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Unit 8: Relationships Among Numbers

8 Introduction

In this unit, we will look at other ways numbers can be expressed. Whether you are shopping around for the best deal, or reading an article online, understanding percentages and ratios can be useful in your everyday life.

Practice Quiz

Check your understanding of percentages and related calculator skills before you start, by giving the Unit 8 pre quiz a try, then use the feedback to help you plan your study.

The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend most time on. Remember, it doesn’t matter if you get some or even all of the questions wrong—it just indicates how much time you may need for this unit!

Click here for the pre quiz.

8.0.1 What to Expect in this Unit

This unit should take around ten hours to complete. In this unit you will learn about:

  • Understanding percentages.
  • How percentages are related to fractions and decimals.
  • Calculating with percentages by hand and with the calculator.
  • Ratios.
  • Percentages and ratios in everyday life.

In Section 1, percentages and related calculator skills will be explored. Depending on your comfort level, this section might require you to spend extra time. Sale prices, restaurant tipping, and online shopping are a few of the examples you will look at.

In Section 2, you will study ratios. Hopefully, you will find the content of this section to be familiar as you apply ratios to decide whether you can afford to buy a hot tub.

In Section 3, you will have the opportunity explore these topics at a deeper level. Percent change and the Golden Ratio are two of the topics you will discover.

Section 4 provides a self-check, so that you can consider the development of your skills. Remember that your confidence will continue to increase the more you practice. You can do this by working through exercises on purchasing a flat-screen television and baking cookies!

8.1 Percentages

Introduction

Articles you read and newscasts you may watch or listen to often mention the results of surveys and studies in percents. Literally, percent means per one hundred. So 25% (which is read as “twenty-five percent”) means “25 out of one hundred.”

One of the reasons percentages are useful is that you can compare them easily. For example, if a restaurant noted that 42% of customers opted for a meat dish, 35% chose a vegetarian dish, and 23% ordered fish, you can quickly conclude that the meat dish was most popular.

Note: The preferences of the customers in the example above will add up to 100; in this case 42 percent prefix plus of 35 percent prefix plus of 23 percent equals 100 percent.

8.1.1 Writing a Percentage as a Fraction or Decimal

Fractions, decimals, and percentages are all interchangeable, so you can choose to use whichever is most appropriate for your situation. From the definition of a percentage, 75% means 75 divided by 100, which can, as you know, be reduced to three divided by four. Alternatively, 75 divided by 100 means 75 division 100, or 0.75. So, 75 percent equation sequence equals 75 divided by 100 equals three divided by four equals 0.75.

You can see from the pizzas below that dividing them into one divided by four and three divided by four, or 25% and 75%, 0.25 and 0.75 always amounts to the same thing.

Summary

To change a percentage to a fraction or decimal, interpret the percent symbol to mean “divided by 100.” You then have a choice to display your result as a fraction or decimal.

Since dividing a number by 100 has the same effect as moving its decimal point two places to the left, you have a valuable shortcut when you turn a percent into a decimal. [ Looking for the decimal point in a whole number? Remember, the decimal point can be inserted at the end of your number behind the ones place. ]

For example: 6% = .

8.1.2 Writing a Decimal as a Percentage

To turn a decimal back into a percentage, you can move the decimal point two places in the opposite direction to the right.

As an example: 1.245 equals 124.5 percent.

Let’s take a closer look—“percent” means “over 100,” so 100% is actually an equivalent way of expressing 100 divided by 100, which is the number 1. Multiplying by 100% does not change the value of a number; it just makes it look different.

Example

If you’d like, you can show the conversion with one or both of the following steps:

For example: 0.723 equals 0.723 multiplication 100 percent equation left hand side equals right hand side open 0.723 multiplication 100 close percent equals 72.3 percent.

8.1.3 Writing a Fraction as a Percentage

Using the same idea that multiplication by 100% means multiplication by 1, you can also convert fractions into percentages.

