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Unit 10: Review and Test Strategy

Topic 5

Common Traps and Careless-Error Patterns

The SAT is designed to reward careful thinking and punish rushed work. Many questions include trap answers: wrong values that you would get if you made a specific common error. Knowing these traps in advance helps you recognize and avoid them.
Trap 1: Answering the wrong question. The problem asks for 2x+12x + 1, but you solve for xx and enter that. Always re-read the final question after solving.
Trap 2: Sign errors. Distributing a negative, subtracting a quantity with multiple terms, or working with negative exponents are all common sources of sign mistakes. The antidote is to write out every step rather than doing multiple operations in your head.
Trap 3: Percent change confusion. A 20%20\% increase followed by a 20%20\% decrease does not return to the original value. The result is 1.20×0.80=0.961.20 \times 0.80 = 0.96, a 4%4\% decrease. Similarly, to reverse a 25%25\% increase, you divide by 1.251.25, not subtract 25%25\%.
Trap 4: Extraneous solutions. Squaring both sides of an equation (as with radical equations) or multiplying by a variable expression can introduce solutions that do not satisfy the original equation. Always check your answers by substituting back.
Trap 5: Confusing radius and diameter. Many problems give a diameter but the formula requires a radius, or vice versa. If a circle has a diameter of 10, the radius is 5, not 10.
Trap 6: Misreading the units. A problem might give dimensions in inches but ask for the answer in feet, or give a rate in miles per hour but ask for minutes. Always check that your answer is in the units the question requests.
Trap 7: Assuming equal means identical. If two angles are supplementary and one is xx, the other is 180−x180 - x, not xx. If two quantities add to a total, knowing the total does not tell you the individual values without another equation.
Trap 8: Forgetting domain restrictions. Division by zero and negative values under a square root are undefined. If a solution makes a denominator zero, it must be rejected.

Core Skills

  • Re-read the final question before entering an answer to avoid ``right work, wrong answer'' errors.
  • Write out distribution and subtraction steps explicitly to avoid sign errors.
  • Check solutions by substituting back into the original equation, especially for radical and rational equations.
  • Distinguish between radius and diameter, and verify units match the question.
  • Recognize percent-change traps, especially successive increase-decrease problems.

Example 1: Answering the Wrong Question

If 3x−6=153x - 6 = 15, what is the value of x−2x - 2? Trap: Solving gives 3x=213x = 21, so x=7x = 7. The trap is entering 7. The question asks for x−2=5x - 2 = 5. Shortcut: Notice that 3x−6=3(x−2)3x - 6 = 3(x - 2), so 3(x−2)=153(x - 2) = 15, which gives x−2=5x - 2 = 5 directly. 5\boxed{5}

Example 2: Sign Error in Distribution

Expand and simplify: 4−2(x−3)4 - 2(x - 3). Trap: Writing 4−2x−6=−2x−24 - 2x - 6 = -2x - 2 (forgetting to distribute the negative to the −3-3, giving −6-6 instead of +6+6). Correct: 4−2x+6=10−2x4 - 2x + 6 = 10 - 2x. 10−2x\boxed{10 - 2x}

Example 3: Extraneous Solution

Solve x+5=x−1\sqrt{x + 5} = x - 1. Step 1: Square both sides: x+5=x2−2x+1x + 5 = x^2 - 2x + 1, so x2−3x−4=0x^2 - 3x - 4 = 0, giving (x−4)(x+1)=0(x - 4)(x + 1) = 0 and x=4x = 4 or x=−1x = -1. Step 2: Check x=4x = 4: 9=3\sqrt{9} = 3 and 4−1=34 - 1 = 3. Valid. Check x=−1x = -1: 4=2\sqrt{4} = 2 and −1−1=−2-1 - 1 = -2. Since 2≠−22 \neq -2, this is extraneous. Trap: Entering both solutions. Only x=4x = 4 is valid. 4\boxed{4}

Example 4: Radius vs.\ Diameter

A circle has a diameter of 14. What is its area? Trap: Using 14 as the radius: π(14)2=196π\pi(14)^2 = 196\pi. Correct: Radius =7= 7. Area =π(7)2=49π= \pi(7)^2 = 49\pi. 49π\boxed{49\pi}

Example 5: Percent Reversal

A price after a 25%25\% increase is $75\$75. What was the original price? Trap: Subtracting 25%25\% of $75\$75: 75−18.75=56.2575 - 18.75 = 56.25. Correct: The original price times 1.251.25 gives 7575. So original =75÷1.25=60= 75 \div 1.25 = 60. $60\boxed{\$60}

Key Takeaways

  • The number-one careless error on the SAT is answering the wrong question. The last line of the problem tells you what to find. Read it twice.
  • Sign errors are the number-two careless error. When distributing a negative or subtracting a grouped expression, write out every term explicitly.
  • Always check solutions of radical and rational equations by substituting back. The algebra may produce valid-looking answers that fail the original equation.
  • When a problem gives a diameter, halve it before using it in any formula. This trap catches even strong students under time pressure.