**Unitary Method Direct Variation :**

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

First let us come to know what is direct variation.

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

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

or

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

Change in both the quantities must be same.

That is,

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

or

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

**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 30 pens cost $45,

then the cost of one pen is = 45 / 30 = $1.50

Let us look at some practice problems on "Unitary method direct variation"

**Problem 1 :**

75 basketballs cost $1,143.75. Find the cost of 26 basketballs

**Solution :**

This is a situation of direct variation.

Because,

less number of basket balls -----> cost will be less

**Given :** 75 basketballs cost $1,143.75

Cost of one basket ball is

= 1143.75 / 75

= $15.25

Then, the cost of 26 basket balls is

= 26 ⋅ 15.25

= 396.50

Hence, the cost of 26 basket balls is $ 396.50

**Problem 2 :**

If David sells 2 gallons of juice for $4, how much money will he earn by selling 17 gallons of juice ?

**Solution : **

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

**Because, **

**more gallons of juice -----> amount received will be more**

**Given :** 2 gallons of juice cost $4

Cost of one gallon of juice is

= 4 / 2

= $2

Cost of 17 gallons of juice is

= 17 ⋅ 2

= $34

Hence, David will earn $34 by selling 17 gallons of juice.

**Problem 3 :**

The cost of a taxi is $40.50 for 15 miles. Find the cost for 20 miles.

**Solution : **

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

**Because, **

**more miles -----> cost will be more**

**Given :** Cost for 15 miles is $40.50

Cost for one mile is

= 40.50 / 15

= $2.70

Then, the cost for 20 miles is

= 20 ⋅ 2.70

= 54

Hence, the cost for 20 miles is $54.

**Problem 4 : **

In a business, if A can earn $7500 in 2.5 years, At the same rate, find his earning for 4 years.

**Solution : **

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

**Because, **

**more time -----> more earning **

**Given :** Earning for 2.5 years is $7500

Earning for 1 year is

= 7500 / 2.5

= $3000

Then, earning for 4 years is

= 4 ⋅ 3000

= $12000

Hence, the earning for 4 years is $12000

**Problem 5 :**

In 36.5 weeks, Miguel raised $2,372.50 for cancer research. How much money will he raise in 20 weeks ?

**Solution :**

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

**Because, **

**less number of weeks ----> amount raised will be less**

**Given :** Miguel raised $2,372.50 in 36.5 weeks

Amount raised in one week is

= 2372.5 / 36.5

= $65

Amount raised in 20 weeks is

= 65 ⋅ 20

= $1300

Hence, the money raised in 20 weeks is $1300

**Problem 6 : **

Shanel gets 2/5 of a dollar for 1/7 hour of work.How much money does she get for 3 hours ?

**Solution : **

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

**Because, **

**more hours -----> more earning **

Pay for 1/7 hour of work = $2/5

Pay for 1 hour of work is

= (2/5) / (1/7)

= (2/5) ⋅ (7/1)

= 14 / 5

= $2.8

Then, pay for 3 hours of work is

= 2.8 ⋅ 3

= 8.4

Hence, Shanel gets $8.4 for 3 hours of work

**Problem 7 : **

If 3/35 of a gallon of gasoline costs 1/5 of a dollar, find the price of 1 gallon of gasoline.

**Solution : **

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

**Because, **

**more gasoline -----> more cost **

Cost of 3/35 of a gallon = $1/5

Cost of 1 gallon is

= (1/5) / (3/35)

= (1/5) ⋅ (35/3)

= 7 / 3

= 2.3

Hence, the cost of 1 gallon of gasoline is $ 2.30

**Problem 8 : **

Declan would like to hire a call taxi for 300 miles trip. If the cost of the taxi is $2.25 per mile, what is the total cost for his trip ?

**Solution : **

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

**Because, **

**more miles -----> more cost **

Cost for one mile = $2.25

Cost of 300 miles is

= 2.25 ⋅ 300

= $675

Hence, the total cost for the trip is $675.

**Problem 9 : **

John ordered 330 units of a product for $495. Then he reduced his order to 270 units. How much money does John have to pay for 270 units ?

**Solution : **

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

**Because, **

**less units -----> less cost **

Cost of 330 units = $495

Cost of 1 unit is

= 495 / 330

= 1.5

Cost of 270 units is

= 1.5 ⋅ 270

= $405

Hence, John has to pay $405 for 270 units.

**Problem 10 :**

My David earns $416 in 8 hours. How much does earn in 2.8 hours ?

**Solution : **

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

**Because, **

**less hours -----> less earning **

Given : Earning in 8 hours = $ 416

Earning in 1 hour = $ 52

Earning in 2.8 hours = 52 ⋅ 2.8 = 145.6

Hence, Mr. David will earn $145.6 in 2.8 hours.

After having gone through the stuff given above, we hope that the students would have understood "Unitary method direct variation".

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