Comprehensive Study Notes on Percentages and Promilles

Preliminary Mathematical Knowledge

• Fractions and Their Properties:

  • A fraction is a number in the form $\frac{a}{b}$, where $a$ is the numerator (števec) and $b$ is the denominator (imenovalec).
  • The denominator indicates how many equal parts the whole is divided into.
  • The numerator indicates how many of those parts are taken.
  • The fraction bar represents division.

• Expanding and Simplifying Fractions:

  • Expanding (Razširjanje): We expand a fraction by multiplying both the numerator and the denominator by the same non-zero number.
    • Example: $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
  • Simplifying (Krajšanje): We simplify a fraction by dividing both the numerator and the denominator by the same non-zero number.
    • Example: $\frac{8}{12} = \frac{8 : 4}{12 : 4} = \frac{2}{3}$

• Calculating Parts of a Whole:

  • To calculate a part (e.g., $\frac{2}{3}$ of $600$), divide the whole by the denominator and multiply by the numerator:
    • $(600 : 3) \times 2 = 200 \times 2 = 400$

• Converting Fractions to Decimals:

  • Fractions with denominators that are powers of ten ($10$, $100$, $1000$, etc.) are easily written as decimals:
    • $\frac{1}{10} = 0,1$
    • $\frac{1}{100} = 0,01$
    • $\frac{1}{1000} = 0,001$
  • Any fraction can be converted into a decimal by dividing the numerator by the denominator:
    • Example: $\frac{7}{5} = 7 : 5 = 1,4$

Definition and Logic of Percentages

• Etymology and Definition:

  • The word "percent" or "procent" comes from the Latin phrase "PRO CENTO", meaning "of a hundred" (od sto).
  • The symbol for percent is $\%$.
  • A percentage is simply a fraction with a denominator of $100$.

• Core Concepts:

  • A percent ($p\%$) is equal to one-hundredth of a whole: $p\%$.
  • $1\%$
  • $100\%$

• The Percentage Formula:

  • Percentages express a share or part ($d$) of a whole ($C$).
  • $p = \frac{d}{C}$, where $d$ is the numerator (part of the whole) and $C$ is the denominator (the whole).

Pharmaceutical Case Study: Comparing Success Rates

A company tested two products, A and B. To determine which product is better, we must calculate the share of satisfied users relative to the total participants.

• Product A:

  • Data: $36$ satisfied out of $50$ total users.
  • Calculation: $\frac{36}{50} = \frac{36 \times 2}{50 \times 2} = \frac{72}{100} = 72\%$

• Product B:

  • Data: $45$ satisfied out of $75$ total users.
  • Calculation (Simplify then Expand): $\frac{45}{75} = \frac{45 : 3}{75 : 3} = \frac{15}{25} = \frac{15 \times 4}{25 \times 4} = \frac{60}{100} = 60\%$

• Conclusion: Product A is more successful because $72\%$.

Conversions and Periodic Decimals

• Common Conversions to Remember:

  • $\frac{1}{10} = 10\%$
  • $\frac{1}{5} = 20\%$
  • $\frac{1}{4} = 25\%$
  • $\frac{2}{5} = 40\%$
  • $\frac{1}{2} = 50\%$
  • $\frac{3}{5} = 60\%$
  • $\frac{3}{4} = 75\%$
  • $\frac{4}{5} = 80\%$
  • $1 = 100\%$

• Periodic Numbers and Rounding:

  • If a fraction results in a repeating (periodic) decimal, the percentage is expressed as an approximation.
  • Example 1: $\frac{5}{9} = 5 : 9 = 0,555... \approx 0,56 = 56\%$ (Rounded to the nearest whole number).
  • Example 2: $\frac{2}{7} = 2 : 7 = 0,28571... \approx 0,286 = 28,6\%$ (Rounded to one decimal place).
  • Example 3 (Exact): $\frac{3}{8} = 3 : 8 = 0,375 = 37,5\%$

Concept of Promille (Permille)

• Definition:

  • For very small shares, a denominator of $1000$ is used.
  • The term is promil (permille), from Latin "pro mille," meaning "of a thousand."
  • The symbol is $\text{‰}$.

• Formulas:

  • $5\text{‰} = \frac{5}{1000} = 0,005$
  • Examples of permille calculations:
    • $\frac{2}{1000} = 2\text{‰}$
    • $\frac{35}{1000} = 35\text{‰}$
    • $\frac{42}{300} = \frac{42 : 3}{300 : 3} = \frac{14}{100} = 140\text{‰}$

Practical Word Problems and Real-World Applications

• Germination Rates:

  • If $46$ out of $50$ beans germinate, what percentage did NOT germinate?
  • Method 1: $50 - 46 = 4$ beans did not germinate. $\frac{4}{50} = \frac{8}{100} = 8\%$.
  • Method 2: Germinated share is $\frac{46}{50} = 92\%$. Non-germinated is $100\%$.

• Food Composition:

  • In a $300\,g$ chocolate bar, there are $75\,g$ of cocoa.
  • Share of cocoa: $\frac{75}{300} = \frac{1}{4} = 25\%$.

• Sporting Statistics:

  • Handball: If Velenje scored $60\%$ of the goals in a match against Laško, Velenje won.
  • Top Scorers: If a team scored $20$ goals and the top two scorers (Miha with $6$ and Gregor with $5$) together scored $11$ goals, their combined percentage is $\frac{11}{20} = \frac{55}{100} = 55\%$.

• Geometry and Area:

  • Given a rectangle with length $5\,cm$ and width $4\,cm$ (Area = $20\,cm^2$).
  • To color $40\%$ of the rectangle: $\frac{40}{100} \times 20 = 0,4 \times 20 = 8\,cm^2$.

• Liquid Mixtures:

  • A syrup mixture contains $72\,dl$ of syrup in $360\,dl$ of total liquid.
  • Syrup Share: $\frac{72}{360} = \frac{1}{5} = 20\%$.
  • Water Share: $100\%$.

Interactive Questions and Scenarios

• Q: Friends share a bill, each paying $20\%$. How many girls were there?
  • A: $100\%$. There were $5$ girls.
• Q: $15\%$ of eggs are broken. What percentage are whole?
  • A: $100\%$.
• Q: A book is $75\%$ written. How much is empty?
  • A: $25\%$.
• Q: Milk contains $3,5\%$ milk fat. What does this mean?
  • A: It means that in every $100$ units (e.g., grams) of milk, there are $3,5$ units of fat.
• Q: Voter turnout was $45\%$. What does this mean?
  • A: Less than half of the eligible voters cast a ballot.

Permille Applications (Specific Examples)

• Disease Prevalence: $15$ people out of $1000$ are infected with a contagious disease.
  • Expression: $15\text{‰}$.
• Water Filtration: From $500\,dm^3$ of water, $2\,dm^3$ of impurities were collected.
  • Calculation: $\frac{2}{500} = \frac{4}{1000} = 4\text{‰}$.
• Breath Alcohol: $50\,ml$ of alcohol per liter ($1000\,ml$) of exhaled air.
  • Expression: $50\text{‰}$.