Ratios, Rates & Percent Knowledge-Based Study

Study notes

1. Understanding Percents as Rates per Hundred

   A percent represents a ratio comparing a quantity to 100. In real-world contexts, percent problems involve three main values: the base amount (the whole or original value), the percent rate, and the resulting part.

   To solve percent applications, convert the percent into a decimal or fraction before performing calculations:

   • Convert a percent to a decimal by dividing by 100: $r = \frac{P}{100}$. For example, 7.5% becomes 0.075.
   • Calculate the part using the equation: $\text{Part} = \text{Base} \times \text{Rate}$.

   Recognizing whether a scenario adds to or subtracts from the original base amount is the first step in determining the correct mathematical model.

   Calculating Percent Increases

   In retail and service settings, taxes, tips, and markups increase the original price.

   • Sales Tax: A percentage levied by a government on goods or services.
   • Tip (Gratuity): An additional percentage paid for a service, usually calculated from the pre-tax bill.
   • Markup: The amount a store adds to the wholesale cost to establish the retail selling price.

   For an original amount $B$ and an increase rate $r$, the final amount $A$ can be determined in two equivalent ways:

   • Method 1 (Two-step): Calculate the increase $I = B \times r$, then find $A = B + I$.
   • Method 2 (One-step): Multiply directly by the growth factor: $A = B(1 + r)$.

   Example: A restaurant bill is $48.00. An 18% tip on this bill is calculated as $48 \times 0.18 = 8.64$. The total amount paid is $48 + 8.64 = 56.64$ dollars.

   Calculating Percent Decreases

   A discount or markdown reduces the original price of an item. The amount subtracted represents the total savings.

   For an original price $P$ and a discount rate $d$:

   • The discount amount is $\text{Discount} = P \times d$.
   • The sale price $S$ is given by: $S = P - \text{Discount}$, or directly by $S = P(1 - d)$.

   Example: A jacket originally priced at $75.00 is on sale for 30% off. The discount is $75 \times 0.30 = 22.50$ dollars. The sale price is $75 \times (1 - 0.30) = 75 \times 0.70 = 52.50$ dollars.

   The Simple Interest Formula

   Simple interest is the fee paid for borrowing money or the return earned on invested funds over time. It is calculated exclusively on the original principal amount.

   The relationship is modeled by the formula:

   $I = P \cdot r \cdot t$

   • $I$ (Interest): The dollar amount earned or owed.
   • $P$ (Principal): The initial amount of money deposited or borrowed.
   • $r$ (Annual Interest Rate): The annual rate expressed as a decimal.
   • $t$ (Time): The duration in years. If time is given in months, express it as a fraction of a year: $t = \frac{\text{months}}{12}$.

   The total balance $A$ at the end of the term is the sum of the principal and the interest earned: $A = P + I = P(1 + rt)$.

   Measuring Accuracy with Percent Error

   Percent error quantifies the difference between an estimated or measured value and an accepted, exact value, expressed as a percentage of the exact value.

   The percent error formula is:

   $\text{Percent Error} = \frac{|\text{Experimental Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100\%$

   • The numerator uses absolute value because error is reported as a non-negative magnitude of deviation.
   • The denominator must always be the true, accepted, or theoretical value, not the estimate.

   Example: A student estimates the length of a desk to be 114 cm. The actual measured length is 120 cm. The error is $|114 - 120| = 6$ cm. The percent error is $\frac{6}{120} \times 100\% = 0.05 \times 100\% = 5\%$.