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Proportions/Variations




In this lesson, we will learn how to solve direct proportions (variations) and indirect proportions (variations) problems.

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Direct Proportions/Variations

Two values x and y are directly proportional to each other when the ratio x : y or is a constant (i.e. always remains the same). This would mean that x and y will either increase together or decrease together by an amount that would not change the ratio.

Knowing that the ratio does not change allows you to form an equation to find the value of an unknown variable, for example:

If two pencils cost $1.50, how many pencils can you buy with $9.00?

The number of pencils is directly proportional to the cost. 

pencils

This video shows how to solve direct proportion questions
Example 1: F is directly proportional to x. When F is 6, x is 4. Find the value of F when x is 5.
Example 2: A is directly proportional to the square of B. When A is 10, B is 2. Find the value of A when B is 3.



How to use direct proportion





Solving Word Problems using Proportions 1
This video shows how to solve word problems by writing a proportion and solving
1. A recipe uses 5 cups of flour for every 2 cups of sugar. If I want to make a recipe using 8 cups of flour, how much sugar do I use?
2. A syrup is made by dissolving 2 cups of sugar in 2/3 cups of boiling water. How many cups of sugar should be used for 2 cups of boiling water?
2. A school buys 8 gallons of juice for 100 kids. how many gallons do they need for 175 kids?



Solving Word Problems using Proportions 2
1. On a map, two cities are 2 5/8 inches apart. If 3/8 inches on the map represents 25 miles, how far apart are the cities (in miles)?
2. Solve for the sides of similar triangles using proportions




 



Inverse Proportions/Variations

Two values x and y are inversely proportional to each other when their product xy is a constant (always remains the same). This means that when x increases y will decrease, and vice versa, by an amount such that xy remains the same.

Knowing that the product does not change also allows you to form an equation to find the value of an unknown variable for example:

It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they work at the same rate?

The number of men is inversely proportional to the time taken to do the job.

hours.

Usually, you will be able to decide from the question whether the values are directly proportional or inversely proportional.

How to solve inverse proportion questions.
This video shows how to solve inverse proportion questions. It goes through a couple of examples and ends with some practice questions
Example 1: A is inversely proportional to B. When A is 10, B is 2. Find the value of A when B is 8
Example 2: F is inversely proportional to the square of x. When A is 20, B is 3. Find the value of F when x is 5.



How to use indirect proportion to work out problems
How to use a more advanced form of indirect proportion where the use of square numbers is involved.



The following video gives more examples of direct proportions / variations and indirect proportions / variations.



Solving Indirect Proportion Math Problems
In math, an indirect proportion is when an increase in one quantity results in a decrease in another quantity. This video will show how to solve an indirect proportion math problem.






 



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