Example

three divided by 25 multiplication 100 percent equation left hand side equals right hand side open three divided by 25 multiplication 100 close percent equation left hand side equals right hand side open three divided by 21 times five multiplication one times zero times zero super four divided by one close percent equation left hand side equals right hand side open three multiplication four divided by one multiplication one close percent equals 12 percent

Pencast symbol

See this additional example turning a fraction into a percentage (click on “View document”).

View document

As an alternative, you can turn the fraction into a decimal first, and then move the decimal point:

Example: equation sequence seven divided by 20 equals 0.35 equals 35 percent.

Summary

To change a decimal or fraction to a percentage, multiply by 100%, or change to a decimal and then to percent.

There are other alternative ways, such as multiplying the numerator and denominator of the fraction to be converted by 100 divided by 100, shown in this example:

equation sequence two divided by five equals two multiplication 100 super 20 divided by five sub one multiplication 100 equals 40 divided by 100 equals 40 percent

For this last example, note that the fraction can be expanded by multiplication so that the denominator is turned into 100. For some fractions, this gives you another alternative for the conversion. For this last example, it looks like this:

equation sequence two divided by five equals two divided by five multiplication 20 divided by 20 equals 40 divided by 100 equals 40 percent

Pencast symbol

Let’s revisit the different ways in which you can convert decimals and fractions into percents (Click on “View document”).

View document

8.1.4 The Importance of the Percent Sign

A mistake that many people make is that they think of % as a unit, instead as a symbol that actually affects the number itself. 28% is not equal to 28, it is actually28 divided by 100, or 0.28.

This example summarized: 28 percent equation sequence equals 28 divided by 100 equals 0.28, not 28.

8.1.5 Converting

Activity symbolActivity: Fractions, Decimals, and Percentages

[ Want to get more comfortable with converting among fractions, decimals, and percentages? Play this game! ] (a) In your math notebook, turn each of the given percentages into a decimal as well as into a fraction in lowest terms.

  • (i) 10%
  • (ii) 25%
  • (iii) 50%
  • (iv) 125%
  • (v) 0.5%
Hint symbol
Comment

To turn a percentage into a decimal, how far and in which direction should you move the decimal? To turn the percentage into a fraction, which number do you divide by? Don’t forget to reduce the fraction if possible.

Solution symbol
Answer

(a)

  • (i) Decimal: 10 percent equals 0.1 Fraction: 10 percent equation sequence equals 10 divided by 100 equals one divided by 10
  • (ii) Decimal: 25 percent equals 0.25 Fraction: 25 percent equation sequence equals 25 divided by 100 equals one divided by four
  • (iii) Decimal: 50 percent equals 0.5 Fraction: 50 percent equation sequence equals 50 divided by 100 equals one divided by two
  • (iv) Decimal: 125 percent equals 1.25 Fraction: 125 percent equation sequence equals 125 divided by 100 equals five divided by four
  • (v) Decimal: 0.5 percent equals 0.005 Fraction: 0.5 percent equation sequence equals 0.5 divided by 100 equals 0.5 multiplication two divided by 100 multiplication two equals one divided by 200

(b) Write each of the following given numbers as percentages.

  • (i) three divided by four
  • (ii) five divided by eight
  • (iii) one divided by three
  • (iv) 0.4
  • (v) 0.0075
  • (vi) 1.2
Hint symbol
Comment

There are several different ways of doing this. Do you like to go through a decimal? Do you prefer to multiply by 100%? (This last approach will be shown in the solutions.)

Solution symbol
Answer

(b) Multiplying each fraction or decimal by 100% gives the following answers:

  • (i) equation left hand side three divided by four equals right hand side three divided by four multiplication 100 percent equals 75 percent
  • (ii) equation left hand side five divided by eight equals right hand side five divided by eight multiplication 100 percent equals 62.5 percent
  • (iii) equation left hand side one divided by three equals right hand side one divided by three multiplication 100 percent equals percent
  • (iv) 0.4 equals 0.4 multiplication 100 percent equals 40 percent
  • (v) 0.0075 equals 0.0075 multiplication 100 percent equals 0.75 percent
  • (vi) 1.2 equals 1.2 multiplication 100 percent equals 120 percent

By no means did you have to do it this way, but don’t forget that your answer must include a percent symbol.

