Calculating Percent Change, Increase, and Decrease
Core Concepts
Learn the essential formulas for calculating percent change, including percent increase and percent decrease. This is crucial for analyzing data, stock prices, discounts, markups, taxes, and other real-world scenarios.
Calculating Percent Change, Increase, and Decrease
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What is the general formula used to calculate percent change?
$$ \text{Percent Change} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\% $$ It can also be expressed as $$\left( \frac{\text{Amount of Change}}{\text{Original Value}} \right) \times 100\%$$
A stock price increased from \(\$50\) to \(\$60\). What was the percent increase?
Original value = \(\$50\), New value = \(\$60\).
$$ \text{Percent Increase} = \frac{60 - 50}{50} \times 100\% = \frac{10}{50} \times 100\% = 0.20 \times 100\% = 20\% $$
A store reduces the price of a shirt from \(\$25\) to \(\$20\). What is the percent decrease?
Original value = \(\$25\), New value = \(\$20\).
$$ \text{Percent Decrease} = \frac{20 - 25}{25} \times 100\% = \frac{-5}{25} \times 100\% = -0.20 \times 100\% = -20\% $$ The percent decrease is \(20\%\).
A company's profits increased by \(15\%\) from last year. If last year's profits were \(\$200,000\), what are this year's profits?
An increase of \(15\%\) means the new value is \(100\% + 15\% = 115\%\) of the original. Convert \(115\%\) to a decimal (\(1.15\)) and multiply: \(1.15 \times \$200,000 = \$230,000\).
A phone is on sale for \(30\%\) off. If its original price was \(\$800\), what is the sale price?
Method 1: Calculate the discount amount (\(0.30 \times \$800 = \$240\)) and subtract it from the original price (\(\$800 - \$240 = \$560\)).
Method 2: A \(30\%\) discount means you pay \(100\% - 30\% = 70\%\) of the price. Calculate \(0.70 \times \$800 = \$560\).
After a \(20\%\) discount, the price of a jacket is \(\$72\). What was its original price?
A \(20\%\) discount means the sale price is \(80\%\) of the original price (\(P\)). So, \(0.80 \times P = \$72\). To find the original price, divide by \(0.80\): \(P = \frac{72}{0.80} = \$90\).
A restaurant bill is \(\$50\). How much is the total bill after adding a \(6\%\) sales tax?
Adding a \(6\%\) tax is equivalent to finding \(106\%\) of the original bill. Convert \(106\%\) to a decimal (\(1.06\)) and multiply: \(1.06 \times \$50 = \$53\).
What is the formula for percent error?
$$ \text{Percent Error} = \left| \frac{\text{Measured Value} - \text{Actual Value}}{\text{Actual Value}} \right| \times 100\% $$ The absolute value is used because error is typically expressed as a non-negative percentage.
A student measures the length of a desk to be \(145\) cm. The actual length is \(150\) cm. What is the percent error of the measurement?
An item is marked down by \(20\%\), and then a \(10\%\) sales tax is applied to the discounted price. What is the overall percent change from the original price?
Let the original price be \(P\). After a \(20\%\) discount, the price is \(0.80P\). A \(10\%\) tax on this price means you multiply by \(1.10\): \(1.10 \times (0.80P) = 0.88P\). The final price is \(88\%\) of the original, which represents a \(12\%\) overall decrease, not a \(10\%\) decrease (\(20\% - 10\%\)).
The population of a town grew from \(25,000\) to \(30,000\) in a decade. What was the percent increase in population?
