In this section, we are going to see, how problems on inverse variation can be solved using unitary method.

First let us come to know what is inverse variation.

What happens when..................

Thus we can say, if an **increase** in one quantity produces a proportionate **decrease** in another quantity, then the quantities are said to be in inverse variation.

or

If a **decrease** in one quantity produces a proportionate **increase** in another quantity, then the quantities are said to be in inverse variation.

Change in both the quantities must be same.

That is,

**Increase -------> Decrease **

or

**Decrease -------> Increase **

**Unitary Method Definition and Example :**

**Definition :**

Unitary-method is all about finding value to a single unit.

Unitary-method can be used to calculate cost, measurements like liters and time.

**Example :**

If 10 pencils cost $15,

then the cost of one pencil is = 15 / 10 = $1.50

**Problem 1 :**

7 men can complete a work in 52 days. In how many days will 13 men finish the same work ?

**Solution :**

This is a situation of inverse variation.

Because,

more men -----> less days

**Given :** 7 men can complete a work in 52 days

No. of days taken by one man to complete the work is

= No. of men ⋅ No. of days

= 7 ⋅ 52

= 364 days

No. of days taken by 13 men to complete the work is

= 364 / 13

= 28 days

13 men can complete the work in 28 days.

**Problem 2 :**

A book contains 120 pages and each page has 35 lines. How many pages will the book contain if every page has 24 lines per page ?

**Solution : **

**This is a situation of inverse variation.**

**Because, **

less lines --------> more pages

35 lines --------> 120 pages

1 line --------> 35 ⋅ 120 = 4200 pages

24 lines --------> 4200 / 24 = 175 pages

If the book has 24 lines per page, then it will contain 175 pages.

**Problem 3 :**

A truck covers a particular distance in 3 hours with the speed of 60 miles per hour. If the speed is increased by 30 miles per hour, find the time taken by the truck to cover the same distance.

**Solution : **

**This is a situation of inverse variation.**

**Because, **

**more speed -----> less time**

**Given : Time = 3 hours and Speed = 60 mph**

**The formula to find the distance is **

**Distance = Time **⋅** Speed**

**Distance = 3 **⋅** 60 = 180 miles**

**If the given speed 60 mph is increased by 30 mph, ****then the new speed will be 90 mph. **

**The formula to find the time is, **

**Time = Distance / Speed**

Time = 180 / 90

Time = 2 hours

If the speed is increased by 30 mph, time taken by the truck is 2 hours.

**Problem 4 : **

David can complete a work in 6 days working 8 hours per day. If he works 6 hours per day, how many days will he take to complete the work ?

**Solution : **

**This is a situation of inverse.**

**Because,**

** less** hours per day -----> more days to complete the work

8 hours per day --------> 6 days to complete the work

1 hour per day ---------> 8 ⋅ 6 = 48 days

6 hours per day ---------> 48 / 6 = 8 days

David can complete the work in 8 days working 6 hours per day.

**Problem 5 :**

Alex takes 15 days to reduce 30 kilograms of his weight by doing 30 minutes exercise per day. If he does exercise for 1 hour 30 minutes per day, how many days will he take to reduce the same weight ?

**Solution :**

**This is a situation of inverse variation.**

**Because, **

**more minutes per day ----> less days to reduce the weight**

**Given : **Minutes per day = 30 and No. of days = 15

Total minutes in 15 days = 30 ⋅ 15 = 450 minutes

So, 450 minutes of exercise required to reduce 30 kilograms weight.

1 hour 30 minutes = 90 minutes

**Given : **Alex does exercise for 1 hour 30 minutes per day.

1 hour 30 minutes = 90 minutes

Then, no. of days required is

= 450 / 90

= 5 days

If Alex does exercise for 1 hour 30 minutes per day, it will take 5 days to reduce 30 kilograms of weight.

**Problem 6 : **

A store sells a product in the following scheme. For the first 5 units, cost per unit is $10. From 6 to 10 units, 20% discount will be given. For a sale of more than 10 units, another 10% discount will be given. Find the total cost of 50 units.

**Solution : **

**This is a situation of inverse variation.**

**Because, **

**more units -----> cost per unit will be less**

For the first 5 units, cost per unit is $10.

From 6 to 10 units,

cost per unit = 80% of 10

cost per unit = 0.8 ⋅ 10 = $8

For more than 10 units,

cost per unit = 90% of 8

cost per unit = 0.9 ⋅ 8 = $7.2

Then, for a sale of 50 units,

cost per unit = $7.20

Total cost of 50 units is

= 50 ⋅ 7.20

= $360

The total cost of 50 units is $360.

**Problem 7 : **

If 5 men can paint a house in 18 hours, how many men will be able to paint it in 10 hours ?

**Solution : **

**This is a situation of inverse variation.**

**Because, **

**less hours -----> more men **

In 18 hours, the house can be painted by 5 men

No. of hours taken by 1 man to paint the house is

= No. of hours ⋅ No. of men

= 18 ⋅ 5

= 90 hours

No. of men required to paint the house in 10 hours is

= 90 / 10

= 9 men

9 men will be able to paint the house in 10 hours.

**Problem 8 : **

In a fort, 360 men have provisions for 21 days. If 60 more men join them, how long will the provision last ?

**Solution : **

**This is a situation of inversion variation.**

**Because, **

**more men -----> provision will last for less days**

**Given :** 360 men have provisions for 21 days

1 man has provisions for

= 360 ⋅ 21

= 7560 days

If 60 more men join, the total no. of men is

= 360 + 60

= 420

420 men have provisions for

= 7560 / 420

= 18 days

If 60 more men join, provision will last for 18 days.

**Problem 9 : **

A man has enough money to buy 12 lb of apples at $1.50 per lb. How much can he buy, if the price is increased by $0.30 per lb ?

**Solution : **

**This is a situation of inverse variation.**

**Because, **

**more price -----> less pounds of apples **

Cost of 12 lb of apples at $1.50 per pound is

= 12 ⋅ 1.50

= $18

So, the person has $18

If the price is increased is increased by $0.30 per lb, then the new price per lb is

= $1.80

No. of pounds of apples he can buy with $18 is

= 18 / 1.80

= 10

If the price is increased by $0.30 per lb, the person can buy 10 lb of apples.

**Problem 10 :**

A man can type 9 pages of a book everyday and completes it in 50 days. How many days will he take to complete it, if he types 15 pages everyday ?

**Solution : **

**This is a situation of inverse variation.**

**Because, **

**more pages per day-----> less days to complete the book **

9 pages per day ------> 50 days

1 page per day --------> 9 ⋅ 50 = 450 days

15 pages per day ------> 450 / 15 = 30 days

The man will complete the book in 30 days, if he types 15 pages per day.

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