8.1.6 Calculator Exploration: Percentage Conversions

Calculator symbol The calculator can be accessed on the left-hand side bar under Toolkit.

In this exploration, you will use the calculator for percentage conversions.

To Write a Percentage as a Fraction or a Decimal

Remember that the percent sign means “out of one hundred.” So, 5% means five divided by 100, and this is how it is entered on the calculator.

Enter 5% as five divided by 100 on the calculator, and click the equals sign, or hit Enter on your keyboard.

The key sequence isKey sequence showing 5 divided by 100 equals

The calculator gives the fraction in its reduced form with the decimal equivalent.

Sum showing 5 divided by 100 equals 1 divided by 20 equals 0.05
Activity symbolActivity: Converting Percentages to Fractions

Try writing these percentages as fractions and as decimals.

(a) 88%

Hint symbol
Comment

Clear the last calculation before you start, and remember that % means “out of one hundred.”

Solution symbol
Answer

(a) Entering 88% as 88 divided by 100 gives 0.88, so 88 percent equals 0.88.

Sum showing 88 divided by 100 equals 22 divided by 25 equals 0.88

(b) 250%

Solution symbol
Answer

(b) 250 percent equals 2.5

Fraction divided 250 divided by 100 equals 2 and a half equals 2.5

To Write a Decimal Number as a Percentage

To write a decimal number as a percentage, multiply the number by 100%. The calculation is quite easy to do without the calculator because you can move the decimal point two places to the right. For example, 0.8 becomes 0.8 multiplication 100 percent equals 80 percent. Let’s check that the calculator does the same. Enter the calculation 0.8 multiplication 100. The calculator gives the answer 80.

Remember that you must include the % sign when you quote the answer.

Activity symbolActivity: Converting Decimals to Percentages

Use the calculator to write these decimal numbers as percentages:

(a) 0.28

Solution symbol
Answer

(a) To convert 0.28 to a percentage, multiply by 100%. Using the calculator, multiply by 100, and then attach the % sign when you write the answer down, as follows: 0.28 equals 28 percent.

Sum showing 0.28 multiplied by 100 equals 28

Note: We do not use the percent key when converting a decimal to a percentage. When you use the percent key, it is converting the value to its decimal equivalent. Here, you are just trying to find the value that belongs in front of the percent sign.

(b) 1.052

Solution symbol
Answer

(b) 1.052 equals 105.2 percent. (Did you remember to attach the % sign when you wrote the answer down?)

Sum showing 1.052 multiplied by 100 equals 105.2

(c) 0.006

Solution symbol
Answer

(c) 0.006 equals 0.6 percent.

Sum showing 0.006 multiplied by 100 = 0.6

8.1.7 To Write a Fraction as a Percentage

You can use the same method, multiplying by 100%, to write a fraction as a percentage. For example, enter one divided by eight into the calculator, then multiply by 100. It’s useful, though not essential, to put parentheses around the fraction. The calculator shows , or 12.5. Therefore, one divided by eight equals percent equals 12.5 percent.

Sum showing 1 eight multiplied by 100 equals 12 and a half equals 12.5

Activity symbolActivity: Making Percentages From Fractions

Use the calculator to write these fractions as percentages, giving an exact answer followed by a decimal, rounded to 2 decimal places, where necessary.

(a) five divided by six

Hint symbol
Comment

Enter the fraction, then multiply by 100. Remember to attach the percent sign when you give your answer.

Solution symbol
Answer

(a) equation left hand side five divided by six equals right hand side five divided by six multiplication 100 percent equals percent, or 83.33%.

Sum showing five sixths multiplied by 100 equals 83 and one third equals 83.333333333

Note: Remember, do not use the percent key when converting a fraction to a percentage.

(b) one divided by 400

Solution symbol
Answer

(b) equation left hand side one divided by 400 equals right hand side one divided by four percent, or 0.25%.

