UNITARY METHOD DIRECT VARIATION

Unitary Method :

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

Direct Variation :

If an increase in one quantity produces a proportionate  increase in another quantity, then the two quantities are directly proportional to each other

or

If a decrease in one quantity produces a proportionate  decrease in another quantity, then the two quantities are directly proportional to each other

Changes in both the quantities must be same. 

That is, 

Increase ------> Increase 

or

Decrease ------> Decrease 

Solved Problems

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

So, 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

So, 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

So, 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

So, 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

So, 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

So, 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

So, 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   

So, 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

So, 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

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

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