Direct and inverse proportion questions :
Here we are going to see example problems of direct and inverse proportion.
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
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 proportionate increase in another quantity, then the quantities are said to be in direct variation.
Change in the two quantities must be in different ways.
That is,
Increase ---------------> Decrease
or
Decrease ---------------> Increase
Example 1 :
Alex takes 15 days to reduce 30 kilograms of his weight by doing 30 minutes exercise per day. If he does exercise for 1 hour 30 minutes per day, how many days will he take to reduce the same weight ?
Solution :
This is a situation of inverse variation.
Because,
more minutes per day----> less days to reduce the weight
And also
1 hour 30 minutes per day = 90 minutes per day
Let "m" be the required number of days.
No. of Days 15 m |
No. of Minutes 30 90 |
Since this is inverse variation, we have to apply the shortcut "straight multiplication"
15 ⋅ 30 = m ⋅ 90
(15 ⋅ 30) / 90 = m
5 = m
Hence, if Alex does exercise for 1 hour 30 minutes per day, it will take 5 days to reduce 30 kilograms of weight.
Let us look into the next problem on "Direct and inverse proportion questions"
Example 2 :
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
No. of Hours 1/7 3 |
Dollars 2/5 m |
Since this is direct variation, we have to apply the shortcut "cross multiplication"
1/7 ⋅ m = 3 ⋅ 2/5
m = 7 ⋅ 6/5
m = 42 / 5
m = 8.4
Hence, Shanel gets $8.4 for 3 hours of work.
Example 3 :
If 5 men can paint a house in 18 hours, how many men will be able to paint it in 10 hours ?
Solution :
This is a situation of inverse variation.
Because,
less hours -----> more men
Let "m" be the required number of men.
No. of Men 5 m |
No. of Hours 18 10 |
Since this is inverse variation, we have to apply the shortcut "straight multiplication"
5 ⋅ 18 = m ⋅ 10
90 / 10 = m
9 = m
Hence, 9 men will be able to paint the house in 10 hours.
Example 4 :
In a fort, 360 men have provisions for 21 days. If 60 more men join them, how long will the provision last ?
Solution :
This is a situation of inversion variation.
Because,
more men -----> provision will last for less days
Given : 360 men -----> 21 days
If 60 more men join, then
the total number of men = 420
Let "m" be the required number of days.
No. of Men 360 420 |
No. of Days 21 m |
Since this is inverse variation, we have to apply the shortcut "straight multiplication"
360 ⋅ 21 = 420 ⋅ m
(360 ⋅ 21) / 420 = m
18 = m
Hence, if 60 more men join, provision will last for 18 days.
Example 5 :
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
Let "m" be the required amount of money
No. of Units 330 270 |
Money 495 m |
Since this is direct variation, we have to apply the shortcut "cross multiplication"
330 ⋅ m = 270 ⋅ 495
m = (270 ⋅ 495) / 330
m = 405
Hence, John has to pay $405 for 270 units.
Example 6 :
A man can type 9 pages of a book everyday and completes it in 50 days. How many days will he take to complete it, if he types 15 pages everyday ?
Solution :
This is a situation of inverse variation.
Because,
more pages per day-----> less days to complete the book
Let "m" be the required number of days
No. of Days 50 m |
No. of Pages 9 15 |
Since this is inverse variation, we have to apply the shortcut "straight multiplication"
50 ⋅ 9 = m ⋅ 15
450 / 15 = m
30 = m
Hence, the man will complete the book in 30 days, if he types 15 pages per day.
After having gone through the stuff given above, we hope that the students would have understood Direct and inverse proportion questions.
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