Sum showing one 400th multiplied by 100 = one quarter equals 0.25

(c) 11 divided by seven

Solution symbol
Answer

(c) 11 divided by seven equals percent, or 157.14%

Sum showing 11 sevenths multiplied by 100 equals 157 and one seventh equals 157.142857

The Percent Key on the Calculator

You may have noticed that there is a percent key on the calculator in the top row of keys below the number pad. There are some calculations where using this key will save you time, but be careful, as the calculator does not always behave how you may expect it to.

Enter Key sequence showing 5 percent equals on the calculator. The calculator converts 5% to a fraction, but does not give the decimal equivalent.

Sum showing 5 percent equals one twentieth

So the calculator can help you convert a percentage to a fraction, but it does not change the fraction to a decimal. Also, the calculator does not help when you convert a decimal or a fraction into a percentage. For these reasons, it is better to avoid using the % key on the calculator for conversions, and to divide or multiply by 100 instead.

To recap:
  • To write a percentage as a fraction or a decimal, write the percentage as a number divided by 100 and click on the equals sign, or hit Enter on the keyboard.
  • To write a decimal number or a fraction as a percentage, multiply by 100 on the calculator and then attach the % sign when you give your answer.
  • Entering a percentage followed by the percent key and clicking equals gives the percentage as a fraction only.

8.1.8 Find a Percentage of a Number

Suppose a mathematics exam is out of 75 points, and students need to achieve at least 60% to pass. How many points guarantee a passing grade?

To pass, the student has to earn at least 60% of 75. Here, we can change the percentage into a fraction, then calculate 60 divided by 100 multiplication 75 to find the required point amount. You can work out the points necessary by reducing the fraction and canceling further.

Here is one way of reaching the answer:equation sequence 60 divided by 100 multiplication 75 equals six divided by 10 multiplication 75 equals three divided by five multiplication 75 equals three divided by five multiplication 75 divided by one equals three multiplication seven times five super 15 divided by 51 multiplication one equals three multiplication 15 equals 45.

Or, you can multiply 60 by 75, then divide by one hundred:60 divided by 100 multiplication 75 equals 45. Both ways show that the student needs 45 points.

Summary

To find a given percentage of a number, change the percentage to a fraction (or decimal) and multiply by the number.

8.1.9 Going Solo

Activity symbolActivity: Going Solo

(a) A radio program reported the following:

“Our ‘Going Solo’ survey of 4,000 single people found that only 1 in 5 is happy on their own.”

Write “1 in 5” as a fraction, a decimal, and a percentage. Which form do you think is easiest to understand?

Hint symbol
Comment

Which numbers create the fraction? How does this fraction convert to a percent? How does it convert to a decimal?

Solution symbol
Answer

(a) “1 in 5” can be written as one divided by five, or one division five equals 0.2, as well as one divided by five multiplication 100 percent equals 20 percent.

The report could have said “one-fifth” or “20%” of people in the survey, but “1 in 5” makes the proportion easy to visualize for people not comfortable with fractions and percentages. Which form is easiest to understand depends on your way of thinking.

Remember, all of these express the same number, although “0.2 of the people” would probably not be used, because it’s much harder to visualize and put in context. In part (b) you will determine how many people this is.

(b) How many of the 4,000 “Going Solo” survey participants seem to be content with being single?

Hint symbol
Comment

You know that it’s one-fifth of the group, and remember, “of” means “multiply.”

Solution symbol
Answer

(b) one divided by five multiplication four times 000 equals 800. So, 800 out of the 4,000 surveyed singles feel happy on their own.

8.1.10 Which Percent is It?

You probably agree that comparing percentages can be easier than comparing fractions. When we have a certain quantity, how do we write it as a percent of another, usually larger, number?

For example, if 42 people out of a group of 70 people agree to participate in a new community project, what percentage of the group is this?

Whenever you have a part of a group and the whole of the entire group, you can calculate percent by setting up the ratio of “part over whole.” You can then turn this fraction successfully into a percentage.

