Examples

In a survey, 243 out of 400 people state that they like dogs. What percent is this?

Answer: [latex]\displaystyle\frac{243}{400}=0.6075=\frac{60.75}{100}[/latex] This is 60.75%. Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.

 

Example

Write each as a percent:

1. [latex]\displaystyle\frac{1}{4}[/latex]
2. 0.02
3. 2.35

Answer:

1. [latex]\displaystyle\frac{1}{4}=0.25[/latex] = 25%
2. 0.02 = 2%
3. 2.35 = 235%

Example

The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?

Answer: To compute the percent change, we first need to find the dollar value change: $6800 – $7400 = –$600. Often we will take the absolute value of this amount, which is called the absolute change: |–600| = 600. Since we are computing the decrease relative to the starting value, we compute this percent out of $7400: [latex]\displaystyle\frac{600}{7400}=0.081=[/latex] 8.1% decrease. This is called a relative change.

Example

There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.

Answer: When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 – 75 = 140. From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base: Using QFC as the base, [latex]\displaystyle\frac{140}{75}=1.867[/latex]. This tells us Albertsons is 186.7% larger than QFC. Using Albertsons as the base,[latex]\displaystyle\frac{140}{215}=0.651[/latex]. This tells us QFC is 65.1% smaller than Albertsons. Notice both of these are showing percent differences. We could also calculate the size of Albertsons relative to QFC:[latex]\displaystyle\frac{215}{75}=2.867[/latex], which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons:[latex]\displaystyle\frac{75}{215}=0.349[/latex], which tells us that QFC is 34.9% of the size of Albertsons.

 

Example

Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?

Answer: To answer this question, suppose the value started at $100. After one week, the value dropped by 60%: $100 – $100(0.60) = $100 – $60 = $40. In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase: $40 + $40(0.75) = $40 + $30 = $70. In the end, the stock is still $30 lower, or [latex]\displaystyle\frac{\$30}{100}[/latex] = 30% lower, valued than it started.

A video walk-through of this example can be seen here. https://youtu.be/4HNxwYMTNl8

Example

Suppose you’re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room?

Answer: In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms: [latex-display]\displaystyle\frac{100\text{ tiles}}{100\text{ft}^2}=\frac{n\text{ tiles}}{400\text{ft}^2}[/latex-display]

Example

Suppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend $10,000?

Answer: While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a fixed number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers.

Matters of scale in this example and the previous one are explained in more detail here. https://youtu.be/-e2typcrhLE

Examples

The 2010 U.S. military budget was $683.7 billion. To gain perspective on how much money this is, answer the following questions.

1. What would the salary of each of the 1.4 million Walmart employees in the US be if the military budget were distributed evenly amongst them?
2. If you distributed the military budget of 2010 evenly amongst the 300 million people who live in the US, how much money would you give to each person?
3. If you converted the US budget into $100 bills, how long would it take you to count it out - assume it takes one second to count one $100 bill.

Answer: Here we have a very large number, about $683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities.

1. If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000.
2. There are about 300 million people in the U.S. The military budget is about $2,200 per person.
3. If you were to put $683.7 billion in $100 bills, and count out 1 per second, it would take 216 years to finish counting it.

 

Example

Compare the electricity consumption per capita in China to the rate in Japan.

Answer: To address this question, we will first need data. From the CIA[footnote]https://www.cia.gov/library/publications/the-world-factbook/rankorder/2042rank.html[/footnote] website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries.   From the World Bank,[footnote]http://data.worldbank.org/indicator/SP.POP.TOTL[/footnote] we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million. Computing the consumption per capita for each country: China: [latex]\displaystyle\frac{4,693,000,000,000\text{KWH}}{1,344,130,000\text{ people}}[/latex] ≈ 3491.5 KWH per person Japan: [latex]\displaystyle\frac{859,700,000,000\text{KWH}}{127,817,277\text{ people}}[/latex] ≈ 6726 KWH per person While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China.

Working with large numbers is examined in more detail in this video. https://youtu.be/rCLh8ZvSQr8

Example

Guided Example

Mary was out in the yard one day and had her two daughters with her. She was doing some renovations and wanted to know how tall the house was. She noticed a shadow 3 feet long when her daughter was standing 12 feet from the house and used it to set up figure 1. We can take that drawing and separate the two triangles as follows allowing us to focus on the numbers and the shapes. These triangles are what are called similar triangles. They have the same angles and sides in proportion to each other. We can use that information to determine the height of the house as seen in figure 2. To determine the height of the house, we set up the following proportion: [latex-display]\displaystyle\frac{x}{15}=\frac{5}{3}\\[/latex-display] Then, we solve for the unknown x by using cross products as we have done before: [latex-display]\displaystyle{x}=\frac{5\times{15}}{3}=\frac{75}{3}=25\\[/latex-display] Therefore, we can conclude that the house is 25 feet high.