ISP 120

Tips for working with Percentages and Comparisons

General advice for solving a problem

1. Identify the type of problem and choose the correct equation. Absolute difference? Percent change? Proportion of?
2. Identify the variables. What is the original amount being compared to? What is the new amount that is being described in relation to the original amount?
3. Plug the values into the formula and solve.

Also notice whether the problem requires you to calculate an answer just once, or whether it has 2 or more steps (eg: decreased 50% then increased 70%).

Tips for specific problem types and formulas:

Absolute Change or Absolute Difference

Absolute change = new amount - original amount

Phrases that typically signal a comparison involving absolute change or absolute difference:

• "how many more than"
• "as much more than"
• "A has how much more than B"

Identifying the "new" and "original" values:

• the new amount is usually the subject of the sentence
• the original amount will usually be the object of a preposition like "than"
• "(new amount) is x more than (original amount)."
• "(new amount) is x less than (base amount)."

Relative Change or Relative Difference

Proportion change = (absolute change) / (original amount) = (new - original) / original = (new / original) - 1

Phrases that typically signal a comparison involving proportion change or proportion difference:

• "A is x times more than B"
• "increased by a factor of x"
• "A is half as much as B" (or some other fraction as much)

Percent Change = (proportion change) * 100% = ((new - original) / original) * 100%

Phrases that typically signal a comparison involving percentage change or percentage difference:

• "A is x% more than B"
• "A is x% less than B"
• "increased by x%"
• "decreased by x%"

Relative Size or Relative Amount

Proportion of = new / original

Phrases that typically signal a comparison involving proportion of:

• "A is x times as much as B"
• "A is half as much as B"

Percentage of = (proportion of) * 100% = (new / original) * 100%

Phrases that typically signal a comparison involving percentage of:

• "A is x% of B"
• "x% of A are B"

Percent Difference vs. Percentage Point Difference

"Percent higher" means a relative change comparison (percent change or percent difference)

"Percentage points higher" means an absolute difference or absolute change.

Example: A has 10% and B has 12%

• B is 20% higher than A. percent change = ((12% - 10%) / 10%) * 100% = (2% / 10%) * 100% = .2 * 100% = 20%
• B is 2 percentage points higher than A. absolute change = new - original = 12% - 10% = 2%