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Algebraic expressions and solving equations — ISEE Upper practice questions

On the ISEE Upper Quantitative Reasoning section, Algebraic expressions and solving equations covers combining like terms, simplifying polynomials, and turning word problems into equations. A student must translate a relationship between the digits of a two-digit number into algebra, add or subtract polynomials, and apply a constraint such as x greater than one when comparing two expressions. Items often pit Column A against Column B, with choices that one column is greater, that they are equal, or that the relationship cannot be determined, alongside ordinary multiple-choice word problems. Traps include dropped signs, mixed-up place values, and treating an inequality as a single value.

  • Section: Quantitative Reasoning 37 questions on the paper
  • 168 practice questions in the app

Sample questions

Question 1

x>1x>1.
Column AColumn B
x+1xx+\dfrac{1}{x}2
  1. AThe quantity in Column A is greater
  2. BThe quantity in Column B is greater
  3. CThe two quantities are equal
  4. DThe relationship cannot be determined from the information given
Show answer

Answer: A — The quantity in Column A is greater

For x>1x>1, (x−1)2>0(x-1)^2>0, so x2+1>2xx^2+1>2x. Dividing by positive xx gives x+1x>2x+\dfrac{1}{x}>2.

Question 2

A two-digit number has a tens digit that is twice its ones digit. When the digits are reversed, the new number and the original number have a sum of 99. What is the original number?

  1. A36
  2. B42
  3. C63
  4. D84
Show answer

Answer: C — 63

The original number is 20a+a=21a20a+a=21a, and the reversed number is 10a+2a=12a10a+2a=12a. Their sum is 33a=9933a=99, so a=3a=3 and the original number is 63.

Question 3

Column AColumn B
(3x2−4x+6)+(2x2+7x−9)(3x^2-4x+6)+(2x^2+7x-9)5x2+3x−35x^2+3x-3
  1. AThe quantity in Column A is greater
  2. BThe quantity in Column B is greater
  3. CThe two quantities are equal
  4. DThe relationship cannot be determined from the information given
Show answer

Answer: C — The two quantities are equal

Combine like terms: 3x2+2x2=5x23x^2+2x^2=5x^2, −4x+7x=3x-4x+7x=3x, and 6−9=−36-9=-3. The first expression is identical to 5x2+3x−35x^2+3x-3, so the two quantities are equal.

Practice 168 Algebraic expressions and solving equations questions in the app

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