Percent Increase and Percent Decrease - Seventh Grade Mathematics

Percent change measures the relative magnitude of growth or decline in a quantity compared to its initial baseline value, governed by proportional multipliers.

Defining Percent Change

When quantities fluctuate over time, reporting the raw numerical change does not convey how significant that change is relative to the starting size. An increase of ten dollars on a two-dollar book is monumental, whereas an increase of ten dollars on a five-hundred-dollar laptop is minor. Percent change quantifies this relative shift.

               THE UNIVERSAL PERCENT CHANGE FORMULA
   Percent Change = ( |New Value - Original Value| / Original Value ) * 100%
   Percent Increase: New Value > Original Value  (Growth)
   Percent Decrease: New Value < Original Value  (Reduction)

The most vital rule of percent change is that the denominator must always be the original baseline value, never the new or final value.

Example 1: A town's population grows from 12,000 to 15,000.
Amount of Increase = 15,000 - 12,000 = 3,000
Percent Increase   = (3,000 / 12,000) * 100% = (1/4) * 100% = 25%

Example 2: An appliance price drops from $400 to $310.
Amount of Decrease = 400 - 310 = 90
Percent Decrease   = (90 / 400) * 100% = 0.225 * 100% = 22.5%

The Multiplier Approach to Scaling Quantities

While calculating the raw difference and dividing works well, a far more efficient and algebraically powerful technique is the multiplier method. The original quantity is treated as one hundred percent (1.00).

               THE SCALING MULTIPLIER PRINCIPLE
   For a p% Increase:  Multiplier = 1 + (p / 100)
   For a p% Decrease:  Multiplier = 1 - (p / 100)
   New Value = Original Value * Multiplier

Consider increasing seventy-five dollars by twenty percent: The multiplier is one plus zero point twenty, which is one point twenty (1.20). Multiplying seventy-five by one point twenty: seventy-five times one point twenty equals ninety dollars.

Now consider decreasing one hundred twenty dollars by fifteen percent: The multiplier is one minus zero point fifteen, which is zero point eighty-five (0.85). Multiplying one hundred twenty by zero point eighty-five yields one hundred two dollars.

+-----------------------+-----------------------+-----------------------------+
| Scenario Description  | Multiplier            | New Value Equation          |
+-----------------------+-----------------------+-----------------------------+
| 8% Sales Tax          | 1 + 0.08 = 1.08       | Cost = Original * 1.08      |
| 25% Discount          | 1 - 0.25 = 0.75       | Sale = Original * 0.75      |
| 3.5% Wage Increase    | 1 + 0.035 = 1.035     | Wage = Original * 1.035     |
| 40% Clearance Sale    | 1 - 0.40 = 0.60       | Price = Original * 0.60     |
+-----------------------+-----------------------+-----------------------------+

The multiplier method bypasses the need for two separate calculation steps (finding the amount and then adding or subtracting), condensing the operation into a single multiplication.

The Asymmetry of Percent Changes

A frequent conceptual pitfall in mathematics is assuming that equal percentage increases and decreases cancel each other out. They do not, because the baseline whole changes between the two operations.

               THE BASELINE SHIFT PARADOX
   Start with:                  $100.00
   Apply a 50% decrease:        $100 * (1 - 0.50) = $50.00
   Apply a 50% increase:        $50 * (1 + 0.50)  = $75.00
   Result:                      $75.00 != $100.00 (Net loss of 25%!)

When the price dropped by fifty percent, the baseline shrank from one hundred dollars to fifty dollars. When the price subsequently rose by fifty percent, that increase was calculated on the smaller fifty-dollar base, generating only twenty-five dollars of gain. To restore fifty dollars back to one hundred dollars requires a one hundred percent increase, not fifty percent.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. An athletic footwear company increases the retail price of a running shoe from eighty dollars to ninety-two dollars. Calculate the percent increase.

Exercise 2. A digital audio file has an uncompressed size of two hundred fifty megabytes. After compression, its size is sixty megabytes. Find the percent decrease in file size.

Exercise 3. A department store purchases a winter coat for one hundred twenty dollars and prices it with a thirty-five percent markup. Use the multiplier method to find the retail selling price.

Exercise 4. A rare collectible card increases in value by twenty percent during year one, and then drops in value by twenty percent during year two. If its initial value was fifty dollars, calculate its value at the end of year two and determine the overall net percent change.

Solutions

Solution to Exercise 1. Find the amount of increase: ninety-two minus eighty equals twelve dollars. Divide the increase by the original price of eighty dollars: twelve divided by eighty equals zero point fifteen (0.15). Multiply by one hundred to express as a percent: the price experienced a fifteen percent (15%) increase.

Solution to Exercise 2. Find the amount of decrease: two hundred fifty minus sixty equals one hundred ninety megabytes. Divide the reduction by the original baseline size of two hundred fifty megabytes: one hundred ninety divided by two hundred fifty equals zero point seventy-six (0.76). Multiply by one hundred: the file size experienced a seventy-six percent (76%) decrease.

Solution to Exercise 3. The markup represents a thirty-five percent increase on the wholesale price. The multiplier is one plus zero point thirty-five, which is one point thirty-five (1.35). Multiply the wholesale cost of one hundred twenty dollars by one point thirty-five: one hundred twenty times one point thirty-five equals one hundred sixty-two dollars. The retail selling price is one hundred sixty-two dollars.

Solution to Exercise 4. In year one, the fifty-dollar card increases by twenty percent. Its multiplier is one point twenty: fifty times one point twenty equals sixty dollars. In year two, the sixty-dollar card decreases by twenty percent. Its multiplier is one minus zero point twenty, which is zero point eighty (0.80): sixty times zero point eighty equals forty-eight dollars. The card's final value is forty-eight dollars. To find the overall net percent change from the original fifty dollars: fifty minus forty-eight equals two dollars of decrease. Divide two by fifty: two divided by fifty equals zero point zero four, which is a four percent net decrease overall.