Common Mistakes
Learn
Understanding common mistakes in quantitative reasoning helps you avoid them on tests and in real-world applications. This lesson identifies frequent errors and shows how to prevent them.
Category 1: Percentage Errors
Mistake: Confusing percent increase with percent of a number.
Example: "The price increased by 20%" is NOT the same as "20% of the price."
Correct approach: A 20% increase means multiply by 1.20, not 0.20.
Mistake: Assuming percentage changes are reversible.
Example: A 25% increase followed by a 25% decrease does NOT return to the original value.
Correct approach: $100 + 25% = $125. Then $125 - 25% = $93.75 (not $100).
Category 2: Statistical Misinterpretations
Mistake: Using mean when median is more appropriate.
Example: Income data: $30K, $35K, $40K, $45K, $500K. Mean = $130K, but median = $40K.
Correct approach: Use median when outliers are present; it better represents the typical value.
Mistake: Confusing correlation with causation.
Example: Ice cream sales and drowning rates both increase in summer, but ice cream does not cause drowning.
Correct approach: Look for confounding variables (summer heat affects both).
Category 3: Unit and Conversion Errors
Mistake: Mixing units without converting.
Example: Adding 2 hours to 45 minutes as 2 + 45 = 47.
Correct approach: Convert to same units: 120 minutes + 45 minutes = 165 minutes = 2 hours 45 minutes.
Mistake: Squaring or cubing linear units incorrectly.
Example: 1 yard = 3 feet, so 1 square yard = 9 square feet (not 3).
Correct approach: When converting area or volume, apply the conversion factor to each dimension.
Category 4: Rate and Ratio Errors
Mistake: Averaging rates incorrectly.
Example: Driving 60 mph for half the distance and 40 mph for the other half does NOT average 50 mph overall.
Correct approach: Use the harmonic mean or calculate total time/total distance.
Mistake: Setting up ratios incorrectly.
Example: If the ratio of boys to girls is 3:5, there are NOT 3/5 as many boys as total students.
Correct approach: Boys are 3/8 of the total (3 out of 3+5).
Category 5: Financial Calculation Errors
Mistake: Confusing simple and compound interest.
Example: Simple interest: I = Prt. Compound interest: A = P(1+r)^t.
Correct approach: Read the problem carefully to determine which formula applies.
Mistake: Forgetting that tax is added, not subtracted.
Example: A $100 item with 8% tax costs $108, not $92.
Correct approach: Tax increases the price; discounts decrease it.
Examples
Identify and correct the mistakes in these problems.
Example 1: Percentage Error
Problem: A stock drops 50% one day and rises 50% the next. A student says it's back to the original price.
Identify the mistake: Percentages are calculated on different bases.
Correct solution: If stock = $100, after -50% = $50. After +50% of $50 = $75. Lost 25% overall.
Example 2: Average Speed Error
Problem: A car travels 120 miles at 60 mph, then 120 miles at 40 mph. A student calculates average speed as (60+40)/2 = 50 mph.
Identify the mistake: Cannot simply average speeds; must use total distance/total time.
Correct solution: Time 1 = 120/60 = 2 hrs. Time 2 = 120/40 = 3 hrs. Total = 240 miles in 5 hrs = 48 mph.
Example 3: Ratio Misunderstanding
Problem: In a class with a 2:3 ratio of sophomores to juniors, a student says 2/3 of students are sophomores.
Identify the mistake: Confused ratio with fraction of total.
Correct solution: Sophomores are 2/(2+3) = 2/5 of the class, not 2/3.
Practice
Find and correct the error in each problem.
Problem 1: A student says: "If a store marks up prices by 25% and then offers a 25% discount, customers pay the original wholesale price." Is this correct? Explain.
Problem 2: A student calculates: "A room is 12 feet by 15 feet. That's 12 x 15 = 180 square feet = 60 square yards (dividing by 3)." Find the error.
Problem 3: A student says: "The median of 2, 4, 6, 8, 10 is 6, so the median of 2, 4, 6, 8, 10, 12 is also 6." Is this correct?
Problem 4: A student claims: "If I invest $1,000 at 5% simple interest, after 10 years I'll have $1,000 x 1.05^10 = $1,628.89." Find the error.
Problem 5: A student says: "A test has 3 times as many wrong answers as right answers. So 3/4 of the answers are wrong." Verify this claim.
Problem 6: A student writes: "To find 15% of 80, I calculate 80/15 = 5.33." Correct this error.
Problem 7: A student claims: "A 30% discount followed by an additional 20% discount is the same as a 50% discount." Is this true?
Problem 8: A student says: "If data shows that cities with more hospitals have higher death rates, then hospitals must cause deaths." What's wrong with this reasoning?
Problem 9: A student calculates tip: "The bill is $85. A 20% tip is $85 x 0.20 = $17, so I pay $85 - $17 = $68." Find the error.
Problem 10: A student says: "If the probability of rain is 50% on Saturday and 50% on Sunday, there's a 100% chance of rain over the weekend." Correct this.
Check Your Understanding
Test yourself with these review questions.
Question 1: Why can't you simply add or subtract percentages when they're calculated on different bases?
Question 2: Explain when to use mean versus median when summarizing data.
Question 3: What is the harmonic mean, and when should it be used instead of the arithmetic mean?
Next Steps
- Create a personal "error log" to track your own common mistakes
- Proceed to the Unit Quiz to test your overall understanding
- Review any categories where you made errors