Algebraic Reasoning

Setting up algebraic equations to describe percentage relationships with variables

These questions ask you to translate a verbal percentage statement into an algebraic expression. The concept is simple — convert the percent to a decimal and multiply — but the SAT tests whether you can place the decimal point correctly.

The Core Translation

"$X\%$ of $n$" → $\dfrac{X}{100} \times n$

That's it. The only challenge is converting the percentage to the right decimal.

Standard Pattern — Single Relationship

A marketing agency's profit is $43\%$ of its total revenue. Which expression represents the profit, where $r$ is the total revenue?

A) $4.3r$
B) $0.043r$
C) $0.43r$
D) $43r$

$43\%$ of $r = \dfrac{43}{100} \times r = 0.43r$
Answer: C

The Decimal Placement Traps

Every wrong answer is a predictable decimal error:

• $43r$ → forgot to divide by 100 (represents 4,300%)
• $4.3r$ → divided by 10 instead of 100 (represents 430%)
• $0.043r$ → divided by 1,000 instead of 100 (represents 4.3%)

The SAT is testing one skill here: can you correctly convert a percentage to a decimal? The rule is always divide by 100, which moves the decimal point two places to the left.

Harder Pattern — Multi-Variable Relationships

The positive number $x$ is $1{,}260\%$ of the sum of the positive numbers $y$ and $z$, and $y$ is $80\%$ of $z$. What percent of $y$ is $x$?

A) 28.35%
B) 1340%
C) 2268%
D) 2835%

Step 1 — Translate to equations:
$x = 12.6(y + z)$
$y = 0.8z$, so $z = \dfrac{y}{0.8} = 1.25y$

Step 2 — Substitute:
$x = 12.6(y + 1.25y) = 12.6(2.25y) = 28.35y$

Step 3 — Convert to percent:
$x = 28.35y$ means $x$ is $2{,}835\%$ of $y$.
Answer: D

Gotcha: Choice A (28.35%) is the decimal multiplier, not the percentage. To convert a decimal multiplier to a percent, multiply by 100: $28.35 \times 100 = 2{,}835\%$.

What to Do on Test Day

• "X% of $n$" always becomes $\dfrac{X}{100} \times n$. Move the decimal two places left.
• Check your decimal conversion by asking: "Is my answer between 0 and the whole?" If the expression gives a value larger than the whole, you probably placed the decimal wrong.
• For multi-step algebraic problems, translate each relationship into an equation, eliminate variables by substitution, and convert the final decimal multiplier to a percent by multiplying by 100.
• Common decimal conversions to memorize:
• $1\% = 0.01$, $5\% = 0.05$, $10\% = 0.10$
• $25\% = 0.25$, $50\% = 0.50$, $75\% = 0.75$