You will be encountering ratios, rates, and proportions in various contexts. It is to know the differences between them.
The ratio of \(a\) to \(b\) is expressed by the fraction \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator, or by the colon notation \(a:b\) |
When a ratio is used to compare two different kinds of measures, it is called a rate. A unit rate, is the ratio comparing units. |
When two ratios are compared, it is called a proportion. |
Ratios compare the same unit to each other.
Examples:
1. Find the fraction and colon notation for the ratio 3.8 to 7.4.
Solution:
Fraction notation:
\[\frac{3.8}{7.4}\]
multiply numerator and denominator by 10
\[\frac{38}{74}\]
divide numerator and denominator by 2
\[\frac{19}{37}\]
Colon notation: \(19:37\)
2. Ling and Taylor took 12 days and 2 weeks respectively to complete the task. Find the ratio of Ling to Taylor in terms of time it took to complete the task.
Solution:
For ratios, we have to convert them to the same unit. \(2\) weeks equals to \(14\) days
\[\frac{12}{14}=\frac{6}{7}\]
Rates are used to compare different units.
Examples:
1. An airplane traveled 2514 kilometers in 3 hours. What was the rate per hour (or speed)?
Solution:
\[\frac{2514\,km}{3\,hrs}=\frac{838\,km}{1\,hr}\]
The airplane was travelling at \(838\) km/hr.
2. The heart of an elephant, at rest, beats an average of 1500 beats in 60 minutes. What is the rate in beats per minute?
Solution:
\[\frac{1500\,beats}{60\,minutes}=\frac{25\,beats}{1\,minute}\]
The elephant at rest has a heart rate of was travelling at \(25\) beats/minute.
Proportions compare the equality of two ratios or fractions. There may be an unknown variable (e.g., \(x\)) to solve in a proportion.
Examples: Solve for the unknown variable in each proportion
\[\frac{6.3}{0.9}=\frac{0.7}{n}\]
To solve the proportion, cross multiply (multiply the denominator to the other side of the equation).
\begin{align} \frac{6.3}{0.9}\times 0.9&=\frac{0.7}{n}\times n \\ 6.3\times n&=0.7\times 0.9 \end{align}
Now solve for the unknown by dividing the coefficient (number in front of the variable) back to the other side of the equation.
\begin{align} \frac{6.3\times n}{6.3}&=\frac{0.7\times 0.9}{6.3} \\ n&= \frac{0.7\times 0.9}{6.3} \\ n &=0.1\end{align}
\[\frac{y}{\frac{3}{5}}=\frac{\frac{7}{12}}{\frac{14}{15}}\]
cross multiply
\begin{align} \frac{y}{\frac{3}{5}} \times \frac{14}{15}&=\frac{\frac{7}{12}}{\frac{14}{15}}\times \frac{3}{5} \\ y\times \frac{14}{15}&=\frac{7}{12}\times \frac{3}{5} \end{align}
Now solve for the unknown y by dividing the coefficient (number in front of the variable) back to the other side of the equation.
\begin{align} \frac{y\times \frac{14}{15}}{\frac{14}{15}}&=\frac{\frac{7}{12}\times \frac{3}{5}}{\frac{14}{15}} \\y&=\frac{\frac{7}{12}\times \frac{3}{5}}{\frac{14}{15}} \\ y &=\frac{3}{8}\end{align}
A serving of fish steak (cross section) is generally \(\frac{1}{2}\) lb. How many servings can be prepared from a cleaned \(18\frac{3}{4}\)-lb tuna?
Solution:
\begin{align}\frac{1\,serving}{\frac{1}{2}\,lb}=\frac{x\,servings}{18\frac{3}{4}lb}\\ 37\frac{1}{2}\,servings=x\end{align}