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
Change in both the quantities must be same.
That is,
Increase ------> Increase
or
Decrease ------> Decrease
If 'y' is directly proportional to 'x', then
y = kx
where 'k' is the constant of proportionality.
If an increase in one quantity produces a proportionate decrease in another quantity, then two quantities are inversely proportional to each other.
or
If a decrease in one quantity produces proportionate increase in another quantity, then two quantities are inversely proportional to each other.
Changes in the quantities must be opposite.
That is,
Increase ------> Decrease
or
Decrease ------> Increase
If 'y' is inversely proportional to 'x', then
y = k/x
where 'k' is the constant of proportionality.
Problem 1 :
y is directly proportional to x. Given that y = 144 and x = 12. Find the value of y when x = 7.
Solution :
y directly proportional to x.
Then,
y = kx
Substitute y = 144 and x = 12.
144 = 12k
12 = k
Then,
y = 12x
Substitute x = 7.
y = 12(7)
y = 84
Problem 2 :
y is inversely proportional to x. Given that y = 12 and x = 6. Find the value of y when x = 8.
Solution :
y inversely proportional to x.
Then,
y = k / x
Substitute y = 12 and x = 6.
12 = k / 6
12 x 6 = k
72 = k
Then,
y = 72 / x
Substitute x = 8.
y = 72 / 8
y = 9
Problem 3 :
75 basketballs cost $1,143.75. Find the cost of 26 basketballs.
Solution :
This is a situation of direct proportion.
Because,
less number of basket balls -----> cost will be less
Let x be the no. of basket balls and y be the cost.
Since this is direct proportion, we have
y = kx
Substitute x = 75 and y = 1143.75.
1143.75 = 75k
15.25 = k
Then,
y = 15.25x
Substitute x = 26.
y = 15.25(26)
y = 396.50
So, the cost of 2626 basketballs is $396.50.
Problem 4 :
7 men can complete a work in 52 days. In how many days will 13 men finish the same work ?
Solution :
This is a situation of inverse proportion.
Because,
more men -----> less days
Let x be the no. of men and y be the no. of days.
Because this is inverse proportion, we have
y = k / x
Substitute x = 7 and y = 52.
52 = k / 7
364 = k
Then,
y = 364 / x
Substitute x = 13.
y = 364 / 13
y = 28
So, 13 men will finish the same work in 28 days.
Problem 5 :
If David sells 2 gallons of juice for $4, how much money will he get by selling 17 gallons of juice ?
Solution :
This is a situation of direct proportion.
Because,
more gallons of juice -----> more amount of money
Let x be the no. of gallons of juice and y be the cost.
Since this is direct proportion, we have
y = kx
Substitute x = 2 and y = 4.
4 = 2k
2 = k
Therefore, y = 2x
Plug x = 17,
y = 2(17)
y = 34
So, David will earn $34 by selling 17 gallons of juice.
Problem 6 :
A book contains 120 pages and each page has 35 lines . How many pages will the book contain if every page has 24 lines per page ?
Solution :
This is a situation of inverse proportion.
Because,
less lines -----> more pages
Let x be the no. of pages and y be the no. of lines.
Since this is inverse proportion, we have
y = k / x
Substitute x = 120 and y = 35.
35 = k / 120
4200 = k
Then,
y = 4200 / x
Substitute y = 24.
24 = 4200 / x
x = 4200 / 24
x = 175
So, if every page has 24 lines per page, the book will contain 175 pages.
Problem 7 :
The cost of a taxi is $40.50 for 15 miles. Find the cost for 20 miles.
Solution :
This is a situation of direct proportion.
Because,
more miles -----> more cost
Let x be the no. of miles and y be the cost.
Since this is direct proportion, we have
y = kx
Substitute x = 15 and y = 40.50.
40.50 = 15k
2.7 = k
Then,
y = 2.7x
Substitute x = 20.
y = 2.7(20)
y = 54
So, the cost for 20 miles is $54.
Problem 8 :
A truck covers a particular distance in 3 hours with the speed of 60 miles per hour. If the speed is increased by 30 miles per hour, find the time taken by the truck to cover the same distance
Solution :
This is a situation of inverse proportion.
Because,
more speed -----> less time
Let x be the hours and y be the speed.
Since this is inverse proportion, we have
y = k / x
Substitute x = 3 and y = 60.
60 = k / 3
180 = k
Then,
y = 180 / x
If the given speed 60 mph is increased by 30 mph, then the new speed is 90 mph.
So, we have to find x when y = 90.
Substitute y = 90,
90 = 180 / x
x = 180 / 90
x = 2
So, if the speed is increased by 30 mph, time taken by the truck is 2 hours.
Problem 9 :
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 proportion
Because,
more time -----> more earning
Let x be the years and y be the earning.
Since this is direct proportion, we have
y = kx
Substitute x = 2.5 and y = 7500.
7500 = 2.5k
3000 = k
Then,
y = 3000x
Substitute x = 4.
y = 3000(4)
y = 12000
So, the earning for 4 years is $12000.
Problem 10 :
David can complete a work in 6 days working 8 hours per day. If he works 3 hours per day, how many days will he take to complete the work ?
Solution :
This is a situation of inverse proportion.
Because,
less hours per day-----> more days to complete the work
Let x be the days and y be the hours.
Since this is inverse proportion, we have
y = k / x
Substitute x = 6 and y = 8.
8 = k / 6
48 = k
Then,
y = 48 / x
Substitute x = 3.
y = 48 / 3
y = 16
So, David can complete the work in 16 days working 3 hours per day.
Problem 11 :
In 36.5 weeks, Miguel raised $2372.50 for cancer research. How much money will he raise 20 weeks ?
Solution :
This is a situation of direct proportion.
Because,
less number of weeks ----> amount raised will be less
Let x be the weeks and y be the amount of money raised.
Since this is direct proportion, we have
y = kx
Substitute x = 36.5 and y = 2372.50.
2372.50 = 36.5k
2372.50 / 36.5 = k
65 = k
Then,
y = 65x
Substitute x = 20.
y = 65(20)
y = 1300
So, the money raised in 20 weeks is $1300.
Problem 12 :
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 proportion.
Because,
more minutes per day----> less days to reduce the weight
Let x be the minutes and y be the days.
Since this is inverse proportion, we have
y = k / x
Substitute x = 30 and y = 15.
15 = k / 30
450 = k
Then,
y = 450 / x
1 hour 30 minutes = 90 minutes
So, we have to find y when x = 90.
Substitute x = 90.
y = 450 / 90
y = 5
So, if Alex does exercise for 1 hour 30 minutes per day, it will take 5 days to reduce 30 kilograms of weight.
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and Venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6
©All rights reserved. onlinemath4all.com
May 23, 22 01:59 AM
Exponential vs Linear Growth Worksheet
May 23, 22 01:59 AM
Linear vs Exponential Growth - Concept - Examples
May 23, 22 01:34 AM
SAT Math Questions on Exponential vs Linear Growth