Proportion and Percents

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%.