Percent Increase and Decrease

To dominate this topic, you must move away from the "is over of" mentality and embrace the algebraic structure of percentage change.

1. The Percent Change Formula

The most fundamental way to calculate a percentage change between an old value and a new value is: $\text{Percent Change} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100$

When to use it: Use this when the question asks, "By what percent did the value increase?" or "What was the percentage decrease?" SAT Trap: Students often divide by the New Value by mistake. Always remember: the "Old Value" (the original, starting, or reference point) is always the denominator.

2. The Multiplier Method (The $1 \pm r$ Rule)

This is the most important tool in your SAT arsenal. Instead of calculating the "amount of change" and adding it back to the original, you should perform the calculation in one step.

If a value increases by $r\%$, the new value is: $\text{New} = \text{Original} \times (1 + r)$ (Where $r$ is the percent written as a decimal.)

If a value decreases by $r\%$, the new value is: $\text{New} = \text{Original} \times (1 - r)$

Examples:

• An increase of $25\%$: Multiply by $(1 + 0.25) = 1.25$
• A decrease of $15\%$: Multiply by $(1 - 0.15) = 0.85$
• An increase of $110\%$: Multiply by $(1 + 1.10) = 2.10$
• A decrease of $4\%$: Multiply by $(1 - 0.04) = 0.96$

3. Successive Percentage Changes

The SAT loves to apply multiple changes in a row. For example, a price might increase by $20\%$ and then decrease by $10\%$. Crucial Rule: You cannot simply add or subtract the percentages ($20\% - 10\% \neq 10\%$). You must multiply the multipliers.

$\text{Final Value} = \text{Original} \times (1 + r_1) \times (1 - r_2)$

If a price $P$ increases by $20\%$ and then decreases by $10\%$, the math is: $\text{Final} = P \times (1.20) \times (0.90) = P \times 1.08$ This shows the total change is actually an $8\%$ increase, not $10\%$.

4. Working Backward (Finding the Original)

If the SAT gives you the "New Value" and the percentage change, and asks for the "Original Value," do NOT apply the percentage to the new value. Formula: $\text{Original} = \frac{\text{New Value}}{1 \pm r}$

Example: A jacket costs \72$after a$20%$ discount. What was the original price? $72 = \text{Original} \times 0.80$ $\text{Original} = \frac{72}{0.80} = 90$

5. Reference Sheet Note

Important: None of these formulas are on the SAT Math Reference Sheet. You must memorize the multiplier concept ($1 \pm r$) and the percent change formula. The SAT expects you to internalize these as basic arithmetic tools.

1. The "100" Trick

If a question asks for a percentage change but doesn't give you any actual numbers (only variables or percentages), pick 100 as your starting value. Example: "If the length of a rectangle increases by $20\%$ and the width decreases by $20\%$, what is the change in area?"

• Let $L = 10$ and $W = 10$ (Area = 100).
• New $L = 12$, New $W = 8$.
• New Area = $12 \times 8 = 96$.
• Change from 100 to 96 is a $4\%$ decrease.

2. Translate English to Math

The SAT uses specific phrasing to hide the math. Learn to translate:

• "What percent of $x$ is $y$?" $\rightarrow \frac{y}{x} \times 100$
• "$x$ is $30\%$ more than $y$" $\rightarrow x = 1.30y$
• "$x$ is $30\%$ less than $y$" $\rightarrow x = 0.70y$
• "The price was reduced by $20\%$" $\rightarrow \text{New} = 0.80 \times \text{Old}$

3. Desmos is Your Best Friend

For successive changes, don't do them one by one. Type the entire string into the Desmos calculator. If a population of 500 grows by $12\%$ for 5 years, type $500(1.12)^5$. If a price of $x$ is taxed at $8\%$ and then discounted by $15\%$, type $x \cdot 1.08 \cdot 0.85$.

4. Read the Final Question Carefully

The SAT often asks for something slightly different than what you just calculated. Does it ask for the new total, the amount of the increase, or the original value? A common trap is calculating the new price when the question asked "By how many dollars did the price increase?"