Proportion in College Algebra- Practice Problems
What Is a Proportion in College Algebra?
A proportion is simply an equation showing that two ratios are equal. If you've ever said "this is to that as that is to something else," you've worked with proportions.
The formal definition: if a/b = c/d, then a, b, c, and d form a proportion. This is the foundation for solving countless algebra problems.
Most proportion problems in college algebra ask you to find a missing value. You solve them using cross-multiplication—multiply diagonally and set the products equal.
The Cross-Multiplication Method
Given a/b = c/d, cross-multiplication gives you:
ad = bc
That's it. Once you cross-multiply, you have a simple equation to solve. No fancy steps, no complicated logic.
Why Cross-Multiplication Works
You're multiplying both sides of the equation by bd. This eliminates the denominators because bd/b = d and bd/d = b. What you're left with is ad = bc.
Direct vs. Indirect Proportions
You need to know the difference before you start solving problems.
Direct Proportion
When one value increases, the other increases at the same rate. If x doubles, y doubles. The ratio stays constant.
Formula: y = kx, where k is the constant of proportionality.
Indirect (Inverse) Proportion
When one value increases, the other decreases. If x doubles, y is cut in half.
Formula: y = k/x, or xy = k.
Proportion Types Comparison
| Type | Relationship | Formula | Example |
|---|---|---|---|
| Direct | Same direction | y = kx | Speed vs. Distance |
| Inverse | Opposite direction | xy = k | Speed vs. Time |
| Joint | Multiple variables | z = kxy | Work problems |
| Compound | Multiple inversions | Varies | Combined rates |
Practice Problems with Solutions
Problem 1: Finding the Missing Value
Solve for x: 3/4 = x/16
Solution:
Cross-multiply: 3 × 16 = 4 × x
48 = 4x
x = 12
Quick check: 3/4 = 0.75. 12/16 = 0.75. ✓
Problem 2: Word Problem - Recipe Scaling
A recipe needs 2 cups of flour to make 24 cookies. How much flour do you need for 60 cookies?
Solution:
Set up the proportion: 2/24 = x/60
Cross-multiply: 2 × 60 = 24 × x
120 = 24x
x = 5 cups of flour
Problem 3: Inverse Proportion
If 6 workers can complete a task in 15 days, how long will 10 workers take?
Solution:
This is inverse—more workers means less time.
6 × 15 = 10 × x
90 = 10x
x = 9 days
Problem 4: Solving with Variables on Both Sides
Solve for x: (x + 2)/5 = (x - 1)/3
Solution:
Cross-multiply: 3(x + 2) = 5(x - 1)
3x + 6 = 5x - 5
6 + 5 = 5x - 3x
11 = 2x
x = 5.5
Problem 5: Complex Fraction
Solve for x: (2x + 1)/(x - 3) = 5/2
Solution:
Cross-multiply: 2(2x + 1) = 5(x - 3)
4x + 2 = 5x - 15
2 + 15 = 5x - 4x
17 = x
Check: (34 + 1)/(17 - 3) = 35/14 = 5/2 ✓
Getting Started: Step-by-Step Method
Here's how to tackle any proportion problem:
- Identify what you're solving for. Look for "find x" or "how much" language.
- Set up the proportion correctly. Keep same units in same positions. If 5 apples cost $3, then x apples cost $y—keep apples on top, money on bottom.
- Cross-multiply. Multiply diagonally across the equals sign.
- Solve the resulting equation. Use basic algebra to isolate the variable.
- Check your answer. Plug it back into the original proportion and verify.
Common Mistakes to Avoid
- Setting up the ratio wrong—make sure units match across the fraction
- Forgetting to check if it's direct or inverse proportion
- Arithmetic errors when cross-multiplying
- Not simplifying at the end when required
- Skipping the verification step
Tips for Solving Faster
Look for opportunities to cancel before multiplying. If you have 3/6 = x/12, you can simplify 3/6 to 1/2 first, making it 1/2 = x/12, so x = 6.
When numbers are large, reduce fractions early. It makes the math much cleaner.
When Proportions Get Tricky
Some problems involve multiple proportions chained together. Solve one proportion at a time, using the result in the next equation.
Other problems hide the proportion in word form. "A is to B as C is to D" translates directly to A/B = C/D.