Unit 4 Proportional Relationships And Percentages — Unit Plan

There are 12 inches in 1 foot, so we can say that for every 1 foot, there are 12 inches, or the ratio of feet to inches is $1:12$. We can find the unit rates by dividing the numbers in the ratio:

When the numbers in a ratio are fractions, we calculate the unit rates the same way: by dividing the numbers. For example, if someone runs $\frac34$ mile in $\frac{11}{2}$ minutes, the ratio of minutes to miles is $\frac{11}{2}:\frac34$.

If we identify two quantities in a problem and one quantity is proportional to the other, then we can calculate the constant of proportionality and use it to answer other questions about the situation. For example, Andre runs at a constant speed of 5 meters every 2 seconds. How long does it take him to run 91 meters at this rate?

To find a value in the right column, we multiply the value in the left column by $\frac25$ because $\frac25 \boldcdot 5 = 2$. This means that it takes Andre $\frac25$ of a second to run 1 meter.

At this rate, it would take Andre $\frac25 \boldcdot 91 = \frac{182}{5}$, or 36.4, seconds to walk 91 meters. More generally, if $t$ is the time it takes to walk $d$ meters at that pace, then $t = \frac25 d$.

Using the distributive property provides a shortcut for calculating the final amount in situations that involve adding or subtracting a fraction of the original amount.

For example, one day Clare runs 4 miles. The next day, she plans to run that same distance plus half as much again. How far does she plan to run the next day?

Tomorrow she will run 4 miles plus $\frac12$ of 4 miles. We can use the distributive property to find this in one step: $1 \boldcdot 4 + \frac{1}{2} \boldcdot 4 = \left(1 + \frac{1}{2}\right) \boldcdot 4$

Clare plans to run 6 miles, because $1\frac12\boldcdot 4=6$.

This works when we decrease by a fraction, too. If Tyler spent $x$ dollars on a new shirt, and Noah spent $\frac{1}{3}$ less than Tyler, then Noah spent $\frac{2}{3}x$ dollars since $x-\frac{1}{3}x=\frac{2}{3}x$.

Imagine that it takes Andre $\frac34$ more than the time it takes Jada to get to school. Then we know that Andre’s time is $1\frac34$, or 1.75, times Jada’s time. We can also describe this in terms of percentages:

Two tape diagrams. Jada's time, 4 white equal sections, labeled 100 percent.  Andre's time, 7 equal sections labeled 175 percent, 4 white and 3 blue.

We say that Andre’s time is 75% more than Jada’s time. We can also see that Andre’s time is 175% of Jada’s time. In general, the terms percent increase and percent decrease describe an increase or decrease in a quantity as a percentage of the starting amount.

For example, if there were 500 grams of cereal in the original package, then “20% more” means that 20% of 500 grams has been added to the initial amount, $500+(0.2)\boldcdot 500=600$, so there are 600 grams of cereal in the new package.

We can see that the new amount is 120% of the initial amount because $500+(0.2)\boldcdot 500 = (1 + 0.2)500.$

Tape diagram. 6 equal sections, 1 shaded blue. 120 percent labels the entire tape. 100 percent labels 5 sections. 20 percent labels the one blue shaded section.

We can use a double number line diagram to show information about percent increase and percent decrease:

Double number line, 8 evenly spaced tick marks. Top line, cereal, grams. Beginning at first tick mark, labels: 0, 100, 200, 300, 400, 500, 600, 700. Bottom line, beginning at first tick mark, labels: 0 percent, 20 percent, 40 percent, 60 percent, 80 percent, 100 percent, 120 percent, 140 percent.

The initial amount of cereal is 500 grams, which is lined up with 100% in the diagram. We can find a 20% increase by adding 20% of 500:

If the initial amount of 500 grams is decreased by 40%, we can find how much cereal there is by subtracting 40% of the 500 grams:

To solve percentage problems, we need to be clear about what corresponds to 100%. For example, suppose there are 20 students in a class, and we know this is an increase of 25% from last year. In this case, the number of students in the class last year corresponds to 100%. So the initial amount (100%) is unknown and the final amount (125%) is 20 students.

Double number line, 8 evenly spaced tick marks. Top line, number of students. Beginning at first tick mark, labels: 0, 4, 8, 12, 16, 20, 24, 28. Bottom line, beginning at first tick mark, labels: 0 percent, 25 percent, 50 percent, 75 percent, 100 percent, 125 percent, 150 percent, 175 percent. 16 and 100 percent are circled.

