Understanding Direct Variation
Direct variation is a fundamental concept in mathematics, particularly in algebra. It describes a relationship between two variables, where one variable is a constant multiple of the other. In this post, we will delve into five direct variation examples to help you master the concept.
Example 1: Simple Direct Variation
The cost of buying apples is directly proportional to the number of apples purchased. If 3 apples cost $5, how much will 12 apples cost?
Let’s denote the cost of apples as C and the number of apples as A. We can write the direct variation equation as:
C = kA
where k is the constant of variation. In this case, k = 5⁄3, since 3 apples cost $5.
To find the cost of 12 apples, we can plug in A = 12 into the equation:
C = (5⁄3) × 12 C = $20
Therefore, 12 apples will cost $20.
🤔 Note: The constant of variation (k) is the ratio of the output to the input.
Example 2: Real-World Application
A car rental company charges a base fee of 40 plus an additional 0.25 per mile driven. If a customer drives 120 miles, how much will they be charged in total?
Let’s denote the total cost as C and the number of miles driven as M. We can write the direct variation equation as:
C = 40 + 0.25M
To find the total cost, we can plug in M = 120 into the equation:
C = 40 + 0.25 × 120 C = 40 + 30 C = $70
Therefore, the customer will be charged $70 in total.
Example 3: Direct Variation with Decimals
A bakery sells a total of 250 loaves of bread per day. If they make a profit of $0.80 per loaf, how much profit do they make in a day?
Let’s denote the total profit as P and the number of loaves sold as L. We can write the direct variation equation as:
P = 0.80L
To find the total profit, we can plug in L = 250 into the equation:
P = 0.80 × 250 P = $200
Therefore, the bakery makes a profit of $200 per day.
Example 4: Direct Variation with Fractions
A group of friends want to share some candy equally. If they have 3⁄4 of a bag of candy and there are 12 friends, how much candy will each friend get?
Let’s denote the amount of candy each friend gets as C and the total number of friends as F. We can write the direct variation equation as:
C = (3⁄4) / F
To find the amount of candy each friend gets, we can plug in F = 12 into the equation:
C = (3⁄4) / 12 C = (3⁄4) × (1⁄12) C = 1⁄16
Therefore, each friend will get 1⁄16 of the bag of candy.
Example 5: Direct Variation with Variables
A company produces x units of a product per day, and the cost of production is directly proportional to the number of units produced. If the cost of production is $500 per day when 50 units are produced, how much will it cost to produce 100 units per day?
Let’s denote the cost of production as C and the number of units produced as x. We can write the direct variation equation as:
C = kx
where k is the constant of variation. In this case, k = 500⁄50 = 10.
To find the cost of production for 100 units, we can plug in x = 100 into the equation:
C = 10 × 100 C = $1000
Therefore, it will cost $1000 to produce 100 units per day.
In conclusion, direct variation is a fundamental concept in mathematics that describes a relationship between two variables. By understanding how to identify and apply direct variation, you can solve a wide range of problems in various fields. With these five examples, you should now have a solid grasp of the concept and be able to apply it to different scenarios.
What is direct variation?
+Direct variation is a relationship between two variables, where one variable is a constant multiple of the other.
How do I identify direct variation?
+You can identify direct variation by looking for a constant ratio between the input and output variables.
What is the constant of variation?
+The constant of variation (k) is the ratio of the output to the input variables.
