UNITARY METHOD DIRECT VARIATION

About "Unitary method direct variation"

On this web page "Unitary method direct variation", we will learn to solve problems on unitary method using direct 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 increase in another quantity, then the quantities are said to be in direct variation.  

or

If a decrease in one quantity produces a roportionate 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 18 units of a product cost $360,

then  price per unit is  = 360 / 18  =  $20

Unitary method direct variation - Practice problems

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

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  =  1143.75 / 75  =  15.25

Then, cost of 26 basket balls  =  15.25 x 26  =  396.50

Hence, the cost of 26 basket balls is $ 396.50

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

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 cost $4

Then, the price pf one gallon  =  4 / 2  =  $2

Price of 17 gallons  =  2 x 17  =  $34

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

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

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

Cost for 15 miles  =  40.50

Cost for one mile  =  40.50 / 15  =  2.70

Then, cost for 20 miles  =  2.70 x 20  =  54

Hence, the cost for 20 miles is $54

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

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  =  $7500

Earning in 1 year  =  7500 / 2.5  =  3000

Then, earning for 4 years  =  4 x 3000  =  12000

Hence, the earning for 4 years is $12000

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

Problem 5 :

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

Then, amount raised in one week  =  2372.5 / 36.5  =  65

Amount raised in 20 weeks  =  65 x 20  =  1300

Hence, the money raised in 20 weeks is $1300

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

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  =  (2/5) / (1/7)

Pay for 1 hour of work  =  (2/5) x (7/1)

Pay for 1 hour of work  =  14 / 5

Then, pay for 1 hour of work  =  $2.8

Pay for 3 hours  =  2.8 x 3  =  8.4

Hence, Shanel gets $8.4 for 3 hours of work

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

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  =  (1/5) / (3/35)

Cost of 1 gallon  =  (1/5) x (35/3)

Cost of 1 gallon  =  7 / 3

Cost of 1 gallon  =  2.3

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

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

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 for 300 miles  =  2.25 x 300

Cost for 300 miles  =  $675   

Hence, the total cost for the trip is $675

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

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  =  495 / 330

Cost of 1 unit  =  1.5

Cost of 270 units  =  1.5 x 270

Cost of 270 units  =  $405

Hence, John has to pay $405 for 270 units 

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

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 x 2.8  =  145.6

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

Comparing unit prices

Problem 1 :

Which is the best deal,

8 dolls cost $120

or 

6 dolls cost $102 ?

Solution : 

To compare the given measures, convert them in to unit rates. 

8 dolls cost $120

Cost of 1 doll  =  120 / 8

Cost of 1 doll  =  $15

6 dolls cost $102

Cost of 1 doll  =  102 / 6

Cost of 1 doll  =  $17

We get the lowest price per doll is $15 in "8 dolls cost $120"

Hence, "8 dolls cost $120" is the best deal

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

Problem 2 :

Which is the best deal,

10 pencils cost $4

or 

6 pencils cost $2.70 ?

Solution : 

To compare the given measures, convert them in to unit rates. 

Cost of 10 pencils  =  $4

Cost of 1 pencil  =  4 / 10

Cost of 1 pencil  =  $0.40

Cost of 6 pencils  =  $2.70

Cost of 1 pencil  =  2.7 / 6

Cost of 1 pencil  =  $0.45

We get the lowest price per pencil $0.40 in "10 pencils cost $4"

Hence, "10 pencils cost $4" is the best deal

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

Problem 3 :

Which is the best deal,

2 liters of milk at $3.80

or 

1.5 liters of milk at $2.70 ?

Solution : 

To compare the given measures, convert them in to unit rates. 

2 liters of milk at $3.80

Cost of 1 liter  =  3.8 / 2

Cost of 1 liter  =  $ 1.90

2 liters of milk at $2.70

Cost of 1 liter  =  2.7 / 1.5

Cost of 1 liter  =  $ 1.8

From the above unit rates, we get the lower price per liter of milk $1.8 in "2 liters cost $2.70"

Hence, "2 liters cost $2.70" is the best deal

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

Problem 4 :

Who is better in earning,

David earns $57.60 in 8 hours

or 

John earns $90 in 12 hours ?

Solution : 

To compare the given measures, convert them in to unit rates. 

David

Earning in 8 hrs  =  $57.60

Earning in 1 hr  =  57.60 / 8

Earning in 1 hr  =  $7.20

John

Earning in 12 hrs = $90

Earning in 1 hr  =  90 / 12

Earning in 1 hr  =  $7.50

From the above unit rates, John earns more than David per hour. 

Hence, John is earning better

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

Problem 5 :

Which is best,

500 grams cheese cost $ 3.25

or 

1.5 kilograms cheese cost $ 9.9

Solution : 

To compare the given measures, convert them in to unit rates in price per kilogram. 

500 grams -----> $3.25

1 kilogram  =  2 x 500 grams 

Price of 1 kg  =  2 x 3.25

Price of 1 kg  =  $6.5

1.5 kilograms -----> $9.9

Price of 1 kg  =  9.9 / 1.5

Price of 1 kg  =  $6.6

From the above unit rates, we get the lower price per kilogram $6.5 in "500 grams cheese cost $ 3.25"

Hence, "500 grams cheese cost $ 3.25" is the best deal

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