**On this web page "Unitary method inverse variation", we will learn to solve problems on unitary method using inverse variation. **

**First let us come to know what is direct variation.**

**What happen 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 direct variation. **

**or**

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

**Change in the two quantities must be in different ways. **

**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 3 men can complete a work in 18 days, **

**then one man can complete in = 3 x 18 = 54 days**

To have better understanding on unitary method inverse variation, let us look at some practice problems on unitary method

**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

Then, one can complete in = 7 x 52 = 364 days

13 men can complete in = 364 / 13 = 28 days

**Hence, 13 men can complete the work in 28 days**

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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 x 120 = 4200 pages

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

**Hence, ****if every page has 24 lines per page, the book will contain 175 pages**

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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**

**Then, Distance = Time x Speed **

**Distance = 3 x 60 = 180 miles**

**If the given speed 60 mph is increased by 30 mph, **

**then the new speed = 90 mph **

**Then, Time = Distance / Speed**

Time = 180 / 90 = 2 hous

**Hence, if the speed is increased by 30 mph, time taken by the truck is 2 hours. **

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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 x 6 = 48 days

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

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

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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 x 15 = 450 minutes

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

1 hour 30 minutes = 90 minutes

Then, No. of days required = 450 / 90 = 5 days

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

L**et us look at the next problem on ****"**Unitary method inverse variation"

**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 x 10 = $8

For more than 10 units, cost per unit = 90% of 8

Cost per unit = 0.9 x 8 = 7.2

Then, for a sale of 50 units, cost per unit = $7.20

Total cost of 50 units = 50 x 7.20 = 360

**Hence, the total cost of 50 units is $360.**

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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

In 1 hour, the house will be painted by = 18 x 5 = 90 men

In 10 hours, the house can be painted by = 90 / 10 = 9 men

**Hence, 9 men will be able to paint the house in 10 hours**

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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**

360 men have provisions for 21 days

1 man has provisions for = 360 x 21 = 7560 days

If 60 more men join, the total no. of men = 360 + 60 = 420

420 men have provisions for = 7560 / 420 = 18 days

**Hence, i****f 60 more men join, provision will last for 18 days**

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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 = 12 x 1.50 = $18

So, the per son has $18

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

then the new price per lb = $1.80

No. of pounds of apples he can buy with $18 = 18 / 1.80 = 10

**Hence, ****if the price is increased by $0.30 per lb, the person can buy 10 pounds apples. **

**Let us look at the next problem on ****"**Unitary method inverse variation"

**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 x 50 = 450 days

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

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

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