So in this case, turn 42 divided by 70 into a percent, first by multiplying by 100% and then by reducing by 10 and then by 7:

42 divided by 70 multiplication 100 percent equation left hand side equals right hand side 42 divided by 70 sub seven multiplication 100 super 10 divided by one percent equation left hand side equals right hand side 42 super six divided by seven sub one multiplication 10 divided by one percent equation sequence equals six multiplication 10 divided by one multiplication one equals 60 percent

(Note: You also could have reduced 42 divided by 70 to three divided by five first and then turned it into a percent.)

These calculations show us that 60% of the group agree to participate in the community project.

As you know, the calculation 42 division 70 multiplication 100 can be performed even quicker with a calculator, as can the other percent calculations.

Summary

To express one number as a percentage of another, divide the first number by the second and convert the result into percent by multiplying by 100%.

8.1.11 Calculator Exploration: Using Percentages

Calculator icon The calculator can be accessed on the left-hand side bar under Toolkit.

In the following exploration, you will use the calculator for calculations involving percentages.

Finding a Percentage of a Number

Suppose that residents in a town are asked their views on a proposed development of a wind farm near the town. The local paper says that 450 people voted, with 54% in favor of the proposal, 36% against the proposal and 10% saying that they were undecided. How many people were in favor of the proposal?

To find the number of people who were in favor, you need to find 54% of 450. The calculation is: 54 percent equation left hand side prefix multiplication of 450 equals right hand side 54 divided by 100 multiplication 450.

Enter this into the calculator and you will see:

Sum 54 hundredths in parenthesis multiplied by 450 equals 243

So, 243 people were in favor of the proposal.

The percent key is useful here. Enter Key sequence showing 54 percent multiplied by 450 equals.

The calculator shows:

Sum showing 54 percent multiplied by 450 equals 243

Again, you can see that 243 people were in favor of the proposal.

Activity symbolActivity: Methods for Finding Percentages

Now it’s your turn. Try both methods (first, entering the percentage as a fraction, and second, using the percent key) to find the number of people who were against the proposal. Then find the number of people who were undecided about the proposal. (Try this calculation without using the calculator.)

Hint icon

Comment

For the first method, enter the percentage as a fraction, then multiply by 450. For the second method, enter the percentage followed by the percent key, then multiply by 450.

Solution symbol
Answer

Since 36% of the 450 people were against the proposal, you want to calculate 36% of 450 to find the exact number. Here are the two methods on the calculator.

Sum showing 35 hundredths multiplied by 450 equals 162
36 percent multiplied by 450 = 162

So, 162 people were against the proposal.

You know that 10% of the 450 people were undecided, so you need to find 10% of 450. You should be able to do this calculation without the calculator: 10% is one-tenth, and one-tenth of 450 is 45. So, 45 people were undecided. The calculator should give you the same answer, but do watch for easy calculations that you can do in your head—it’s good practice!

As a check, the three numbers you have calculated should add to 450. The number in favor was 243, the number against was 162 and the number undecided was 45, and these do add to 450.

8.1.12 To Write One Number as a Percentage of Another

[ Source: U.S. Census (state data, “Search”; national data, “Map”) ]

Activity symbolActivity: Writing One Number as a Percentage of Another

(a) In 2010, the population of Maryland was about 5,774,000, and the total population of the United States was about 308,746,000. What was the population of Maryland as a percentage of the total population of the United States?

Hint symbol
Comment

To find the percentage, you take the ratio of “part to whole” and multiply by 100%.

Solution symbol
Answer

(a) The calculation required is 5,774,000 divided by 308,746,000 multiplication 100 percent.

Try this on the calculator. The calculator should show:

Sum showing 5,774,00 over 308,746,000 multiplied by 100 equals 1.870145686

You need to round the answer. How many decimal places would make sense here?

This rather depends on who the answer is for. Probably, either one or two decimal places are sufficient. Round the answer to 2 decimal places.

Rounded to 2 decimal places, the population of Maryland was 1.87% of the total population of the United States in 2010.

(b) California has the largest population of any state in the US, about 37,254,000 in 2010 and the total population of the United States was about 308,746,000. Find the population of California as a percentage of the total population of the United States, rounding your answer to 2 decimal places.

