Rational Expressions · Sub-skill drill
Adding and Subtracting Rational Expressions
To add or subtract rational expressions, find a common denominator (the least common multiple of the denominators), rewrite each fraction with that denominator, then combine the numerators. The denominators of SAT rational addition problems are usually small enough that the common denominator is just the product of the two denominators. After adding, factor the numerator and check whether anything cancels with the denominator.
How this sub-skill is tested on the SAT
To add or subtract rational expressions, find a common denominator (the least common multiple of the denominators), rewrite each fraction with that denominator, then combine the numerators. The denominators of SAT rational addition problems are usually small enough that the common denominator is just the product of the two denominators. After adding, factor the numerator and check whether anything cancels with the denominator.
This sub-skill sits inside the broader Rational Expressions topic, which is part of the College Board's Passport to Advanced Math content domain. Passport to Advanced Math is the SAT's bridge to the kind of algebraic manipulation you will see in a college precalculus or calculus course. The questions are not about memorizing identities — they are about fluency with structure. Can you factor a quadratic by inspection? Can you read the vertex of a parabola off its standard form? Can you simplify a rational expression without losing a domain restriction? Can you translate between f(x), a graph, and a table of values without panicking? Most students who plateau between 650 and 720 plateau here, because the section rewards algebraic intuit
Practice questions in this drill set
Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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For x != 0, the expression 5x^2 - 10x / 5x is equivalent to which of the following?
- A x - 2 1
- B 5x + -10
- C x - 2
- D x + -1
Worked solution
Answer: C — x - 2
Factor the numerator: 5x^2 + (-10)x = 5x(x + (-2)). Divide by 5x (allowed because x ≠ 0): the result is x + (-2).
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For x != 0, the expression 6x^2 + 12x / 6x is equivalent to which of the following?
- A x + 3
- B 6x + 12
- C x - 2
- D x + 2
Worked solution
Answer: D — x + 2
Factor the numerator: 6x^2 + (12)x = 6x(x + (2)). Divide by 6x (allowed because x ≠ 0): the result is x + (2).
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For x != 0, the expression 3x^2 - 15x / 3x is equivalent to which of the following?
- A x - 5 1
- B x + -4
- C x - 5
- D 3x + -15
Worked solution
Answer: C — x - 5
Factor the numerator: 3x^2 + (-15)x = 3x(x + (-5)). Divide by 3x (allowed because x ≠ 0): the result is x + (-5).
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For x != 0, the expression 6x^2 - 6x / 6x is equivalent to which of the following?
- A x - 1 1
- B x + 0
- C 6x + -6
- D x - 1
Worked solution
Answer: D — x - 1
Factor the numerator: 6x^2 + (-6)x = 6x(x + (-1)). Divide by 6x (allowed because x ≠ 0): the result is x + (-1).
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For x != 0, the expression 5x^2 - 25x / 5x is equivalent to which of the following?
- A x - 5 1
- B x + -4
- C 5x + -25
- D x - 5
Worked solution
Answer: D — x - 5
Factor the numerator: 5x^2 + (-25)x = 5x(x + (-5)). Divide by 5x (allowed because x ≠ 0): the result is x + (-5).
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For x != 0, the expression 3x^2 + 6x / 3x is equivalent to which of the following?
- A x + 3
- B 3x + 6
- C x - 2
- D x + 2
Worked solution
Answer: D — x + 2
Factor the numerator: 3x^2 + (6)x = 3x(x + (2)). Divide by 3x (allowed because x ≠ 0): the result is x + (2).
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.