Looking at the double number line, if 20 students is a 25% increase from the previous year, then there were 16 students in the class last year. 

We can use equations to express percent increase and percent decrease.

For example, if $y$ is 15% more than $x$, we can represent this by using any of these equations:

$y = x + 0.15x$

$y = (1 + 0.15)x$

$y = 1.15x$

So if someone makes an investment of $x$ dollars, and its value increases by 15% to reach $1,250, then we can write the equation $1.15x =1,250$ to find the value of the initial investment.

Here is another example: if $a$ is 7% less than $b$, we can represent this by using any of these equations: 

$a = b - 0.07b$

$a = (1-0.07)b$

$a = 0.93b$

So if the amount of water in a tank decreased 7% from its starting value of $b$ to its ending value of 348 gallons, then we can write $0.93b = 348$.

Often, an equation is the most efficient way to solve a problem involving percent increase or percent decrease.

Many places have sales tax. A sales tax is an amount of money that a government agency collects on the sale of certain items. For example, a state might charge a tax on all cars purchased in the state. Often, the tax rate is given as a percentage of the cost. For example, a state’s tax rate on car sales might be 2%, which means that for every car sold in that state, the buyer has to pay a tax that is 2% of the sales price of the car.

Fractional percentages often arise when a state or city charges a sales tax on a purchase. For example, the sales tax in Arizona is 7.5%. This means that when someone buys something, they have to add 0.075 times the amount on the price tag to determine the total cost of the item.

For example, if the price tag on a T-shirt in Arizona says $11.50, then the sales tax is $(0.075) \boldcdot 11.5 = 0.8625$, which rounds to 86 cents. The customer pays $11.50 + 0.86$, or $12.36 for the shirt.

The total cost to the customer is the item price plus the sales tax. We can think of this as a percent increase. For example, in Arizona, the total cost to a customer is 107.5% of the price listed on the tag.

A tip is an amount of money that a person gives someone who provides a service. It is customary in many restaurants to give a tip to the server that is between 10% and 20% of the cost of the meal. If a person plans to leave a 15% tip on a meal, then the total cost will be 115% of the cost of the meal.

There are many everyday situations where a percentage of an amount of money is added to or subtracted from that amount in order to be paid to some other person or organization:

For example,

• If a restaurant bill is $34 and the customer pays $40, they left $6 dollars as a tip for the server. That is 18% of $34, so they left an 18% tip. From the customer’s perspective, this can be thought of as an 18% increase of the restaurant bill.
• If a realtor helps a family sell their home for $200,000 and earns a 3% commission, then the realtor makes $6,000, because $(0.03) \boldcdot 200,000 = 6,000$, and the family gets $194,000, because $200,000 - 6,000 = 194,000$. From the family's perspective, this can be thought of as a 3% decrease on the sale price of the home.

If we know the initial amount and the final amount, we can also find the percent increase or percent decrease. For example, a plant was 12 inches tall and grew to be 15 inches tall. What percent increase is this? Here are two ways to solve this problem:

Consider this new example: A rope was 2.4 meters long. Someone cut it down to 1.9 meters. What percent decrease is this? Here are two ways to solve the problem:

Percent error can be used to describe any situation where there is a correct value and an incorrect value, and we want to describe the relative difference between them. For example, if a milk carton is supposed to contain 16 fluid ounces, and it only contains 15 fluid ounces:

• The measurement error is 1 oz.
• The percent error is 6.25% because $1 \div 16 = 0.0625$.

We can also use percent error when talking about estimates. For example, a teacher estimates there are about 600 students at their school. If there are actually 625 students, then the percent error for this estimate is 4%, because $625 - 600 = 25$ and $25 \div 625 = 0.04$.

Percent error is often used to express a range of possible values. For example, if a box of cereal is guaranteed to have 750 grams of cereal, with a margin of error of less than 5%, what are possible values for the actual number of grams of cereal in the box? The error could be as large as $(0.05) \boldcdot 750 = 37.5$ and could be either above or below the correct amount.

Tape diagram. One longer white section and 2 smaller blue sections both labeled 37 point 5 and 5 percent. Long white section and one blue section are labeled 750.

Therefore, the box can have anywhere between 712.5 and 787.5 grams of cereal in it, but it should not have 700 grams or 800 grams, because both of those are more than 37.5 grams away from 750 grams.