RATIO & PROPORTION

**GETTING STARTED**

☛ **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**

** **

Solving,

** 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**‘.

AGE PROBLEMS

**GETTING STARTED**

☛ 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?**** **

**Soln:**

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.**** Soln:**

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

X =6

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

**GETTING STARTED**

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 **

** = Rs.840**

☛ 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°

**Inverse Variation**

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**