## 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 = | ||

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

∴ | _{1} |
= | _{2} |
= | _{3} |
= .......... = | _{n} |

y_{1} |
y_{2} |
y_{3} |
y_{n} |

**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 = | ||

9 |

Price of 12 bananas = | × 12 = ₹ 96 | |

9 |

*By Formula,*More bananas, More price so it is direct proportion

Let the price of 12 bananas be ₹ x

∴ | _{1} |
= | _{2} |
= | _{3} |
= .......... = | _{n} |

y_{1} |
y_{2} |
y_{3} |
y_{n} |

⇒ | = | ||

72 | x |

⇒ x = 12 × | = 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.

⇒ | _{1} |
= | _{2} |

y_{1} |
y_{2} |

⇒ | = | ||

1 | 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 ∝ | ||

second quantity |

**x**_{1}**× y**_{1 }**= x**_{2 }**× y**_{2}**= x**_{3}**× y**_{3}**=…….x**_{n}**× y**_{n}**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 = | = | = 5 hours | ||

speed | 80 |

**By formula,**

More speed, Less time so it is indirect proportion

x

_{1}× y

_{1}= x

_{2 }× y

_{2}= x

_{3}× y

_{3}=…….x

_{n}× y

_{n}

or, 50 × 8 = 80 × x

⇒ x = 50 × | = 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 = | ||

30 |

cost of 50 chains = | × 50 = 155 × 50 = ₹ 7750 | |

30 |

*By formula,*More chains, More cost so it is direct proportion

Let cost of 50 chairs be ₹ x.

∴ | _{1} |
= | _{2} |
= | _{3} |
= .......... = | _{n} |

y_{1} |
y_{2} |
y_{3} |
y_{n} |

⇒ | = | ||

4650 | x |

⇒ | = | ||

4650 | x |

x = | = 155 × 50 = ₹ 7750 | |

30 |

**Formula**- If M_{1} can do W_{1} work in D_{1} days T_{1} hours per day and M_{2} person can do W_{2} work in D_{2} days T_{2 }hours per day, then

∴ | _{1}D_{1}T_{1} |
= | _{2}D_{2}T_{2} |

W_{1} |
W_{2} |

**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, M_{1} = 15, D_{1} = 24, T_{1} = 12, M_{2} = 27, T_{2} = 10, D_{2} = ?

∴ | _{1}D_{1}T_{1} |
= | _{2}D_{2}T_{2} |

W_{1} |
W_{2} |

⇒ | = | ||

1 | 1 |

⇒ x = 15 × 24 × | × 10 | |

27 |

**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, M_{1} = 22, D_{1} = 16, M_{2} = 32, D_{2} = ?

∴ | _{1}D_{1}T_{1} |
= | _{2}D_{2}T_{2} |

W_{1} |
W_{2} |

⇒ x = 22 × | = 11 | |

32 |