EQN-06-02

Comparison Strategy: Plug In & Spot the Trap

The number-testing method for variable comparisons, and the classic traps that point to "cannot be determined".

~6 min read•Difficulty 3/5

When the quantities contain a variable, test smart values and watch what happens to the relationship.

Test the special numbers that change behaviour:

If two different test values give two different relationships, the answer is D.

A : x^{2} B : x

x = 2 \to A = 4 > B = 2 (A greater)

x = \frac{1}{2} \to A = \frac{1}{4} < B = \frac{1}{2} (B greater)

Different results \to D

A fraction between 0 and 1 gets smaller when squared $(\frac{1}{2}^{2} = \frac{1}{4}) .$
A negative number squared becomes positive.
Multiplying by a number between 0 and 1 shrinks a value.

These break the "bigger exponent means bigger value" intuition — exactly what the trap relies on.

Worked examples

Example 1 (difficulty 3/5)

No restriction on n. Quantity A: n². Quantity B: n³. Which is greater?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

Given

0 < x < 1.

Quantity A: x. Quantity

B : x .

Which is greater?

◎ Predict the answer first:

Always test 0, 1, a negative, and a fraction — these expose hidden "D" cases.
Respect the given restriction $(e . g .0 < x < 1)$ — it can turn a "D" into a definite answer.
If one tested value already gives both orders across tests, stop and choose D.

Testing only positive whole numbers

Why: They are the easiest to picture.

Fix: Negatives and fractions are where comparisons flip; always include them unless excluded.

Assuming a higher power is always larger

Why: True for numbers above 1, but not for fractions or negatives.

Fix: Between 0 and 1 a higher power is smaller; for negatives the sign can change.