Percentages: Fundamentals and Applications

Percentages: Fundamentals and Applications

• A percentage is a portion of a whole.

• Percentages appear in daily life in several ways:

  • Percent off a total (sale discounts, e.g., 15% off).
  • Percentage increases (e.g., salary increased by 10%).
  • Percentage decreases (e.g., setting a goal to cut spending by a certain percent).

• The first step in any percentage problem: determine what the question is asking for.

• Key representations of percentages:

  • 100% represents the whole.
  • 100% can be written by moving the decimal two places left: $100\% = 1$.
  • Common percentages in decimal form:
  • $100\% = 1$
  • $25\% = 0.25$
  • $75\% = 0.75$
  • $120\% = 1.20$
  • $80\% = 0.80$

• Pictorial representation (rectangle) to model a total (the whole, 100%):

  • When the whole is 100%, the entire rectangle represents the total.
  • To show 25% of the total, divide the rectangle into four equal sections (since $25\% \times 4 = 100\%$), with each section representing $25\%$ of the total.
  • Removing 25% corresponds to removing one section, leaving $75\%$ of the total, i.e., $0.75 \times \text{Total}$.
  • To find 25% of a total: use $0.25 \times \text{Total}$.

• Examples with percentages on the rectangle:

  • Increasing by a percent:
  • If you increase by 20%, then $100\% + 20\% = 120\%$ of the total.
  • In decimal form: $120\% = 1.2 \times \text{Total}$.
  • Decreasing by a percent:
  • If you decrease by 20%, then $100\% - 20\% = 80\%$ of the total.
  • In decimal form: $80\% = 0.8 \times \text{Total}$.

• Real-world context: Brazil and the 2014 World Cup planning (brief excerpt from transcript):

  • Brazil, known for soccer fervor, hosts the 2014 World Cup; discussions about money spent and readiness.
  • Arena Pernambuco (outside Recife) used as an example stadium under consideration for hosting games.
  • A local club game on the night of the visit attracted 6,800 fans, leaving the stadium about 85% empty (i.e., 15% filled).

• Problem setup: find the total capacity (total number of seats) of the arena.

  • Given: 6,800 fans attended, and this represents 15% of the total capacity.
  • Therefore, 15% of the total equals 6,800, and 85% would be the remaining portion (the portion not attended).
  • Let the total capacity be denoted by a variable (e.g., $T$).

• Two methods to solve the stadium problem:

  • Method 1: Proportions (cross-multiplication)\
  • Relationship observed in the transcript: 15% corresponds to 6,800, and 85% corresponds to the remaining portion.
  • Set up the proportion (as described):
    • $\frac{0.15}{0.85} = \frac{6{,}800}{T}$
    • Cross-multiplying gives: $0.15 \times T = 6{,}800 \times 0.85$
    • Solving for $T$ yields approximately $T \approx 38{,}533$.
    • Interpreting the result: 85% of this stadium would be about $38{,}533\$\,\text{seats}$.
    • Note: This method is described in the transcript, but the final interpretation shows that the total should be larger because 6,800 is only 15% of the total.
    • The transcript then points out that the total seats should be the 6,800 plus the 85% portion, giving a total of roughly $6{,}800 + 38{,}533 = 45{,}333$ seats.
  • Method 2: Direct equation (simpler, as described in the transcript):\
  • Set up the equation for the total $T$ directly from the statement that 15% of the total equals 6,800:
    • $0.15 \times T = 6{,}800$
    • Solve: $T = \frac{6{,}800}{0.15} = 45{,}333.$
  • This directly gives the total capacity (no need to separate 85% as a partial value).

• Results and interpretation from the stadium problem:

  • Using the equation approach: $T = 45{,}333$ seats.
  • Sanity check:
  • 15% of 45,333 is 6,800 seats (since $45{,}333 \times 0.15 = 6{,}800$).
  • Therefore, 85% of 45,333 is $0.85 \times 45{,}333 \approx 38{,}533$, which matches the earlier computation for the 85% portion.
  • The total capacity of the arena is therefore $45{,}333$ seats, and the observed attendance of 6,800 corresponds to the 15% occupied portion.

• Practical implications and reflections:

  • The problem illustrates how percentages can relate to real-world capacity and utilization questions (e.g., stadium planning, event attendance, occupancy rates).
  • It highlights the importance of correctly identifying what percentage of what quantity you are dealing with (the “total” vs. a portion of the total).
  • Rounding and interpretation matter in real-world contexts; small rounding differences can occur, but the core relationships remain clear: if 15% corresponds to a value, the total is that value divided by 0.15.

• Connections to foundational principles and real-world relevance:

  • Percentages are a way of expressing ratios with a common base (the whole).
  • Converting between percentages and decimals is a fundamental skill for solving practical problems (discounts, increases/decreases, budgeting).
  • The two-method approach demonstrates flexibility in problem solving: using a direct equation is often simplest, while proportional reasoning can provide a check or intuition.

• Ethical and practical considerations:

  • When planning large events, accurate capacity calculations affect safety, logistics, and financial planning.
  • Public communication about attendance and capacity should be precise to avoid misinterpretation.

• Summary of key formulas and steps in this lesson:

  • Percent to decimal conversions:
  • $100\% = 1$, $25\% = 0.25$, $75\% = 0.75$, $120\% = 1.20$, $80\% = 0.80$
  • Percentage of a total: $\text{Portion} = \text{Decimal} \times \text{Total}$
  • Increase by a percent: $1.0 + p$ where $p$ is the percent as decimal, so e.g., $1.0 + 0.20 = 1.20\;\Rightarrow\; \text{new total} = 1.20 \times \text{Total}$
  • Decrease by a percent: $1.0 - p$ where $p$ is the percent as decimal, so e.g., $0.80 = 1.0 - 0.20\;\Rightarrow\; \text{new total} = 0.80 \times \text{Total}$
  • Proportion approach (example): if $6{,}800$ seats represent 15% of total $T$, then $0.15 T = 6{,}800$ and $T = \frac{6{,}800}{0.15} = 45{,}333$
  • Direct equation approach (example): to find total when 15% equals 6,800, solve $0.15 \times T = 6{,}800\Rightarrow T = \frac{6{,}800}{0.15} = 45{,}333$

• Quick practice prompts (to reinforce concepts):

  • If a shirt is 25% off and the original price is $P$, what is the sale price? (Answer: $P \times (1 - 0.25) = 0.75 P$)
  • If a population increases by 10% from a baseline of $N$, what is the new population? (Answer: $1.10 \times N$)
  • If a budget decreases by 12%, what is the new budget? (Answer: $0.88 \times ext{Original Budget}$)