Skip to content# Direct and Inverse Proportion Graphs, Formula with Worksheet

## What is Direct and Inverse Proportion?

## Formula for Direct and Inverse Proportions

## Steps to Solve Questions with Direct Proportion

**Solution:**

## Steps to Solve Questions with Inverse Proportion

**Solution:**

## Direct and Inverse Proportion Graphs

## Inverse Proportion Graphs

## Conclusion

## Worksheet on Direct and Inverse Proportion

**Direct and Inverse Proportion**

- ‘Direct & inverse proportion’ are the two terms about the relationship between two variables.
- These ideas are the foundation of mathematics and they are often incorporated in everyday mathematical applications.

**What is Direct and Inverse Proportion?****Formula for Direct and Inverse Proportions****Direct and Inverse Proportion Graphs**

Here is one more link to practice a few extra questions: Maths Genie Direct and Inverse Proportion Questions

**Direct Proportion:**

- A direct proportion is a relationship between two variables that either increase or decrease at the same time and in the same numerical value.
- Thus, with the elevation of one of the variables, the second variable rises and the fall of one variable results decline of the other.

**Inverse Proportion:**

- Inverse proportion is a type of relationship between two variables whereby both variables change in parallel in the opposite direction, the more one of the variables increases the more the other one decreases.
- So, to say differently, the increase of one variable results in the decrease of the other and vice versa.

**For Direct Proportion,**

- The equation is y=kx, where y represents the dependent variable, x is the independent variable and k is the rate of proportionality.

**For Inverse Proportion,**

- The equation is y = k/x, and it has y dependent and x independent variables while k is the constant of proportionality.

**Step #1:**Identify the two variables and demonstrate if they have a direct proportional correlation.**Step #2:**The formula for the constant of proportionality defining a linear function as y = kx is used to calculate the value of the constant (k).**Step #3:**As x and y change their value and substitute them in the formula then the value of k is calculated.**Step #4:**After you have discovered the value of k, use it to derive x or y, if there is any unknown value in x or y.

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**Solved Example: **

**Question 1: If a box of 10 pencils is $2, what is the price of a box of 20 pencils?**

Solution:

**Step #1:**These are two variables, which are the pencils’ count and price.**Step #2:**We are aware of the direct proportionality, which ensures that we can use the formula y = kx.

**Step #3:**Substituting the values we get

**2 = 10k, **

which gives

**k = 0.2**

**Step #4:**Through the value of k, we can find the price for 20 pencils. Thus.

**y = 0.2 x 20 **

**= $4.**

**Question 2: A force is directly proportional to the square of the distance between two objects. If a force of 12 N is required to move an object 4 meters away from another object, how much force is required to move the object 8 meters away?**

Solution:

**Step #1:****Define variables and relationships**

Let F be the force required to move the object.

Let d be the distance between the objects.

The force is directly proportional to the square of the distance, so **F ∝ d ^{2}.**

**Step #2:****Express the proportional relationship with a constant**

Write the equation:

**F = k x d ^{2}, **

where k is the proportionality constant.

**Step #3:****Use the given values to find the constant (k)**

Given force (12 N) and distance (4 meters):

**12 N = k x 4 ^{2}.**

Solve for k:

**k = 12 / 16 = 0.75.**

**Step #4:****Substitute the constant into the proportional relationship**

The equation now becomes:

**F = 0.75 x d ^{2}.**

**Step #5:****Find the force required for the new distance (8 meters)**

Substitute the new distance (8 meters) into the equation:

**F = 0.75 x 8 ^{2}.**

Calculate:

**F = 0.75 x 64 **

**= 48 N.**

Therefore, a force of 48 N is required to move the object 8 meters away.

**Practice Questions**

Question 1: If a car travels 120 miles on 6 gallons of gas, how far can it travel on 10 gallons, assuming the relationship between distance and fuel consumption is directly proportional?

Answer :**Step #1:** Identify the two variables involved - distance and gallons of gas.

**Step #2:** Recognize the direct proportionality, represented by the formula y = kx.

**Step #3:** Apply the given values to the formula:

**120/6 = k**

**Step #4:** Solve for k:

**K = 20**

**Step #5:** Use the determined k value to find the distance for 10 gallons:

**20 x 10 = 200 miles.**

Question 2: If a machine produces 30 units in 5 hours, how many units can it produce in 8 hours, assuming a direct proportionality between production and time?

