Whenever you have two variables that share in a simple relationship, they can take the form of a direct variation. We can say that one variable varies directly with another variable. This usually happens when one variable increases or decreases and then the related variable that we defined does as well. These lessons and worksheets help your students to understand the concept of direct variation with regard to equations with variables. These are a set of unique worksheets that students can add to their collection of activity sheets. They will help them learn how to solve for variables in different algebraic expressions by using direct variation. Students will also use grids in order to determine the relationships of variables.
Learn how to solve problems like the following: x varies directly with y. If x = 4 when y = 14, find y when x = 2
You will solve these word problems dealing with direct variation between variables. Example: The distance of a train from a station, varies directly with the time, t. If d = 100 miles when t = 2 hours, find d when t = 3.
The problems on this worksheet are much different than worksheet 1. You will find a mix of word and number based problems. Example:
If x varies directly with y and x is 21 when y is 10, find the constant of variation.
Review the steps to solving equations dealing with direct variation between variables. Example: In the following chart, does one variable vary directly with the other.
Practice solving these direct variation problems that are all number based. Example: p varies directly with q". If p = 3 when q = 21, find p when q = 3.
Solve these 10 problems and then score how many answers you got correct. Example:
If x varies directly with y and x is 24 when y is 10, find the constant of variation. This is a great way to see how much work you might or might not need to put in.
Complete these 3 problems, then put your answer in the "My Answer" box for each. This can help you review or introduce this skill to students.
What are Direct Variations in Math?
Direct variations are an integral part of mathematics you regularly encounter when studying. Many people find direct variations to be confusing. If you are one of them, read here to learn what direct variations are and how to solve them.
Direct variations are a category of proportionality. They show how one variable relates varies with another. Direct variation indicates a linear relationship between two variables. Direct variation is also called direct proportionality.
Two variables that increase and decrease by the same amount are directly proportional. If one quantity increases or decreases, the other follows suit.
For example, the height of the wall is directly proportional to the number of bricks. If the number of bricks in the wall increase, so does the height.
Direct variation is formulated using the symbol ‘∝,’ which shows that two values follow direct variation. For example, if two values, x, and y, are directly proportional, they are expressed as y ∝ x.
The two values, y, and x, increase and decrease by the same factor. This factor is constant. This means that it will not change even if the values themselves do. We can denote this factor as the constant 'k.' so the formula y ∝ x becomes y =kx or x = y/k (where the constant becomes 1/k).
Solving Direct Variation Problems
Solving direct variation problems is easy once you understand the formula. Since the constant does not change, you need to figure out the constant, and the values of y or x can be found, given that one of the values is given. To better understand the concept, let’s look at some examples:
Example 1: The number of cookies made varies directly with the flour used. If 12 cookies require 2 cups of flour, how many cookies can you make with 6 cups of flour?
Let y = number of cookies made and x = flour used. Since the two are directly proportional, we can write them as
y = kx
to find k, we use y = 12 and x= 2
12 = k (2)
k = 12/6
k = 2.
Now we used this value of k and the x = 6 to find y,
y = kx
y = 2(6)
y = 12
Twelve cookies can be made using six cups of flour.
Example 2: The number of cookies made varies directly with the flour used. If making 36 cookies requires 3 cups of flour, how much flour would you need to make 72 cookies?
Let y = number of cookies and x = flour used. Since the two are directly proportional, we can write them as,
y = kx
we use y = 36 and x = 3 to find the value of k,
36 = k (3)
k = 12
now, we use the value of k and y = 72 to find x,
y = kx
x = y/k
x = 72/12
x = 6
You would need six cups of flour to make seventy-two cookies.
Direct variations are easy to do once you get the concept. But understanding the concept only gets you halfway to mastering direct variation questions. The other half you will cover after practicing these questions. Practice makes perfect!