into word problem in one variable
Eg. The ratio of Ram and Rahim’s age is 2:3 today. After 9 years, their ages will be in the ratio 7:9.
Find their present ages.
Let Ram’s age be 2x. Thus Rahim’s present age is 3x
After 9 years, their ages will be 2x+9 and 3x+9 respectively.
(2X+9)/(3x +9) = 7/9
18x+81 = 21x+63
3x = 18
Hence, Ram’s age is 12 years and Rahim’s age is 18 years.
TIP: The best way to solve the age problems is to apply options given as per the conditions mentioned in question.
Variation is one of the most fundamental concept which will be a building block for other concepts.
☛ Suppose you go to a nearby shop and see that 10 books cost Rs. 400. You want to buy 21 such books. What would be the cost of 21 such books?
Since 10 books cost Rs. 400,
Cost of 1 book should be 400/10 = Rs.40
Cost of 21 books = Cost of 1 book x 21
= 40 x 21
☛ The method of finding the value for n units given that you know the value for 1 unit is known as ‘Unitary Method’.
Here, we noticed that with increase in number of books, total money spent also increased. This is an example of ‘Direct Variation’. With an increase of X, there is also an increase of Y.
☛ Another example can be angle traced by Clocks.
Minute hand of a clock traces 6° in one minute. How much angle does it cover in 40 minutes?
More minutes, more angle.
Hence in 40 minutes, minute hand would cover 40 x 6° = 240°
To find the value of ?, we cross multiply,
? = 40 x 6/1 = 240°
There are cases where the value of Y decreases with the increase in value of X.
☛ Let us take an example of 10 men working on a project and completing it in 5 days. If the same work is to be done by 20 men, how much time will they take?
☛ By logic, we know, more men, less the time taken. So, this is a case of inverse variation.
Considering 1 person completes 1 unit of work in a day, 10 men working 5 days complete 50 units of work. Let us say the entire job consists of 50 smaller jobs.
So, 1 man will complete the entire job (with 50 units of work) in 50 days.
☛ 20 men will complete 20 units of work in 1 day. Hence, they will take 50/20 = 2.5 days to complete the entire work.
? = 10 x 5/20 = 2.5
☛ If we notice the arrow marks in direct variation and inverse variation, direct variation had cross multiplication of values while inverse variation, we multiply values in the same row.
The questions will not be directly asked on this topic but it is the foundation for all the topics