Basics of Proportion:
A proportion is a statement that two ratios are equal. A ratio is simply a comparison of two quantities.
For example, if we have two quantities A and B, we can express their ratio as A:B or A/B.
To set up a proportion, we compare two ratios and set them equal to each other.
This can be written as follows: A/B = C/D In this equation,
A and B are one ratio, while C and D are another ratio.
The key is that these two ratios are equal, so if we know the value of three of the variables, we can solve for the fourth.
Example 1:
If 2 apples cost $1.50, how much will 5 apples cost?
we can set up a proportion as follows:
2 apples / $1.50 = 5 apples / x
where x is the cost of 5 apples.
To solve for x, we can cross-multiply and simplify:
2 apples * x = 5 apples * $1.50
2x = $7.50
x = $3.75
So 5 apples will cost $3.75.
Example 2:
If a car travels 200 miles in 4 hours, how long will it take to travel 300 miles?
we can set up a proportion as follows:
200 miles / 4 hours = 300 miles / x hours
where x is the time it takes to travel 300 miles.
To solve for x, we can cross-multiply and simplify:
200 miles * x = 4 hours * 300 miles
200x = 1200
x = 6
So it will take 6 hours to travel 300 miles.
Types of Proportion
There are two main types of proportion
direct proportion and inverse proportion
Direct proportion
When one quantity increases, the other quantity also increases in the same proportion.
This can be expressed as: A/B = C/D (where A and C are directly proportional to each other)
For Example
If the price of 3 apples is $1.50, then the price of 6 apples will be $3.00.
The price increases in direct proportion to the number of apples.
Inverse proportion
When one quantity increases, the other quantity decreases in the same proportion.
This can be expressed as: A/B = D/C (where A and D are inversely proportional to each other)
For Example
if it takes 5 workers 10 days to complete a project, then it would take 10 workers 5 days to complete the same project.
The number of workers and the number of days are inversely proportional to each other.
Direct Proportion Question
Question 1:
If a car travels at a constant speed of 60 miles per hour, how long will it take to travel 240 miles?
Solution:
Distance is directly proportional to time.
Let’s call the time t.
distance = speed x time
240 = 60 x t
t = 240/60
t = 4 hours
So,it will take the car 4 hours to travel 240 miles at a speed of 60 miles per hour.
Requested by: Dawood
Question 2:
If 12 apples cost $3, how many apples can be purchased for $6?
Solution:
Number of apples is directly proportional to the amount of money spent.
Let’s call the number of apples a.
cost = price per unit x quantity
3 = p x 12
p = 3/12
p = 0.25
Now we can calculate the number of apples that can be purchased for $6.
cost = price per unit x quantity
6 = 0.25 x a
a = 6/0.25
a = 24
So, 24 apples can be purchased for $6.
Requested by: Faisal
Question 3:
If it takes 10 minutes to fill a tank with water using a certain hose, how long will it take to fill the same tank using a hose that has half the diameter?
Solution:
The time is directly proportional to the volume of water that flows through the hose per unit time.
Let’s call the time t.
volume = area x velocity x time
The area of the hose is proportional to the square of its diameter, so if the diameter is halved, the area is divided by 4.
velocity is constant in both cases, so we can ignore it.
volume = area x t
t1 = 10 (original hose)
t2 = t (new hose)
area2 = 1/4 x area1
volume1 = volume2
area1 x t1 = area2 x t2
area1 x 10 = 1/4 x area1 x t
t = 40
Therefore, it will take 40 minutes to fill the same tank using a hose that has half the diameter.
Requested by: Imran
Question 4:
A company produced 5000 units of a product in 10 days. How many days will it take for the company to produce 8000 units of the same product if the production rate remains constant?
Solution:
The number of days is directly proportional to the number of units produced.
Let’s call the number of days d.
units = rate x time
5000 = rate x 10
rate = 500 units/day
Now we can calculate the number of days it will take to produce 8000 units.
8000 = 500 x d
d = 8000/500
d = 16
Therefore, it will take 16 days for the company to produce 8000 units of the same product if the production rate remains constant.
Requested by: Mahmood
Question 5:
If it takes 5 hours to mow a lawn that is 1 acre in size, how long will it take to mow a lawn that is 2 acres in size?
Solution:
The time is directly proportional to the area to be mowed.
Let’s call the time t.
area = rate x time
1 = rate x 5
rate = 1/5 acres/hour
Now we can calculate the time it will take to mow a lawn that is 2 acres in size.
2 = (1/5) x t
t = 10
Therefore, it will take 10 hours to mow a lawn that is 2 acres in size.
Requested by: Rashid
Question 6:
A car can travel 320 miles on 16 gallons of gasoline. How many miles can the car travel on 10 gallons of gasoline?
Solution:
The distance is directly proportional to the amount of gasoline used.
Let’s call the distance d.
distance = rate x amount of gasoline
rate = distance/amount of gasoline
rate = 320/16
rate = 20 miles/gallon
Now we can calculate the distance the car can travel on 10 gallons of gasoline.
distance = rate x amount of gasoline
distance = 20 x 10
distance = 200
Therefore, the car can travel 200 miles on 10 gallons of gasoline.
