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# Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

## Know the difference

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

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

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

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}