Percent Knowledge-Based Study

Study notes

1. What Is a Percent?

   The word percent comes from the Latin phrase per centum, meaning "by the hundred" or "for every hundred." In mathematics, a percent represents a ratio in which the whole is always divided into 100 equal parts.

   We use the symbol % to denote a percent. For example, 35% literally means 35 parts out of 100, which can be written as the ratio 35 : 100 or the fraction $\frac{35}{100}$.

   Key Points:

   • A percent compares a part to a total base of 100.
   • 100% represents the entire quantity (a whole).
   • Percents greater than 100% represent values greater than the original whole.

   Connecting Percents, Decimals, and Fractions

   Every percentage can be expressed as a fraction or a decimal, and vice versa. Being able to transition between these three representations is essential for solving percent problems.

   1. Converting a Percent to a Fraction or Decimal

   • To a fraction: Place the percent value over a denominator of 100 and simplify. For instance, $45\% = \frac{45}{100} = \frac{9}{20}$.
   • To a decimal: Divide by 100 by shifting the decimal point two places to the left. For instance, $45\% = 0.45$ and $7.5\% = 0.075$.

   2. Converting a Fraction or Decimal to a Percent

   • From a decimal: Multiply by 100 by shifting the decimal point two places to the right and append the % sign. For instance, $0.68 = 68\%$.
   • From a fraction: Convert the fraction so its denominator is 100, or divide the numerator by the denominator to obtain a decimal first. For instance, $\frac{3}{5} = \frac{60}{100} = 60\%$.

   Calculating the Part

   A fundamental task in percent calculations is finding a specific percentage of a given total quantity. The word "of" in these contexts indicates multiplication.

   To find $P\%$ of a number $N$, convert the percent to a fraction or decimal, then multiply:

   Part = $\frac{P}{100} \times N$  or  Part = $\text{decimal form} \times N$

   Worked Example:

   Find 15% of 240.

   1. Convert 15% to a decimal: $15\% = 0.15$.
   2. Multiply by 240: $0.15 \times 240 = 36$.

   Therefore, 15% of 240 is equal to 36.

   Solving for the Base or Rate

   In many situations, you know the part and need to determine either the percentage rate or the original total (the base).

   1. Finding the Percent (Rate)

   When given a part and a whole, calculate the percent using the proportion:

   $\text{Percent} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\%$

   Example: What percent of 80 is 28?
   $\frac{28}{80} = \frac{7}{20} = 0.35$
   $0.35 \times 100\% = 35\%$.

   2. Finding the Whole (Base)

   When given the part and the percent, calculate the whole using the formula:

   $\text{Whole} = \frac{\text{Part}}{\text{Percent in decimal form}}$

   Example: 18 is 30% of what number?
   $\text{Whole} = \frac{18}{0.30} = 60$.