Direct Variation - Definition, Examples, Quiz, FAQ, Trivia

What is Direct Variation?

Direct variation describes a simple relationship between two quantities: when one quantity increases, the other increases at a constant rate. Similarly, when one decreases, the other decreases at that same constant rate.

For example, think about buying apples at the grocery store. If each apple costs 50 cents, then:

• 1 apple costs 50 cents
• 2 apples cost 100 cents
• 3 apples cost 150 cents

The Direct Variation Formula

We can express direct variation with a simple formula:

The Direct Variation Graph

The graph of a direct variation relationship is always a straight line that passes through the origin (0,0). The slope of this line is equal to k, the constant of proportionality.

Characteristics of the graph:

• Always starts at (0,0) - if x=0, then y=0
• Straight line (never curves)
• Slopes upward (positive k) or downward (negative k)
• The steeper the slope, the larger the constant k

Solving Direct Variation Problems

Let's learn how to solve direct variation problems step by step:

Example Problem: If y varies directly with x, and y = 15 when x = 3, what is y when x = 7?

Step 1: Write the direct variation formula: y = kx

Step 2: Plug in the values we know to find k: 15 = k × 3

Step 3: Solve for k: k = 15 ÷ 3 = 5

Step 4: Now use k to find y when x = 7: y = 5 × 7 = 35

So when x = 7, y = 35.

Another Example: The distance a car travels varies directly with time. In 2 hours, it travels 120 miles. How far will it travel in 5 hours?

Solution:

• Distance = k × Time
• 120 = k × 2 → k = 60 miles per hour
• Distance = 60 × 5 = 300 miles

Real Life Examples

Direct variation is all around us! Here are common examples:

1. Wages: Money earned = hourly rate × hours worked
If you earn $10 per hour, your earnings vary directly with hours worked.

2. Travel: Distance = speed × time
At constant speed, distance varies directly with time.

3. Recipes: Ingredients = servings × amount per serving
If a cookie recipe needs 2 cups of flour for 12 cookies, then for 24 cookies you need 4 cups.

4. Shopping: Total cost = price per item × number of items
The more items you buy, the higher the total cost.

5. Geometry: Circumference = π × diameter
For any circle, circumference varies directly with diameter.

Practice Quiz

Test your understanding with these 5 questions. Choose the correct answer for each.

Frequently Asked Questions

Here are answers to common questions about direct variation:

Math Trivia

Discover interesting facts about proportional relationships: