DAT Quantitative Reasoning: Simplify Both Columns Before You Compare

DAT High-Yield Concept

Additive Invariance: Adding or subtracting the same amount from both columns preserves the inequality relation ( $A > B \iff A + c > B + c$ ).
Positive Multiplicative Invariance: Multiplying or dividing both columns by a positive number maintains the comparative relation ( $A > B \iff cA > cB$ for $c > 0$ ).
Variable Sign Trap: Never multiply or divide both columns by an expression whose sign is unknown. A negative factor reverses the inequality sign ( $A > B \iff cA < cB$ for $c < 0$ ).
Strategic Cancellation: Cancel identical terms and identical positive factors in both columns before evaluating or testing values.

1. Addition and Subtraction Properties (Additive Invariance)

In DAT Quantitative Reasoning comparison questions, treat Column A and Column B like the two sides of an inequality. You can add or subtract any real number—whether positive, negative, or containing variables—from both columns simultaneously without altering which column is larger.

Worked Example 1: Subtracting Identical Expressions

Problem: Compare Column A: $3x + 14$ and Column B: $3x - 2$ for any real number $x$ .

Reasoning: Subtract $3x$ from both columns:

Column A: $3x + 14 - 3x = 14$
Column B: $3x - 2 - 3x = -2$

Since $14 > -2$ , Column A is strictly greater than Column B for all real values of $x$ .

Exam Trap: Plugging in test values prior to simplifying. Evaluating expressions manually for multiple test numbers consumes valuable time when basic subtraction eliminates the variable entirely.

2. Multiplying and Dividing by Positive Quantities

Multiplying or dividing both columns by a known positive number does not change which column is greater. This is especially useful for clearing fractions or eliminating common scalar multipliers.

Worked Example 2: Clearing Fractional Denominators

Problem: Compare Column A: $\frac{7}{12}$ and Column B: $\frac{5}{9}$ .

Reasoning: The least common denominator is $36$ , which is positive. Multiply both columns by $36$ :

Column A: $36 \times \frac{7}{12} = 3 \times 7 = 21$
Column B: $36 \times \frac{5}{9} = 4 \times 5 = 20$

Because $21 > 20$ , Column A is greater.

3. The Unknown Sign Trap: Illegal Multipliers

Multiplying or dividing both columns by a negative quantity flips the relationship between them. If you multiply or divide by an algebraic expression whose sign is unknown (for example, a variable $x$ without restrictions), the true comparison cannot be preserved.

What you may do to both columns, and the one move that breaks — A quantitative comparison asks only which column is larger, so any operation that preserves that relationship is fair game. Adding or subtracting the same amount from both columns is always safe, and so is multiplying or dividing both by a number known to be positive. Dividing both columns by an expression whose sign is unknown is not safe, because a negative multiplier reverses the comparison. In the third row cancelling x would say B is greater, yet x equal to negative 2 makes A greater, so the correct answer is that the relationship cannot be determined.

Rule of Operations: You may add or subtract any term across columns, and you may multiply or divide by any strictly positive value. You must never multiply or divide both columns by a variable expression unless its sign is guaranteed to be positive.

Worked Example 3: Dividing by a Variable of Unknown Sign

Problem: Given no restrictions on $x$ , compare Column A: $x^2$ and Column B: $5x$ .

Reasoning: A common mistake is dividing both columns by $x$ to get $x$ vs $5$ . This step is illegal because $x$ could be negative or zero.

Instead, subtract $5x$ from both columns (a valid move) to compare $x^2 - 5x$ against $0$ . Factoring yields $x(x - 5)$ . Testing ranges (covered in detail in dedicated lessons on plug-in strategies) shows that if $x = 6$ , Column A is larger ( $36 > 30$ ); if $x = 2$ , Column B is larger ( $4 < 10$ ); if $x = 0$ , both columns are equal ( $0 = 0$ ). Thus, the relationship cannot be determined.

4. Cancelling Identical Terms and Positive Factors

Simplification relies on removing matching terms and positive factors across both columns before attempting evaluation.

Worked Example 4: Cancelling Positive Factors

Problem: Given $y > 0$ , compare Column A: $\frac{4y + 12y^2}{4y}$ and Column B: $1 + 3y$ .

Reasoning: Because $y > 0$ , the term $4y$ is strictly positive. Factor $4y$ out of the numerator in Column A:

$\frac{4y(1 + 3y)}{4y} = 1 + 3y$

Column A simplifies directly to $1 + 3y$ , which is identical to Column B. Thus, Column A equals Column B.

Comparison of Column Operations

Operation	Allowed Condition	Impact on Inequality Direction	Validity
Addition / Subtraction	Any real expression or variable	Preserves comparison direction	Valid
Multiplication / Division	Guaranteed positive ( $c > 0$ )	Preserves comparison direction	Valid
Multiplication / Division	Guaranteed negative ( $c < 0$ )	Reverses comparison direction	Requires sign reversal
Multiplication / Division	Unknown sign / Variable	Unknown impact	Forbidden

P-A-S-S the Comparison Rules:
• Positive multiplier preserves direction.
• Addition and subtraction are always safe.
• Sign unknown? Stop—do not multiply or divide.
• Simplify first by cancelling identical terms.

Critical Thinking Questions

Why is subtracting $7x$ from both columns always algebraically valid, whereas dividing both columns by $7x$ is invalid when $x$ has an unrestricted real domain?
If $ab > 0$ , does this guarantee that you can divide both columns by $a$ without reversing the comparison direction? Explain why or why not.
Under what circumstances is squaring both columns a valid operation in quantitative comparisons?

References

Sources returned by Google Search grounding during lesson verification.

501 Quantitative Comparison Questions - AWS — vertexaisearch.cloud.google.com
GRE Math: Solving Quantitative Comparisons - Magoosh — vertexaisearch.cloud.google.com
Manipulating inequalities | Mathcentre — vertexaisearch.cloud.google.com

Simplify Both Columns Before You Compare

Summary

Practice Questions

AI Tutor