Percentages and Comparisons – Detailed Study Notes
Percentages: Basic Concepts
- Percent etymology and meaning
- per means divided by, cent means 100; hence percent means divided by 100.
- Example: 30% means 30 out of 100, i.e., the percentage of a whole is \%\quad\text{of the whole} = \frac{\text{part}}{\text{whole}}\times 100.
- Penny analogy: a cent is 1/100 of a dollar, reinforcing the 100-base.
- Calculator pitfall to avoid
- Do not press the percent button as part of a calculation to obtain a percent.
- Instead, compute the ratio or difference, then write the percent symbol {
with your pen}. - Example practice: if 684 out of 1069 people support the president:
- Percent of supporters = \frac{684}{1069} \times 100 \approx 64.1\%.
- Example: polling result interpretation
- Given a poll of 1069 people, 684 support the president.
- Hence, approximately 64.1% support the president.
- Starting value vs reference value in change calculations
- Starting value (also called reference value in some texts) is the initial value.
- New value is the value after the change.
- Example: stock price from 7 to 14.
- Calculation: absolute change and percent change.
- Percentage change (change over starting value)
- If a value doubles, it increases by 100%, not 200%
- Reason: you add 100% of the starting value to the starting value.
- Example: starting value = 7, new value = 14.
- Absolute change: \Delta = 14 - 7 = 7.
- Relative change: \%\Delta = \frac{14 - 7}{7} \times 100\% = 100\%.
- Interpreting decrease and sign conventions
- When something decreases, you should either:
- use the word "decreased by" and omit the negative sign, or
- keep a negative sign and not say "decreased" explicitly.
- Do not write both a negative sign and say "decreased" (that would imply a double negative and could be confusing).
- Example: a value goes from 1000 to 300.
- Absolute change: \Delta = 300 - 1000 = -700.
- Relative change: \%\Delta = \frac{300 - 1000}{1000} \times 100\% = -70\%.
- Interpretation: the current value is 700 dollars lower, i.e., a 70% decrease from the starting value.
- Percentages for comparisons: new concept name and helpful cues
- When comparing two values, label them as:
- comparison value (the one being compared to) and
- reference value (the baseline you compare against).
- The sentence structure helps identify which is which:
- Look for a phrase like the price of the Lexus; this often marks the reference value (the value after the more-than/more than).
- The value before the word "than" is the comparison value; the value after is the reference value.
- In sentences with a percentage, you are describing a relative difference.
- In sentences with a dollar amount, you are describing an absolute difference.
- How to read and set up a comparison problem
- Example framework: average incomes in two states (e.g., Maryland and Mississippi).
- Task: How much lower is Mississippi's average income than Maryland's?
- Identify:
- Reference value (the value mentioned after the comparative phrase, usually after "than").
- Comparison value (the other state).
- Given values: Maryland = 86,700 (reference value).
- Determine which type of difference to compute:
- If the sentence contains a dollar amount, compute an absolute difference: |\text{comparison} - \text{reference}|.
- If the sentence contains a percentage, compute a relative difference: \frac{\text{comparison} - \text{reference}}{\text{reference}} \times 100\%.
- Quick rule of thumb from the transcript:
- Presence of a percentage => relative difference.
- Presence of a dollar amount => absolute difference.
- Formulas to remember (two contexts always in parallel)
- Change context (starting to new):
- Absolute change: \Delta_{abs} = \text{new} - \text{starting}.
- Relative change: \%
\Delta = \frac{\text{new} - \text{starting}}{\text{starting}} \times 100\%. - Comparison context (comparison value C vs reference value R):
- Absolute difference: \Delta_{abs} = |C - R|.
- Relative difference: \%
\Delta = \frac{C - R}{R} \times 100\%.
- Worked examples from the transcript
- Poll example
- Given: 1069 total, 684 supporters.
- Percent supporters: \%
= \frac{684}{1069} \times 100 \approx 64.1\%. - Stock price change example
- Starting value: 7; New value: 14.
- Absolute change: \Delta = 14 - 7 = 7.
- Relative change: \frac{14 - 7}{7} \times 100\% = 100\%.
- Computer value decrease example
- Initial value: 1000; New value: 300.
- Absolute change: \Delta = 300 - 1000 = -700.
- Relative change: \frac{300 - 1000}{1000} \times 100\% = -70\%.
- Interpretation: 70% decrease in value from the starting value to the current value.
- Understanding the sign and phrasing for decreases
- You can say "decreased by 70%" (preferred) or use a negative percentage, but avoid saying "decreased by -70%".
- Remainder percent example (remainder idea)
- If a value is 47.2% of a whole, the remainder is 100% − 47.2% = 52.8%.
- Example calculation: 100\% - 47.2\% = 52.8\%.
- Quick practice prompts to solidify understanding
- If a value doubles from 600 to 1200, what is the percent increase?
- \Delta = 1200 - 600 = 600, \%
\Delta = \frac{600}{600} \times 100\% = 100\%. - If a value drops from 8500 to 5100, what is the percent decrease and the absolute decrease?
- Absolute change: \Delta = 5100 - 8500 = -3400.
- Relative change: \frac{5100 - 8500}{8500} \times 100\% \approx -40\%.
- Practical tips for exam readiness
- Always identify units first (dollars vs percentages) to decide between absolute vs relative calculations.
- Use the starting/reference terminology consistently; know which one your sentence indicates.
- When reporting changes, prefer wording that avoids ambiguous signs (e.g., "increased by X%" or "decreased by Y%").
- Remember: absolute change uses a dollar amount if the sentence uses dollars; relative change uses a percentage if the sentence uses a percent.
- Summary cheat sheet
- Percent meaning: \% = \tfrac{\text{part}}{\text{whole}} \times 100\%.
- Change (starting to new): \Delta_{abs} = \text{new} - \text{starting};\quad \%
\Delta = \tfrac{\text{new} - \text{starting}}{\text{starting}} \times 100\%. - Comparison (C vs R): \Delta_{abs} = |C - R|;\quad \%
\Delta = \tfrac{C - R}{R} \times 100\%.
- Final note on mixed phrases
- The transcript ends with a brief calculation prompt hinting at a simple subtraction of percentages: 100\% - 47.2\% = 52.8\%. If a percentage remains after removing a portion, the remainder is computed similarly.