**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 â‰ˆ 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 â‰ˆ 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**

**Click the Link Below For Practice Problems**

**On Inverse proportion**

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