Requested by: Saad
Question 7:
A recipe calls for 2 cups of sugar for 20 cookies. How many cups of sugar are needed for 30 cookies?
Solution:
The amount of sugar needed is directly proportional to the number of cookies.
Let’s call the amount of sugar s.
sugar = rate x number of cookies
rate = sugar/number of cookies
rate = 2/20
rate = 0.1 cups/cookie
Now we can calculate the amount of sugar needed for 30 cookies.
sugar = rate x number of cookies
sugar = 0.1 x 30
sugar = 3 cups
Therefore, 3 cups of sugar are needed for 30 cookies.
Requested by: Qasim
Question 8:
A truck travels 500 miles in 10 hours. How many miles will the truck travel in 20 hours at the same speed?
Solution:
The distance is directly proportional to the time taken.
Let’s call the distance d.
distance = rate x time
rate = distance/time
rate = 500/10
rate = 50 miles/hour
Now we can calculate the distance the truck will travel in 20 hours.
distance = rate x time
distance = 50 x 20
distance = 1000 miles
Therefore, the truck will travel 1000 miles in 20 hours at the same speed.
Requested by: Salman
Question 9:
A factory can produce 600 units of a product in 8 hours. How many units of the same product can the factory produce in 12 hours?
Solution:
The number of units produced is directly proportional to the time taken.
Let’s call the number of units produced u.
units = rate x time
rate = units/time
rate = 600/8
rate = 75 units/hour
Now we can calculate the number of units the factory can produce in 12 hours.
units = rate x time
units = 75 x 12
units = 900
Therefore, the factory can produce 900 units of the same product in 12 hours.
Requested by: Ahsan
Question 10:
A truck can carry 5000 pounds of cargo for 100 miles on a full tank of gas. How far can the truck travel with 7500 pounds of cargo on a full tank of gas?
Solution:
The distance is directly proportional to the weight of the cargo.
Let’s call the distance d.
distance = rate x weight of cargo
rate = distance/weight of cargo
rate = 100/5000
rate = 0.02 miles/pound
Now we can calculate the distance the truck can travel with 7500 pounds of cargo.
distance = rate x weight of cargo
distance = 0.02 x 7500
distance = 150
Therefore, the truck can travel 150 miles with 7500 pounds of cargo on a full tank of gas.
Requested by: Ali Raza
Question 11:
A construction crew can build a wall 20 feet long in 4 hours. How long will it take the crew to build a wall that is 40 feet long at the same rate?
Solution:
The time taken is directly proportional to the length of the wall. Let’s call the time taken t.
length of wall = rate x time
rate = length of wall/time
rate = 20/4
rate = 5 feet/hour
Now we can calculate the time taken to build a wall that is 40 feet long.
length of wall = rate x time
40 = 5 x t
t = 8
Therefore, it will take the construction crew 8 hours to build a wall that is 40 feet long at the same rate.
Requested by: Ali Raza
Click the Link Below For Practice Problems
On Direct proportion
Inverse Proportion
Question 1:
If 4 workers can complete a task in 10 days, how many days will it take for 8 workers to complete the same task?
Solution:
The time taken to complete a task is inversely proportional to the number of workers.
Let’s call the time taken t.
time taken = constant/number of workers
constant = time taken x number of workers
constant = 4 x 10
constant = 40
Now we can calculate the time taken for 8 workers to complete the task.
time taken = constant/number of workers
time taken = 40/8
time taken = 5 days
Therefore, it will take 5 days for 8 workers to complete the task.
Requested by: Ahmed
Question 2:
If a car travels 60 miles in 2 hours, how long will it take to travel 120 miles at the same speed?
Solution:
The time taken to travel a distance is inversely proportional to the distance.
Let’s call the time taken t.
time taken = constant/distance
constant = time taken x distance
constant = 2 x 60
constant = 120
Now we can calculate the time taken to travel 120 miles.
time taken = constant/distance
time taken = 120/60
time taken = 2 hours
Therefore, it will take 2 hours to travel 120 miles at the same speed.
Requested by: Ilyas
Question 3:
If 4 men can build a wall in 8 hours, how many hours will it take for 8 men to build the same wall?
Solution:
Number of hours is inversely proportional to the number of men.
Let’s call the number of hours h.
men x hours = constant
4 x 8 = 8 x h
32 = 8h
h = 32/8
h = 4
So, it will take 4 hours for 8 men to build the same wall.
Requested by: Qadir
Question 4:
If 5 people can paint a house in 10 days, how many people are needed to paint the house in 6 days?
Solution:
Number of people is inversely proportional to the number of days. Let’s call the number of people needed p.
people x days = constant
5 x 10 = p x 6
50 = 6p
p = 50/6
p ≈ 8.33
So,approximately 9 people are needed to paint the house in 6 days.
