## Unitary Method

The unitary method is a method to solve arithmetic problems based on variation in quantities. In this method we find the value of a unit and then the value of a required number of units.
Ex- If the price of 6 apples is ₹ 30 and you have to buy 10 apples, then find the price of 10 apples.
first you have to find the price of 1 apple, then find the price of 10 apples
∵ Price of 6 apples = ₹ 30
∴ Price of 1 apple = 30 ÷ 6 = ₹ 5
∴ Price of 10 apples = 5 × 10 = ₹ 50

 Value of one article = Value of the given number of articles Number of articles

Value of required number of articles = Value of one article × Required number of articles

### Direct Proportion :-

Direct proportion is the relation between two quantities if on increasing a quantity, then the other quantity also increase, then both quantities are said to be in direct proportion to each other.
For example- If the speed of a car is increase then the distance covered is also increase.
i.e, first quantity ∝ second quantity
Let the first quantity be x and other y, then

 ∴ x1 = x2 = x3 = .......... = xn y1 y2 y3 yn

Ex- If the price of 9 bananas is ₹ 72, then find out the price of 12 bananas.
Solution:- Price of 9 bananas = 72

 Price of 1 banana = 72 9

 Price of 12 bananas = 72 × 12 = ₹ 96 9

By Formula,
More bananas, More price so it is direct proportion
Let the price of 12 bananas be ₹ x

 ∴ x1 = x2 = x3 = .......... = xn y1 y2 y3 yn

 ⇒ 9 = 12 72 x

 ⇒ x = 12 × 72 = 12 × 8 = ₹ 96 9

Ex– Mohan walks 120 m everyday, how many kilometers will he walk in 5 weeks.
Solution:- 1 day walk = 120 m
Total days = 5 × 7
= 35 days
Total walking distance = 120 × 35
= 4200 m
= 4.2 km ( 1 km = 1000 m )
By formula,
Let the walking distance in 5 weeks or 35 days = x m
More days, more walking distance so it is direct proportion.

 ⇒ x1 = x2 y1 y2

 ⇒ 120 = x 1 35

⇒ x = 120 × 35
∴ x = 4200 = 4.2 km

### Indirect Proportion :-

Indirect proportion is the relation between two quantities if on increasing a quantity, then the other quantity decreases then both quantities are said to be in indirect proportion.
For example- If the speed of car is increase, then time taken is decrease.

 i.e, first quantity ∝ 1 second quantity

Let the first quantity be x and other y, then

x1 × y1 = x2 × y2 = x3 × y3 =…….xn × yn

Ex- If A travels at a speed of 50 km/h and covers a distance in 8 hours, then how much time will he take to travel the same distance at 80 km/h ?
Solution:- Distance = Speed × time
= 50 × 8
= 400 km
Now to cover the same distance at speed of 80 km/h

 Time taken = Distance = 400 = 5 hours speed 80

By formula,
More speed, Less time so it is indirect proportion
x1 × y1= x2 × y2 = x3 × y3 =…….xn × yn
or, 50 × 8 = 80 × x

 ⇒ x = 50 × 8 = 5 hours 80

Ex- If 30 chains cost ₹ 4650, then how much do 50 chains cost?
Solution:- cost of 30 chains = 4650

 cost of 1 chain = 4650 30

 cost of 50 chains = 4650 × 50 = 155 × 50 = ₹ 7750 30

By formula,
More chains, More cost so it is direct proportion
Let cost of 50 chairs be ₹ x.

 ∴ x1 = x2 = x3 = .......... = xn y1 y2 y3 yn

 ⇒ 30 = 50 4650 x

 ⇒ 30 = 50 4650 x

 x = ( 50 × 4650 ) = 155 × 50 = ₹ 7750 30

Formula- If M1 can do W1 work in D1 days T1 hours per day and M2 person can do W2 work in D2 days T2 hours per day, then

 ∴ M1D1T1 = M2D2T2 W1 W2

Ex- If 15 men working 12 h per day can reap a field in 24 days, in how many days can 27 men reap the field working 10 h per day ?
Solution:- Let the work completed in x days.
Given, M1 = 15, D1 = 24, T1 = 12, M2 = 27, T2 = 10, D2 = ?

 ∴ M1D1T1 = M2D2T2 W1 W2

 ⇒ 15 × 24 × 12 = 27 × x × 10 1 1

 ⇒ x = 15 × 24 × 12 × 10 27

∴ x = 16 days

Ex- 22 men can complete a job in 16 days. In how many days, will 32 men complete that job?
Solution:- Let the work completed in x days.
given, M1 = 22, D1 = 16, M2 = 32, D2 = ?

 ∴ M1D1T1 = M2D2T2 W1 W2

⇒ 22 × 16 = 32 × x

 ⇒ x = 22 × 16 = 11 32

∴ x = 11 days