## Unitary Method and Variation

Subject: Compulsory Maths

#### Overview

The unitary method is a technique in elementary algebra for solving a class of problems in variation. It consists of altering one of the variables to a single unit, i.e. 1, and then performing the operation necessary to alter it to the desired value.

#### Unitary Method

The unitary method is a method, in which the value of a quantity is first obtained to find the value of any required quantity.

There are two types of variation while solving the problem of unitary method. They are

• Direct variation
• Inverse variation

### Direct Variation

When the increase or decrease in the value of one quantity causes the increase or decrease in the value of other quantity than they are said to be direct variation. For example, the cost of goods varies directly to the numbers of goods. More good, more cost. Less good, less cost.

### Inverse Variation

When the increase or decrease in the value of one quantity causes the decrease or increase in the value of other quantity then they are said to be inverse variation. For example, time and work. More time consumes to complete a piece of work with less number of men's.

##### Things to remember
• The Unitary Method provides an alternative approach to solving problems.
• Percentages and simple percentage problems would normally precede the unitary method approach.
• Ratio calculations are an aspect of the unitary method.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

Solution:

12 men can make a furniture in 9 days.

1man can make it in 9 × 12 days = 108 days.

18 men can make the furniture in $\frac{108}{18}$ days = 6 days

Solution:

Since 6 pens cost = Rs 450.

So, cost of 1 pens = Rs $\frac{450}{6}$ = Rs 75

So, cost of 8 pens = Rs 75×8 = Rs 600

Solution:

10 Japanese Yen = Rs 6.60

1 Japanese Yen = Rs $\frac{6.60}{10}$

65,300 Japanese Yen = Rs $\frac{6.60}{10}$× 65300 = Rs 43,098

Solution:

Given information,

From the table,

Cost of 5 Chairs = Rs 150

Cost of 1 Chair = Rs $\frac{150}{5}$ = Rs 30

Cost of 10 Chairs = Rs 30 × 10 = Rs 300

Cost of 6 Chairs= Rs 30 × 6 = Rs 180

Cost of 4 Chairs = Rs 30 × 4 = Rs120

So, Cost of x Chairs = Rs 30 × x = Rs 30x

Solution:

10 Kg Oranges cost Rs 750

1 kg Orange cost Rs $\frac{750}{10}$ = Rs 75

6 kg Oranges cost Rs 75×6 = Rs 450

∴ 6 kg Oranges cost Rs 450

Solution:

18 days is required to complete the work for 10 workers.

1 day is required to complete the same work for = 10×18 = 180 workers.

15 days is required to complete the work = $\frac{180}{15}$ = 12 workers.

∴ 12 workers can do the same work in 15 days.

Solution:

 Quantity Distance (D) 7 112 12 X

Here,

$\frac{7}{12}$ = $\frac{112}{x}$

or, 7x = 112× 12

or, x = $\frac{112×12}{7}$

or, x = 192 km

∴ 12 litre can travel 192 km.

Solution:

 Days Total Workers 24 20 x 15

Here, $\frac{24}{x}$ = $\frac{15}{20}$

or, 15 x = 480

or, x = $\frac{480}{15}$

or, x = 32

∴ 15 men do the work in 32 days.

Solution:

300 students required = 12 rooms

1 student required = $\frac{12} {300}$ room

375 students required = $\frac{12}{300}$× 375 = 15 rooms

∴ Required rooms = (15 -12) = 3 rooms.

Alternative,

 Number of Students Class Room 300 12 375 x=?

$\frac{300}{375}$ = $\frac{12}{x}$

Or, 300×x = 12×375

Or, x = $\frac{12×375}{300}$

0r, x = 15

∴ 15 rooms are required for 375 students.

Solution:

Rs 576 = 4 Dozen(48) pens

Rs 1 = $\frac{48}{576}$pens

Rs 228 = $\frac{48}{576}$× 228 = 19 pens

Alternatively,

 Pen's cost Quantity of Pen 576 4×12 = 48 228 x=?

$\frac{576}{228}$ = $\frac{48}{x}$

or, x = $\frac{48×228}{576}$

or, x = 19 pens

∴19 pens cost Rs 228

Solution:

A truck travels 48km per hour and covers the distance in 6 hours

If a truck travels 1 km per hour and covers the distance in 6×48 =288 hours

If a truck travels 36 km per hour and cover the distance = $\frac{288}{36}$ hours = 8 hours

∴ 8 hours is required to cover the distance if the speed of truck decreases by 36 km per hour.

Solution:

Since, cost of 4 Computer chairs = Rs 1,544

So, cost of 1 Computer chairs = Rs $\frac{1544}{4}$ = Rs 386

And cost of 14 Computer Chairs = Rs 386×14 = Rs 5,404

Solution:

In 10 days, a worker earns = Rs 1,850

In 1 day, a worker earns = Rs $\frac{1850}{10}$ = Rs 185

In 3 days, a worker earns = Rs 185× 3 = Rs 555

Hence, he will earn Rs 555 in 3 days.

Solution:

Cost of 93m cloth = Rs 1395

Cost of 1m cloth = Rs $\frac{1395}{93}$ Rs 15

Cost of 105m cloth = Rs 15×105 = Rs 1575

∴ The Cost of 105m cloth is Rs 1575

Solution:

Since, 32 men can reap a field in 15 days.

1 men can reap a field in 15× 32 = 480 days

20 men can reap a field in $\frac{480}{20}$ = 24 days

∴ 20 men can reap a field in 24 days.

Solution:

If, 6 hours is required to finish the work for 4 men

1 hour is required to finish the work for 4×6 = 24 men

4 hours is required to finish the work for $\frac{24}{4}$ = 6 men

∴ 6 men is required to finish the work in 4 hours.

Solution:

5 Euro = Rs 625

1 Euro = Rs $\frac{625}{5}$ = Rs 125

Rs 8,946 = $\frac{8,946}{125}$ = 71.568 Euro

Solution:

Cost of 5 goats and 2 Cows = Rs 1,35,000

Cost of 1 cow = Rs 17,500

Cost of 2 cows = Rs 17,500×2 = Rs 35,000

Cost of 5 goats = Total Amount - 2 Cows Amount

=Rs 1,35,000 - Rs 35,000

= Rs 100,000

∴ The Cost of 1 goat = Rs $\frac{1,00,000}{5}$

= Rs 20,000

∴ The Cost of 1 goat = Rs 20,000

Solution:

Since, Cost of 5 computer mouse = Rs1250.

So, Cost of 1 computer mouse = Rs $\frac{1250}{5}$ = Rs 250

And Cost of 10 computer mouse = Rs 250×10 = Rs 2,500

∴ The cost of 10 computer mouse is Rs 2,500

Solution:

Since 15 days is required to finish the work for 20 workers

1 day is required to finish the work = 20 $\times$15 = 300

10 days is required to finish the work $\frac{20\times15}{10}$ = $\frac{300}{10}$ for 30 workers

Hence, 30 workers should be added to finish the work in 10 days.

Solution:

The price of 3 chairs and 4 tables = Rs 7,540

The price of 1 chair = Rs 220

The price of 3 chairs = Rs 220$\times$3 = Rs 660

Now,

Price of 4 tables = Total amount - Price of 3 chairs

= Rs 7540 - Rs 660

= Rs 6880

Hence, Price of 1 table = $\frac{Rs 6880}{4}$

= Rs 1720