DIRECT VARIATION & INVERSE VARIATION -
Direct Variation -
Please look at the followingsituations.
You buy more pens ---------------> Costs you more
More no. of students -------------> More no. of teachers
Travel less distance ----------------> Time taken is less
No. of books reduced --------------> Weight of bagis less
Thus we can say, if an increase in one quantity produces a proportionate in crease in an other 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
Decrease --------------> Decrease
or
Inverse Variation -
Please look at the following situations.
More men ----------------> Less days to complete a work
More speed --------------> Less time to cover the distance
More vehicles on the road --------------> Less road space
Less time per day --------------> More days to complete the work
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 a proportionate increase in another quantity, then the quantities are said to be in direct variation.
Change in both the quantities must be same. That is,
Example.1) 75 basket balls cost $ 1143.75. Find the cost of 26 basketballs.
This is a situation of direct variation. Because,
less number of basket balls -------------> cost will be less
Let "m" bethe cost of 26 basket balls.
No.of Basket Balls Cost
75 1143.75
26 m
Since this is direct variation, we have to apply the shortcut "cross multiplication"
75 x m = 26 x 1143.75
26 x 1143.75
m = -------------------------
75
m = $ 396.5
Example.2) 7 men can complete a work in 52days. In how many days will 13 men finish the same work?
Ans.) This is a situation of inverse variation. Because,
more men -----------------> less days
Let"m"be there quired no. of days.
No. of Men No. of Days
7 52
13 m
Since this isinverse variation, we have to apply the shortcut "straight multiplication"
7 x 52 = 13 x m
(7 x 52) / 13 = m
28 = m
So, 13 men can complete the work in 28 days. (Ans.)
Example.3) A book contains120pagesand eachpagehas35 lines. Howmanypageswill thebookcontain if every page has 24 lines per page ?
Solution :
This is a situation of inverse variation. Because,
less lines ----------------> more pages
Let"m"be therequirednumberof pages.
No. of Pages No.ofLines
120 35
m 24
Since this is inverse variation, we have to apply the shortcut "straight multiplication"
120 x 35 = m x 24
(120 x 35) / 24 = m
175 = m
So, if every page has 24 lines per page, the book will contain 175 pages. (Ans.)
Example.4) If 5 men can paint a house in18 hours, how many men will be able to paint it in10 hours?
Solution :
Thisisasituationofinversevariation. Because,
less hours ---------------> more men
Let "m"be the required number of men.
No. of Men No. of Hours
5 18
m 10
Since this is inverse variation, we have to apply the shortcut "straight multiplication"
5 x 18 = m x 10
90 / 10 = m
9 = m
So, 9 men will be able to paint the house in 10 hours. (Ans.)