7.3 Percent Change, Profit, and Cost Problems

Understanding Percentages on the SHSAT

Percentages represent fractions of a hundred, where "percent" literally translates to "per 100." The basic relationship for any percentage calculation is: $\text{Part} = \text{Percent} \times \text{Whole}$

Since calculators are prohibited on the SHSAT, your speed and accuracy depend on your ability to translate percentages into fractions or decimals. Converting to fractions is often the fastest path to simplifying calculations through cross-cancellation. The table below displays common fraction-to-percentage conversions that you should memorize:

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Percent Increase and Decrease

Percent change measures the relative change of a quantity over time. The formula for percent change is: $\text{Percent Change} = \frac{\text{Amount of Change}}{\text{Original Value}} \times 100\%$ $\text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$

The Original Value Trap

The most common error when calculating percent change is using the New Value in the denominator instead of the Original Value. Always remember: change is measured relative to the starting point.

Let's examine a percent increase problem: A student's score on an algebra quiz increased from 80 points to 96 points. What was the percent increase in the student's score? First, calculate the amount of change: $\text{Change} = 96 - 80 = 16 \text{ points}$ Second, divide by the original score (80): $\text{Fractional Change} = \frac{16}{80} = \frac{1}{5}$ Third, convert the fraction to a percentage: $\frac{1}{5} \times 100\% = 20\%$ The score increased by 20%.

Now let's examine a percent decrease problem: The price of a shirt was reduced from $40 to $35. What was the percent discount? First, calculate the change: $\text{Change} = 40 - 35 = 5 \text{ dollars}$ Second, divide by the original price (40): $\text{Fractional Change} = \frac{5}{40} = \frac{1}{8}$ Third, convert to a percentage using the conversion table: $\frac{1}{8} \times 100\% = 12.5\%$ The percent discount was 12.5%.

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Sequential Percent Changes (Compound Changes)

Another major trap on the SHSAT is adding or subtracting percentages when changes occur in sequence. For example, if a price is increased by 20% and then decreased by 20%, the net change is not 0%. This is because the second percentage change is calculated using the new, increased value as its baseline, not the original value.

To solve sequential percent problems, either:

1. Assume an original value of 100 (which makes calculating percentages easy) and work through the steps.
2. Multiply the original variable by the successive percentage multipliers.

Let's work through the 20% markup and 20% discount problem: A retailer marks up the price of a jacket by 20%. Later, the retailer offers a 20% discount on the marked-up price. What is the net percent change from the original price? Assume the original price of the jacket is $100. First step: Mark up by 20%: $\text{New Price} = 100 \times (1 + 0.20) = 120 \text{ dollars}$ Second step: Discount by 20% on the new price: $\text{Sale Price} = 120 \times (1 - 0.20) = 120 \times 0.80 = 96 \text{ dollars}$ Third step: Calculate the net percent change: $\text{Net Change} = \frac{96 - 100}{100} \times 100\% = -4\%$ The net change is a 4% decrease.

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Profit, Cost, Markup, and Discount Terminology

Many SHSAT word problems use business terminology. Understanding the relationship between these terms is essential for setting up the correct equations:

• Cost Price (Cost): The price a merchant pays to acquire an item.
• Markup: The amount added to the cost price to determine the selling price. Usually expressed as a percentage of the cost.
• Selling Price (Retail Price): The price at which the merchant offers the item to customers. $\text{Selling Price} = \text{Cost} \times (1 + \text{Markup Percent})$
• Discount: A reduction in the selling price, usually expressed as a percentage of the selling price. $\text{Sale Price} = \text{Selling Price} \times (1 - \text{Discount Percent})$
• Profit: The financial gain made by selling an item. $\text{Profit} = \text{Revenue (or final Sale Price)} - \text{Cost Price}$

Let's solve a multi-step business word problem: A store owner buys a book for $20. He marks up the cost by 50% to establish the retail price. Later, the book is placed on a discount rack for 10% off the retail price. If a customer buys the book and pays an additional 8% sales tax, what is the total amount the customer pays? First, calculate the retail price: $\text{Retail Price} = 20 \times (1 + 0.50) = 20 \times 1.50 = 30 \text{ dollars}$ Second, calculate the sale price after the 10% discount: $\text{Sale Price} = 30 \times (1 - 0.10) = 30 \times 0.90 = 27 \text{ dollars}$ Third, calculate the sales tax on the sale price: $\text{Sales Tax} = 27 \times 0.08 = 2.16 \text{ dollars}$ Fourth, calculate the total cost paid by the customer: $\text{Total Paid} = \text{Sale Price} + \text{Sales Tax} = 27.00 + 2.16 = 29.16 \text{ dollars}$ The customer pays a total of $29.16.

Note that the profit made by the store owner (excluding tax) is $27.00 - $20.00 = $7.00.

Visualizing Sequential Multipliers

Using decimal multipliers is the most efficient algebraic way to handle sequential percent changes. Each percent increase of $p%$ can be represented by the multiplier $(1 + \frac{p}{100})$, and each percent decrease of $d%$ by the multiplier $(1 - \frac{d}{100})$. If a product undergoes a series of changes, you simply multiply these terms together: $\text{Final Price} = \text{Original Price} \times \text{Multiplier}_1 \times \text{Multiplier}_2 \times \dots$

For instance, in the previous markup and discount example, the retail price was first multiplied by $1.50$ (a 50% increase) and then by $0.90$ (a 10% decrease). The compound multiplier is: $1.50 \times 0.90 = 1.35$ This single compound multiplier tells us that the final sale price ($27) is exactly 135% of the original cost ($20), which represents a net 35% increase. Master this technique to skip intermediate step calculations and save valuable time on the SHSAT.