DIRECT AND INVERSE VARIATIONS

Direct and inverse variations - Unitary Method consists of two types of variations- DIRECT AND INVERSE. You can solve any question of the unitary method after reading my article-Problems/examples carefully.

Note- Inverse Variation is also called as Indirect Variation.

DIRECT AND INVERSE VARIATIONS - PROBLEMS & EXAMPLES

(PROBLEM / EXAMPLE )

TYPE - 1

Q1.) 15 stamps of equal value cost Rs. 9.00. How many stamps of same value can be bought for Rs 36.00?

Solution. Since the number of stamps and their cost are directly proportional to each other.
Therefore, the ratio between the stamps=the ration between the corresponding costs.
Let the number of stamps=X
Stamps : Stamps = Cost:Cost
15 : X = 9 : 36

=> 9 × x = 15 × 36
=> 9X=15×36
=> 9X= 540
=> x=540/9 = 60 STAMPS.
Hence the number of stamps=60.

Q2.) A machine takes 5 hours to cut 120 tools.How many tools will it cut in 20 hours?

Solution. Number of tools = 120 Time = 5 hours.
Number of tools = ? Time = 20 hours.
Since time and tools are directly proportional to each other.
Therefore the ratio between the hours = ratio between the corresponding tools cut
by machines .
Let number of tools = X
Tools : Tools = Hours : Hours
120 : X = 5 : 20
OR

=> 120 × 20 = 5 × x
=> 2400=5X
=> 2400/5=X
=> 480=X
Thus the number of tools cut by the machine in 20 hours =480.

DIRECT AND INVERSE VARIATIONS

(PROBLEM / EXAMPLE )

TYPE - 2

Q3.) Anupama takes 125 steps in walking a distance of 100metres. What distance would
she covers in 315 steps?

Solution. Let the distance covered = X metres.
STEPS DISTANCE( IN METERS)
125 100
315 X
Hence the number of steps taken and distance covered is directly proportional
to each other.
Hence, the ratio between steps taken = the ratio between the corresponding distance covered
Steps : Steps = Distance : Distance
125 : 315 = 100 : X
OR

By cross-multiply we have
125 × X = 100 × 315
125X= 31500
X=31500/125
X=252 metres.

Q4.) Suneeta types 1080 words in one hour. What is her GWAM(gross words a minute) rate?
Solution: Let her GWAM = X words a minute.
MINUTES WORDS
one hour= 60min 1080
1min X
Because the time and words typed are directly proportional to each other.
Therefore the ratio between minutes = the ratio between corresponding words
TIME-MINUTES: TIME-MINUTES = WORDS: WORDS
60 : 1 = 1080 : x
OR

By cross multiplying we have,
60 × x = 1080 × 1
60X=1080
X=1080/60
X=18
Thus her GWAM = 18 words a minute.

REMEMBER:-

A) Pairs of terms above and below are in the same ratio.
B) The ratio of any two terms in the row is same as the ratio corresponding terms in the other row.

DIRECT AND INVERSE VARIATIONS

(PROBLEM / EXAMPLE )

TYPE - 3

Q5.) Shalu cycles to her school at an average speed of 12 km/h. It takes her 20 minutes to
reach the school. If she wants to reach her school in 15 minutes, what should be
her average speed?

Solution. Let her average speed = X km/h
SPEED( in km/h) Time (in minutes)
12 20
X 15
Here speed and time vary inversely.
therefore this is the question of inverse variation.
Hence,

By cross-multiplication, we have
15 × X = 12 × 20
15X= 240
X=240/15
X=16
Thus her average speed = 16km/h.

Q6.) A shopkeeper has just enough money to buy 52 cycles worth Rs.525.00 each. If
Each cycle cost Rs. 21.00 more, then how many cycles would he be
able to buy with that amount of money?

Solution: Let the required number of cycles be X.
NUMBER OF CYCLES COST OF CYCLE(Rs)
52 525
X 546(new cost of cycles = 525+21=Rs546)
Because number of cycles that can be purchased and cost of cycles
Vary inversely.
Therefore this is the question of inverse variation.
Hence,

546 × X= 525 × 52
546X=27300
X=27300/546
X = 50
Thus the number of cycles that can be purchased =50.

Since this is my second article( First article link: Unitary Method in Maths)of the unitary method under- Direct and inverse variation- problems and examples.

For more problems and examples visit