Understanding when to substitute unknowns with numbers

Some questions can be solved much faster by choosing your own numbers for the unknowns instead of doing the algebra.
The idea is to substitute an easy value, work out the answer, and then see which answer choice matches.

Here is the whole method in one example.
Suppose a question asks which expression is equivalent to $2(x + 3)$, with choices $2x + 3$, $2x + 6$, $x + 6$, and $2x + 5$.
Since everything is written in terms of $x$, pick a simple value like $x = 1$.
The expression becomes $2(1 + 3) = 8$.
Now test the choices at $x = 1$: they give $5$, $8$, $7$, and $7$.
Only $2x + 6$ gives $8$, so it is the answer.

This works whenever two things are true.
First, the answer is a relationship — a fraction, percent, ratio, or an expression in the variables — not one fixed number.
For instance, if a question asks what fraction of a class passed and gives only fractions, the class size does not matter: $\frac{18}{30}$ and $\frac{6}{10}$ are the same fraction, so you can just assume $30$ students.
Second, nothing fixes the unknowns to particular values.
When both hold, any numbers you pick give the same answer, so choose easy ones.

It does not work when the question asks for a specific value that the information pins down.
For example, "a number increased by $5$ equals $12$" forces the number to be exactly $7$.
Guessing $n = 3$ gives $8$, not $12$, so a random value fails.
Whenever a constraint (an equation you must solve) fixes the unknown, solve it instead of assuming.

For questions built entirely on percentages, $100$ is the perfect number to assume.
Say a price rises $20\%$ and then falls $10\%$: start at $\$100$, go up to $\$120$, then down to $\$108$.
Reading the final answer off $100$ is far easier than tracking the percentages in symbols.

Two tips make this reliable.
Pick small, easy numbers, but avoid $0$ and $1$, because they often make several choices give the same result.
And if two choices happen to match your first number, just try a second number to break the tie.

Which of the following is equivalent to $3(x - 2) + 4$?
(A) $3x - 2$
(B) $3x + 2$
(C) $x + 2$
(D) $3x + 6$
Since the question and every choice are in terms of $x$, pick a simple value like $x = 2$.
The expression becomes $3(2 - 2) + 4 = 4$.
Testing the choices at $x = 2$: (A) gives $4$, (B) gives $8$, (C) gives $4$, and (D) gives $12$.
Both (A) and (C) give $4$, so try $x = 3$: the expression is $3(1) + 4 = 7$, (A) gives $7$, and (C) gives $5$.
Only (A) matches, so (A) is the answer.

A shirt's price is raised by $30\%$ and then lowered by $10\%$. What percent of the original price is the final price?
(A) $120\%$
(B) $117\%$
(C) $121\%$
(D) $90\%$
Since it is all percentages with no fixed price, assume the original is $\$100$.
A $30\%$ rise gives $\$100 \to \$130$.
A $10\%$ drop gives $\$130 \to \$130 - \$13 = \$117$.
So the final price is $117\%$ of the original, which is (B).

Can you solve $2n + 5 = 17$ by assuming any value for $n$?
No — this equation constrains $n$ to one specific value, so you must solve it.
Subtracting $5$ from both sides gives $2n = 12$, so $n = 6$.
A guess like $n = 3$ would give $2(3) + 5 = 11$, not $17$, so only $n = 6$ works.