Math Problem Statement

The value of a house varies over a five-year period. The percentage change over this period has been +0.8%, -1.0%, -1.6%, +2.9% and -2.1%. What is the total percentage change over the five-year period?

(Let + represent an increase in value and - represent a decrease in value).

Solution

To find the total percentage change over the five-year period, we need to multiply the percentage changes year by year. However, since the percentage changes are given as percentages (e.g., +0.8%, -1.0%, etc.), we must first convert these percentages into their decimal equivalents before applying them.

The formula for calculating the overall change over multiple periods is:

Total change=(1+percentage change1)×(1+percentage change2)××(1+percentage changen)1\text{Total change} = (1 + \text{percentage change}_1) \times (1 + \text{percentage change}_2) \times \dots \times (1 + \text{percentage change}_n) - 1

Let's apply this formula step-by-step using the percentage changes:

  • +0.8% = 1 + 0.008 = 1.008
  • -1.0% = 1 - 0.010 = 0.990
  • -1.6% = 1 - 0.016 = 0.984
  • +2.9% = 1 + 0.029 = 1.029
  • -2.1% = 1 - 0.021 = 0.979

Now, multiply these factors together:

1.008×0.990×0.984×1.029×0.9791.008 \times 0.990 \times 0.984 \times 1.029 \times 0.979

Finally, subtract 1 to find the total percentage change.

Let’s calculate that: Define the percentage changes in decimal form factors = [1.008, 0.990, 0.984, 1.029, 0.979]

Multiply the factors together

total_change = 1 for factor in factors: total_change *= factor

Subtract 1 and convert to percentage

total_percentage_change = (total_change - 1) * 100 total_percentage_changeThe total percentage change over the five-year period is approximately -1.08%. This means the value of the house decreased by about 1.08% over the five years.

Would you like any further clarification on this process or any additional questions?

Here are some related questions you might find interesting:

  1. How would the total percentage change differ if the percentage changes were all positive?
  2. What is the difference between simple percentage change and compounded percentage change?
  3. How can we calculate the average annual percentage change over this period?
  4. How would the total percentage change change if the changes were larger in magnitude?
  5. What would the effect be if there was no negative percentage change in this sequence?

Tip: When dealing with percentage changes over multiple periods, always convert percentages to decimals before multiplying to avoid errors in calculation!

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Math Problem Analysis

Mathematical Concepts

Percentage Change
Compound Percentage
Decimal Conversion
Multiplication of Decimals

Formulas

Total change = (1 + percentage change_1) × (1 + percentage change_2) × ... × (1 + percentage change_n) - 1

Theorems

Compound percentage change theorem

Suitable Grade Level

Grades 9-11