Ratios, Rates and Percentages

To check if two ratios are equivalent, we can compare their fractions. If the fractions are equivalent, then the ratios are equivalent.

To share an amount proportionally to given numbers: count the parts, find the value of one part, then multiply.
Example. Share \(\textcolor{colorprop}{100}\) dollars between two people proportionally to their ages, \(\textcolor{colordef}{20}\) and \(\textcolor{colordef}{30}\) years: $$\begin{aligned}[t] \text{Number of parts} &= \textcolor{colordef}{20} + \textcolor{colordef}{30} = \textcolor{colordef}{50} && \text{(add the numbers)} \\ \text{Value of one part} &= \textcolor{colorprop}{100} \div \textcolor{colordef}{50} = \textcolor{olive}{2} && \text{(divide the amount)} \\ \text{Shares} &= \textcolor{colordef}{20} \times \textcolor{olive}{2} = \textcolor{colorprop}{40} \text{ and } \textcolor{colordef}{30} \times \textcolor{olive}{2} = \textcolor{colorprop}{60} && \text{(multiply)} \end{aligned}$$ The shares \(\textcolor{colorprop}{40}\) and \(\textcolor{colorprop}{60}\) are in the ratio \(20:30 = 2:3\).

If \(\textcolor{colordef}{5}\) apples cost \(\dollar\textcolor{colorprop}{10}\), how much do \(\textcolor{colordef}{8}\) apples cost?

• Step 1: Find the cost of 1 apple (the unit cost).
  Divide the total cost by the number of apples. One apple costs \(\dollar\textcolor{colorprop}{2}\).
• Step 2: Find the cost of 8 apples.
  Multiply the unit cost by the number of apples you want. So, \(\textcolor{colordef}{8}\) apples cost \(\dollar\textcolor{colorprop}{16}\).
• 

If \(\textcolor{colordef}{5}\) apples cost \(\dollar\textcolor{colorprop}{10}\), how much do \(\textcolor{colordef}{8}\) apples cost?

• Step 1: Set up the proportion. Let \(x\) be the unknown cost. \(\dfrac{\dollar\textcolor{colorprop}{10}}{\textcolor{colordef}{5} \text{ apples}} = \dfrac{\dollar\textcolor{colorprop}{x}}{\textcolor{colordef}{8} \text{ apples}}\)
• Step 2: Cross-multiply and solve for \(x\). Multiply the numbers that are diagonal to each other.
• So, \(\textcolor{colordef}{8}\) apples cost \(\dollar\textcolor{colorprop}{16}\).

To convert a percentage to a fraction, write it as a fraction over 100, then simplify if possible.

To convert a fraction like \(\dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}}\) to a percentage, you have two common methods:

• Method 1: Equal Fractions. Find an equal fraction with denominator \(100\) (when possible, in particular when the denominator divides \(100\); it is not possible for \(\dfrac{1}{3}\)).
• Method 2: Multiply by \(100\pourcent\). This works because multiplying by \(100\pourcent\) is the same as multiplying by 1. $$ \begin{aligned} \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} &= 0.75 \quad (\text{since } 3 \div 4 = 0.75) \\ &= 0.75 \times \textcolor{colordef}{100\pourcent} \\ &= \textcolor{colorprop}{75}\textcolor{colordef}{\pourcent} \end{aligned} $$

To convert a percentage to a decimal, divide by 100. A quick way to do this is to move the decimal point two places to the left. $$ \textcolor{colorprop}{45}\textcolor{colordef}{\pourcent} = \textcolor{colorprop}{45} \div \textcolor{colordef}{100} = 0.45 $$

To convert a decimal to a percentage, multiply by 100. A quick way to do this is to move the decimal point two places to the right and add the percent sign (\(\pourcent\)). $$ 0.68 = 0.68 \times 100\pourcent = \textcolor{colorprop}{68}\textcolor{colordef}{\pourcent} $$

To find a part of a total, multiply the percentage by the whole. $$\textcolor{colorprop}{\text{Part}} = \text{Percentage} \times \textcolor{colordef}{\text{Whole}}$$ Taking \(x\pourcent\) of a number means taking the fraction \(\dfrac{x}{100}\) of it. Remember to convert the percentage to a decimal or fraction before calculating.

1. Calculate the change amount: $$\text{Change} = \text{Percentage} \times \text{Original Value}$$
2. Calculate the new value:
   • For an increase: $$\text{New Value} = \text{Original Value} + \text{Change}$$
   • For a decrease: $$\text{New Value} = \text{Original Value} - \text{Change}$$

When the starting value and the final value are both known, the starting value being strictly positive:

• compute the change: final value \(-\) starting value;
• compare that change to the starting value, as a percentage:

$$\text{percentage change} = \frac{\text{final value} - \text{starting value}}{\text{starting value}} \times 100\pourcent.$$ The result comes out positive for an increase and negative for a decrease: the sign gives the direction (\(+12\pourcent\), \(-25\pourcent\)), and the answer is stated with the word that goes with it — “a decrease of \(25\pourcent\)”.