SAT Trick – Expressions
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Mastering SAT Advanced Algebra: Standard vs. Alternative Solution Methods
Watch The Quick SAT Math Trick
Beat the clock on test day! Check out our quick visual guide showing how substitution can save you precious minutes.
SAT Math Practice Question
Which of the following is equivalent to the given complex fraction where \(x > 5\)?
\[ \frac{1}{\frac{1}{x+5} + \frac{1}{x+4}} \]
Answer Choices:
- A) \(\displaystyle \frac{2}{2x+9}\)
- B) \(\displaystyle \frac{x^2 + 9x + 20}{2x+9}\) (Correct Answer)
- C) \(\displaystyle \frac{1}{2x+9}\)
- D) \(2x + 9\)
Method 1: Standard Algebraic Approach
Tackle this question systematically by simplifying the denominator first.
- Step 1: Find a Common Denominator. The denominators inside the complex fraction are \((x+5)\) and \((x+4)\). Their common product is \((x+5)(x+4)\).
- Step 2: Combine the Denominator Terms. \[ \frac{1}{x+5} + \frac{1}{x+4} = \frac{(x+4) + (x+5)}{(x+5)(x+4)} = \frac{2x+9}{x^2+9x+20} \]
- Step 3: Apply the Fraction Rule. Using the property \(\frac{1}{\frac{a}{b}} = \frac{b}{a}\), flip the denominator fraction to the numerator: \[ \frac{1}{\frac{2x+9}{x^2+9x+20}} = \frac{x^2+9x+20}{2x+9} \]
Result: Matches Answer Choice B.
Method 2: The Substitution Trick (Time-Saver)
An SAT trick that can help you save time when you get stuck with ordinary algebra.
- Step 1: Choose a Test Value. Since the condition states \(x > 5\), let us pick \(x = 6\).
- Step 2: Evaluate the Original Expression. Substitute \(6\) into the complex fraction: \[ \frac{1}{\frac{1}{6+5} + \frac{1}{6+4}} = \frac{1}{\frac{1}{11} + \frac{1}{10}} = \frac{1}{\frac{21}{110}} = \frac{110}{21} \]
- Step 3: Test the Answer Choices. Plug \(x = 6\) into the choices to see which one evaluates to \(\frac{110}{21}\):
- Choice A: \(\frac{2}{2(6)+9} = \frac{2}{21}\)
- Choice B: \(\frac{6^2 + 9(6) + 20}{2(6)+9} = \frac{36+54+20}{12+9} = \frac{110}{21}\ \checkmark\)
- Choice C: \(\frac{1}{21}\)
- Choice D: \(2(6)+9 = 21\)
Note: Knowing alternative strategies like plug-in substitution is a brilliant way to beat the clock on test day if algebraic manipulation feels heavy.
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