Answer :**Solution:**

**Step #1:** Identify the variables – units produced and time.

**Step #2:** Recognize the direct proportionality, represented by the formula y = kx.

**Step #3:** Apply the given values to the formula:

**30/5 = k**

**Step #4:** Solve for k:

**K = 6**

**Step #5:** Use the determined k value to find the units produced in 8 hours:

**6 x 8 = 48 units**

**Step #1:**Identify the two variables and see if they have the nature of an inverse proportion.**Step #2:**Employing the formula y = k/x to obtain the constant of proportionality (k).**Step #3:**Substitute x and y for the same values in the formula and solve for the k.**Step #4:**When you have derived the value of k, you can utilize it to find out then any of the variables x or y.

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**Solved Example**

**Question 1: ****If it takes 6 hours for 2 workers to complete a task, how long would it ****take for 3 workers to complete the same task?**

Solution:

**Step #1:**The two variables are the number of workers and the time taken to complete the task.

**Step #2:**We know that they are inversely proportional, so we can use the formula

**y = k/x.**

**Step #3:**Substituting the values we get

**6 = k/2, **

which gives

**k = 12.**

**Step #4:**Using this value of k, we can find the time taken by 3 workers as

**y = 12/3 **

**= 4 hours.**

**Question 2: The time taken to complete a task is inversely proportional to the square of the number of workers. If a task takes 16 hours to complete with 4 workers, how long will it take to complete the task with 8 workers?**

Solution:

**Step #1:****Define variables and relationships**

Let T be the time taken to complete the task.

Let w be the number of workers.

The time taken is inversely proportional to the square of the number of workers, so T ∝ 1 / w^{2}.

**Step #2:****Express the inverse proportional relationship with a constant**

Write the equation:

**T = k / w ^{2}, **

where k is the proportionality constant.

**Step #3:****Use the given values to find the constant (k)**

Given time (16 hours) and number of workers (4):

**16 hours ∝ 1 / 4 ^{2}.**

Solve for k:

**16 hours = k / 16, **

**k = 16 x 16 **

**= 256.**

**Step #4:****Substitute the constant into the inverse proportional relationship**

The equation now becomes:

** T = 256 / w ^{2}.**

**Step #5:****Find the time required for the new number of workers (8 workers)**

Substitute the new number of workers (8) into the equation:

**T = 256 / 8 ^{2}.**

Calculate:

**T = 256 / 64 **

**= 4 hours.**

Therefore, it will take 4 hours to complete the task with 8 workers.

**Practice Questions**

Question 1: If a water tank can be filled by 4 pipes in 2 hours, how long will it take for the tank to be filled if only 2 pipes are used, assuming an inverse proportionality between the number of pipes and the time required?

Answer :**Step #1:** Identify the variables - number of pipes and time.

**Step #2:** Recognize the inverse proportionality, represented by the formula y = k/x

**Step #3:** Apply the given values to the formula:

**2 = k/4**

**Step #4:** Solve for k:

**K = 8**

**Step #5:** Use the determined k value to find the time for 2 pipes:

**8/2 = 4 hours.**

Question 2: If a garden can be weeded by 5 gardeners in 4 hours, how many hours will it take for 8 gardeners to complete the same weeding task, assuming an inverse proportionality between the number of gardeners and the time required?

Answer :**Solution:**

**Step #1:** Identify the variables - number of gardeners and time.

**Step #2:** Recognize the inverse proportionality, represented by the formula y = k/x

**Step #3:** Apply the given values to the formula:

**4 = k/5**

**Step #4:** Solve for k:

**K = 20**

**Step #5:** Use the determined k value to find the time for 8 gardeners:

**20/8 = 2.5 hours.**

**Direct Proportion:**

- On a direct proportion graph, a straight line crosses the origin. It means that if a certain variable is going up, the other one is going up proportionally.
- The curve of the directly proportional graph is the constant of proportionality. The larger the slope, the more the constant of proportion.

**Inverse Proportion:**

- A hyperbola is demonstrated by the association between the x and y-axis, which approaches and never touches the x or y-axis.
- This implies that with the rise of one variable the other variable will go down proportionally as well.
- The more graph is closer to either the x or y-axis, the bigger the constant of proportionality is.

**In direct proportion graphs, the y-intercept is always zero, while in inverse proportion graphs, the x and y intercepts are never zero.**

Such graphs us as a direct and inverse proportional tool to model real-world relationships.

Preferable, they can present a relationship between distance travelled and time taken during constant-speed driving along the travel.

Also, these diagrams are suitable for demonstrating the proportion between the amount of ingredients and the number of servings in cooking recipes.

**Question 1:** Suppose x and y are in inverse proportion. If y = 12 then x = 4, find the value of y when x = 8..

**Question 2:** If two cardboard boxes occupy 500 cubic centimetres of space, then how much space is required to keep 200 such boxes?

**Question 3:** If 35 men can finish a piece of work in 8 days, in how many days 20 men complete the same work?

**Question 4:** If 270 kg of corn would feed 42 horses for 21 days, for how many days would 360 kg of corn feed 21 horses?

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