Introduction

Using knowledge of equivalent fractions and cross product to solve for unknown values, students can use ratios and proportions to solve algebra and geometry problems.

Ratios and Proportions

A ratio is the comparision of amounts, and a proportion is two equal ratios.

Ratios can be written using three different formats:

  • $a : b$
  • $a~\text{to}~b$
  • $\frac{a}{b}$

A proportion can be verified using cross product. Also, use cross product to solve for unknown values.

$\begin{array}{rcl} \frac{a}{b}&=&\frac{c}{d}\\ b\times c&=&a\times d \end{array}$

Solving Proportions Example

For this problem, use the given information to set up a proportion, and then use cross product to solve for the unknown value.

Rosita makes gift baskets to give to party guests. For every 3 pieces of chocolate she includes in the basket, she also includes 5 pieces of caramel. If she includes 10 caramels, how many chocolates will she add to the basket?

$\begin{array}{rcl} \frac{3}{5}&=&\frac{x}{10}\\ 3\times 10&=&5\times x\\ 30&=&5x\\ x&=&6 \end{array}$

Percent, Decimals, and Fractions

For rational numbers, percent, decimals, and fractions are simply different formats to represent the same value.

Examples:

$\begin{array}{rcccl} 25\%&=&0.25&=&\frac{1}{4}\\ 33\%&=&0.33&=&\frac{1}{3}\\ 50 \%&=&0.50&=&\frac{1}{2} \end{array}$

Percent Equations

What is the percent of a number? There’s more than one strategy to solve this problem.

What is 20% of 1,860? To solve, change the percent to a decimal and multiply. Another strategy is to make a proportion, and then use cross product to calculate the unknown value.

$20\%=0.20$

$0.20\times1,860=372$

$\begin{array}{rcl} \frac{20}{100}&=&\frac{x}{1,860}\\ 100\times x&=&20\times1,860\\ 100x&=&37,200\\ x&=&372 \end{array}$

Multi-Step Word Problems with Percent

Sometimes more than one step is required to solve a percent problem.

For this problem, first calculate the discounted price then add the tax. A computer costing $750.00 is on sale for 25% off and sales tax is 8%. What is the total price to purchase the computer? Before calculating, change all percents to decimals.

$\begin{align} 20\%&=0.20\\ 8\%&=0.08 \end{align}$

$\begin{align} 750-(0.20\times750)&= ?\\ 750-187.5&=562.50\\ 562.50+(0.08\times562.50)&=?\\ 562.50+45&=607.5 \end{align}$

Here’s another strategy to solve the problem. This method uses less steps but results in the same answer.

$\begin{align} 100\% -25\%&=75\%&=0.75\\ 100\% + 8\%&= 108\%&=1.08 \end{align}$

$\begin{align} 0.75\times750&= 562.50\\ 1.08\times562.50&=607.50 \end{align}$

Percent Change

To calculate the change of a percent, follow a series of steps.

  • Calculate the absolute value of the change
  • Divide by the orginal number
  • Convert the decimal answer to a percent

Follow the steps to determine the percent of change. Use common sense to determine if the change is an increase or a decrease.

Example 1: From February to March, the average temperature increased from 35 degrees to 47 degrees. What is the percent of change?

$\begin{align} |35-47| &= 12\\ 12\div35&=0.34\\ 0.34&=34\% \end{align}$

The percent of change is a 34% increase.

Example 2: Attendance at a high school football game was 3,288 people for the first game and 2,686 people for the second game. What is the percent of change?

$\begin{align} |3,288-2,686| &= 602\\ 602\div3,288&=0.183\\ 0.183&=18.3\% \end{align}$

Attendance at the second game decreased by 18.3%.