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In these lessons, we will learn how to solve direct proportions (variations) and inverse proportions (inverse variations) problems. (Note: Some texts may refer to inverse proportions/variations as indirect proportions/variations.)

Related Topics:

Direct Variations

Proportion Word Problems

More Algebra Lessons

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.

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?

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

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.

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.

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