Common Mistakes
Learn to identify and avoid common errors with proportions.
Learn
Common Mistakes to Avoid
- Setting up proportions incorrectly: Keep units consistent on each side
- Confusing direct and inverse proportion: Not all relationships are proportional the same way
- Forgetting to check if the relationship passes through origin: True proportions have y = kx form
- Cross-multiplying incorrectly: Remember a/b = c/d means ad = bc
- Misidentifying the constant of proportionality: k = y/x, not x/y
Practice
Question 1: What is wrong with this setup? "3 apples cost $2. How much do 9 apples cost?" Setup: 3/2 = x/9
Answer
The units are mismatched. It should be 3/2 = 9/x or apples/dollars = apples/dollars. Correct: 3/2 = 9/x, so x = $6.
Question 2: Is this relationship proportional? A taxi charges $3 base fee plus $2 per mile.
Answer
No. Because of the $3 base fee, the relationship does not pass through the origin. At 0 miles, cost is $3, not $0. This is a linear relationship but not proportional.
Question 3: Find the error: To solve 4/x = 12/15, a student wrote 4 x 15 = 12x.
Answer
The cross-multiplication is incorrect. It should be 4 x 15 = 12 x x, which gives 60 = 12x, so x = 5.
Question 4: If y is proportional to x and y = 20 when x = 4, what is k?
Answer
k = y/x = 20/4 = 5. A common mistake is calculating x/y = 4/20 = 0.2.
Question 5: Why does "5 workers finish in 10 days, so 10 workers finish in 5 days" represent inverse proportion?
Answer
As workers increase, time decreases. The product stays constant: 5 x 10 = 10 x 5 = 50. This is inverse (not direct) proportion.
Question 6: Table: x = 2, y = 6; x = 4, y = 12; x = 5, y = 16. Is this proportional?
Answer
No. Check ratios: 6/2 = 3, 12/4 = 3, but 16/5 = 3.2. The ratio is not constant, so not proportional.
Question 7: A student says "y = 2x + 1 is proportional because y increases as x increases." Is this correct?
Answer
No. A proportional relationship must be y = kx (passing through origin). The "+1" means when x = 0, y = 1, not 0.
Question 8: What is the unit rate if 15 items cost $45?
Answer
Unit rate = 45/15 = $3 per item. A common error is 15/45 = 0.33 items per dollar.
Question 9: Solve: If 2/3 = x/12, find x.
Answer
Cross multiply: 2 x 12 = 3 x x. 24 = 3x. x = 8.
Question 10: A graph passes through (0, 2) and (3, 8). Is this a proportional relationship?
Answer
No. Proportional relationships must pass through the origin (0, 0). This line has a y-intercept of 2.
Next Steps
- Review your work for these common errors
- Continue to Lesson 6: Unit Checkpoint