Requested by: Yasir
Question 5:
If it takes 6 workers 8 days to build a house, how many days will it take for 9 workers to build the same house?
Solution:
Number of days is inversely proportional to the number of workers.
Let’s call the number of days d.
workers x days = constant
6 x 8 = 9 x d
48 = 9d
d = 48/9
d ≈ 5.33
Requested by: Mustafa
Question 6:
If 6 workers can dig a ditch in 12 hours, how many workers are needed to dig the same ditch in 8 hours?
Solution:
Number of workers is inversely proportional to the number of hours. Let’s call the number of workers w.
workers x hours = constant
6 x 12 = w x 8
w = 9
Therefore, 9 workers are needed to dig the same ditch in 8 hours.
Requested by: Omar
Question 7:
If 6 machines can produce 1200 units in 10 hours, how long will it take for 3 machines to produce 600 units?
Solution:
The time taken to produce units is inversely proportional to the number of machines used.
Let’s call the time taken t.
time taken = constant/number of machines
constant = time taken x number of machines
constant = 6 x 10 x 1200
constant = 72000
Now we can calculate the time taken for 3 machines to produce 600 units.
time taken = constant/(number of machines x number of units)
time taken = 72000/(3 x 600)
time taken = 40 hours
Therefore, it will take 40 hours for 3 machines to produce 600 units.
Requested by: Raza
Question 8:
If a leak can be fixed by 2 workers in 8 hours, how long will it take for 4 workers to fix the leak?
Solution:
The time taken to fix the leak is inversely proportional to the number of workers.
Let’s call the time taken t.
time taken = constant/number of workers
constant = time taken x number of workers
constant = 2 x 8
constant = 16
Now we can calculate the time taken for 4 workers to fix the leak.
time taken = constant/number of workers
time taken = 16/4
time taken = 4 hours
Therefore, it will take 4 hours for 4 workers to fix the leak.
Requested by: Aakash
Question 9:
If a truck can carry 12 tons of goods for 240 miles on 6 gallons of fuel, how many gallons of fuel will it need to carry 24 tons of goods for 480 miles at the same rate?
Solution:
The fuel needed is inversely proportional to the amount of goods carried and the distance traveled.
Let’s call the fuel needed f.
fuel needed = constant/(amount of goods x distance)
constant = fuel needed x amount of goods x distance
constant = 6 x 12 x 240
constant = 17280
Now we can calculate the fuel needed to carry 24 tons of goods for 480 miles.
fuel needed = constant/(amount of goods x distance)
fuel needed = 17280/(24 x 480)
fuel needed = 3.58 gallons
Therefore, it will need 3.58 gallons of fuel to carry 24 tons of goods for 480 miles at the same rate.
Requested by: Suleiman
Question 10:
If a car can travel 100 miles on 4 gallons of fuel, how far can it travel on 6 gallons of fuel?
Solution:
The distance traveled is directly proportional to the amount of fuel used.
Let’s call the distance traveled d.
distance traveled = constant x fuel used
constant = distance traveled/fuel used
constant = 100/4
constant = 25
Now we can calculate the distance the car can travel on 6 gallons of fuel.
distance traveled = constant x fuel used
distance traveled = 25 x 6
distance traveled = 150 miles
Therefore, the car can travel 150 miles on 6 gallons of fuel.
Requested by: Yasir
Question 11:
If 12 men can build a wall in 24 days, how many days will it take for 6 men to build the same wall?
Solution:
The time taken to build the wall is inversely proportional to the number of men working on it.
Let’s call the time taken t.
time taken = constant/number of men
constant = time taken x number of men
constant = 12 x 24
constant = 288
Now we can calculate the time taken for 6 men to build the wall.
time taken = constant/number of men
time taken = 288/6
time taken = 48 days
Therefore, it will take 48 days for 6 men to build the same wall.
Requested by: Kantesh
Question 12:
If a machine can fill 40 bottles in 5 minutes, how many bottles can it fill in 10 minutes?
Solution:
The number of bottles filled is directly proportional to the time taken.
Let’s call the number of bottles filled b.
bottles filled = constant x time taken
constant = bottles filled/time taken
constant = 40/5
constant = 8
Now we can calculate the number of bottles the machine can fill in 10 minutes.
bottles filled = constant x time taken
bottles filled = 8 x 10
bottles filled = 80
Therefore, the machine can fill 80 bottles in 10 minutes.
Requested by: Ishfaque
Question 13:
If it takes 15 minutes to fill a swimming pool using a single pipe, how long will it take to fill the same pool using 3 pipes of equal capacity?
Solution:
The time taken to fill the pool is inversely proportional to the number of pipes used.
Let’s call the time taken t.
time taken = constant/number of pipes
constant = time taken x number of pipes
constant = 15 x 1
constant = 15
Now we can calculate the time taken to fill the pool using 3 pipes of equal capacity.
time taken = constant/number of pipes
time taken = 15/3
time taken = 5 minutes
Therefore, it will take 5 minutes to fill the same pool using 3 pipes of equal capacity.
Requested by: Ali Nawaz