Subject: Compulsory Maths

**Unitary Method**

It is a technique in mathematics for solving a problem finding the value of the single unit and then finding the necessary value by multiplying the single unit value. If the quantities are in direct proportion, unit value is obtained by division. Similarly, if the quantities are in inverse proportion, unit value is obtained by multiplication. For examples,

Suppose the cost of 5 pens is Rs. 50.

Then, the cost of 1 pen is Rs. \(\frac{50}{5}\) = Rs. 10

Here, 1 pen is the unit of quantity and Rs. 10 is the unit cost. The number of pens and their cost is in direct proportion.

Again, suppose 4 men can finish a work in 6 days.

Then, 1 man can finish the work in 4 × 6 days = Rs. 24 days.

Here, the number of men and their working days are in inverse proportion.

**Time and Work**

In less time we do less amount of and in more time we do the greater amount of work. So, Time and amount of work done are in direct proportion. In this case, the amount of work done in unit time is obtained by division. For example,

In 8 days, Hari can do 1 work.

In 1 day, Hari can do \(\frac{1}{8}\) part of work.

\(\frac{1}{8}\) parts of work are done in 1 day.

Similarly, in 2 days, Hari can do\(\frac{2}{8}\) =\(\frac{1}{4}\) parts of the work

In 3 days, Hari can do\(\frac{3}{8}\) parts of the work.

In 4 days, Hari can do\(\frac{4}{8}\) =\(\frac{1}{2}\) parts of work and so on.

- If the quantities are in direct proportion, unit value is obtained by division.
- If the quantities are in inverse proportion, unit value is obtained by multiplication.
- The amount of work done in unit time is obtained by division.

- 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.

The cost of 12 m of clothes is Rs 600, find the cost of 8 m of clothes.

Solution:

The cost of 12 m of clothes = Rs 600

The cost od 1 m of clothes = Rs \(\frac{600}{12}\)

= Rs 50

The cost of 8 m of clothes = 8 × Rs 50

= Rs 400

So, the required cost of 8 m of clothes is Rs 400.

The floor of a room is 14 m long and 8.5m broad. Find the cost of carpeting the floor at Rs 75 per sq. m.

Solution:

Here,

the length of the floor (l) = 14 m

the breadth of the floor (b) = 8.5m

∴ Area of the floor = l × b

= 14m × 8.5m

= 119 m^{2}

Now,

the cost of carpeting ^{ }119 m^{2} is 119 × Rs 75

= Rs 8925

So, the required cost of carpeting the floor is Rs 8925.

A man pays Rs 525 for income tax at the rate of Rs 2 per Rs 100, Find his income.

Solution:

Here,

Rs 2 is the tax for the income of Rs 100

Re 1 is the tax for the income of Rs \(\frac{100}{2}\)

= Rs 50

Rs 525 is the tax for the income of Rs 525 × 50

= Rs 26250

So, the required income of the man is Rs 26250

If \(\frac{3}{4}\) parts of a land costs Rs 15,000, find the cost of \(\frac{2}{5}\) parts of the land.

Solution:

Here,

the cost of \(\frac{3}{4}\) parts of a land = Rs 15,000

The cost of 1 (whole) land = Rs \(\frac{15000}{\frac{3}{4}}\)

= Rs \(\frac{15000 × 4}{3}\)

= Rs 20,000

The cost of \(\frac{2}{5}\) parts of land = Rs 20,000 × \(\frac{2}{5}\)

= Rs 8,000

so, the required cost is Rs 8,000.

7 workers build a wall in 20 days. how many more workers should be employed to build it in 14 days?

Solution:

Here,

In 20 days, 7 workers build a wall

In 1 days, 20 × 7 workers build the wall

In 14 days, \(\frac{20 × 7}{14}\) workers build the wall = 10 workers build the wall

∴ The required number of workers to be added = 10 - 7

= 3 workers

X and Y together can do a piece of work in 10 days and X alone can do it in 15 days. In how many days would Y alone can finish the work?

Solution:

In 10 days, (X + y) can finish 1 work

In 1 days, (X = y) can finish \(\frac{1}{10}\) parts of the work

Also,

In 15 days, X can finish 1 work

In 1 days , X can finish \(\frac{1}{15}\) parts of the work.

Now,

In 1 days (X + Y - X) can finish (\(\frac{1}{10}\) - \(\frac{1}{15}\)) parts of the work

In 1 days Y can finish (\(\frac{3 - 2}{30}\)) parts of the work = \(\frac{1}{30}\) parts of the work.

∴ Y can finish \(\frac{1}{30}\) parts of the work in 1 day.

Y can finish 1 work in \(\frac{1}{\frac{1}{30}}\) days = 1 × \(\frac{30}{1}\) = 30 days.

So, Y alone can finish the work in 30 days

Hari can do a piece of work in 8 days. He worked for 6 days and left. If Gopal finished the remaining work, how much work did he do ?

Solution:

Here,

In 8 days, Hari can do 1 work

In 1 day, Hari can do \(\frac{1}{8}\) parts of the work.

In 6 days, Hari can do \(\frac{6}{8}\) parts of the work = \(\frac{3}{4}\) parts of the work.

Now,

Remaining parts of work = (1 - \(\frac{3}{4}\))

= \(\frac{1}{4}\) parts of the work

So, Gopal finished \(\frac{1}{4}\) part of the work

A can do a piece of work in 6 days and B can do it in 12 days, In how many days would they finish it working together?

Solution:

In 6 days, A can do 1 work

In 1 day, A can do \(\frac{1}{6}\) parts of the work

Also,

In 12 days, B can do 1 work

In 1 day, B can do \(\frac{1}{12}\) parts of the work

Now,

In 1 day, (A + b) can do (\(\frac{1}{6}\) + \(\frac{1}{12}\)) partss of the work

= (\(\frac{2 + 1}{12}\)) parts of the work = \(\frac{3}{12}\) = \(\frac{1}{4}\) parts of the work

∴ (A + B) can do \(\frac{1}{4}\) parts of the work = 1

(A + B) can do 1 work in \(\frac{1}{\frac{1}{4}}\) day = 1 × \(\frac{1}{4}\) = 4 days

So, A and B finish the work in 4 days working together.

A group of 40 people had provisions for 60 days. If 10 more people joined the group, how long would the provisions last?

Solution:

Here,

after joining 10 more people, total number of the people in the group = 0 + 10 = 50

40 people had provisions for 60 days

1 people had provisions for 40 × 60 days

50 people had provisions for \(\frac{40 × 60}{50}\) days = 48 days

So, the provisions would long last for 48 days.

If the cost of 5 pens is Rs. 100. Then find the cost of 2 pens.

Solution:

Here, Cost of 5 pens = Rs. 100

or, Cost of 1 pen = \(\frac{Rs.100}{5}\) = Rs. 20

Then, The cost of 2 pens = 2 × Rs. 20 = Rs. 40 ans.

If 6 men finish a piece of work in 5 days. Then, In how many days does the 3 men finish the same work?

Solution:

6 men finish a piece of work in 5 days.

Then, 1 man finishes the same piece of work in 6 × 5 days = 30 days.

Now, 3 men finish the same piece of work in \(\frac{30}{3}\) = 10 days ans.

If the cost of 12 litres of milk is Rs. 432. Then, find the cost of 5 litres of milk.

Solution:

Here, The cost of 12 litres of milk = Rs. 432

or, The cost of 1 litres of milk = Rs. \(\frac{432}{12}\) = Rs. 36

The cost of 5 litres of milk = 5 × Rs. 36 = Rs. 180

So the required cost of 5 litres of milk is Rs. 180.

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