Hint symbol
Comment

Write the part divided by the whole, then multiply by 100%.

Solution symbol
Answer

(b) The calculator shows:

37, 254,000 over 308,746,000 multiplied by 100 equals 12.06622919

The population of California, rounded to 2 decimal places, was about 12.07% of the total population of the United States in 2010.

To recap:

  • To find a percentage of a number, write the percentage as a fraction (by dividing by 100) and multiply by the number. Or use the percent key, by entering the percentage followed by the percent key and multiply by the number.
  • To write one number as a percentage of another, write the part divided by the whole and multiply by 100%.

References

U.S. Census Bureau. “2010 Census Interactive Population Map.” U.S. Census Bureau. http://2010.census.gov/ 2010census/ popmap/ index.php

U.S. Census Bureau. “2010 Census Interactive Population Search.” U.S. Census Bureau. http://2010.census.gov/ 2010census/ popmap/ ipmtext.php

8.1.13 World Populations

This activity will require the use of a calculator.

Calculator icon The calculator can be accessed on the left-hand side bar under Toolkit.

Activity symbolActivity: Different Nationalities

In 2011 it was estimated that 1 in 6 of the world’s population were living in India and 1 in 23 were living in the USA.

Convert these values to percentages using your calculator to make them easier to compare.

Solution symbol
Answer

1 in 6 is the fraction one divided by six. In percentages, this becomes open one divided by six close multiplication 100 percent almost equals 17 percent.

1 in 23 is the fraction one divided by 23. In percentages, this becomes open one divided by 23 close multiplication 100 percent almost equals four percent.

Note here that we have rounding to the nearest whole number as population figures are necessarily estimates.

8.1.14 Money

Activity symbolActivity: Where Does the Money Go?

Let’s suppose your gross monthly income (that is, your income before taxes) is $3,960. Financial experts recommend that no more than $1,386 of this amount should go toward your mortgage. Based on your monthly salary, you should also spend no more than $316.80 per month on car payments. Additionally, it is advisable to place $792 of your gross monthly income into a savings account.

If you are putting these exact amounts toward these categories each month, which percentage of your gross monthly income will be spent on the following expenses?

(a) Mortgage

Hint symbol
Comment

Did you try rewording the given information, then translating it? For example, $1,386 out of $3,960 should go toward mortgage payments. This translates into the quotient dollar times 1386 divided by dollar times 3960. The $ symbols cancel (because they are units) and the quotient can be worked out. Remember, you will need to convert your unit-less decimal to a percentage.

Solution symbol
Answer

(a) The percentage spent on the mortgage is equation sequence dollar times 1386 divided by dollar times 3960 equals 1386 divided by 3960 equals 0.35 equals 35 percent.

(b) Car Payments

Solution symbol
Answer

(b) The percentage spent on car payments is equation sequence dollar times 316.80 divided by dollar times 3960 equals 316.8 divided by 3960 equals 0.08 equals eight percent.

(c) Savings

Solution symbol
Answer

(c) The percentage put into savings is equation sequence dollar times 792 divided by dollar times 3960 equals 792 divided by 3960 equals 0.2 equals 20 percent.

Of course, these numbers represent general advice. It might be better to purchase a used vehicle that is not too old and has a reasonable number of miles on it, and try to pay for it outright from your savings account, rather than to have a car payment.

8.1.15 Saving for Retirement

Activity symbolActivity: Saving for Retirement

You know that saving for retirement is important, and there are many resources available to tell you how much you should save per paycheck.

[ For many employed people in the U.S., participating in a company’s 401(k) plan with a company match or setting up a Roth IRA with a broker firm is a good first step. ]

Do you know which percentages of pretax income (gross income) many reputable financial advisors recommend to put away for retirement if a client starts to save for retirement in his or her:

(a) Twenties?

Hint symbol
Comment

According to financial advisors,

(a) people who start in their twenties should put 15% of their pretax income toward retirement savings,

(b) Thirties?

Hint symbol
Comment

(b) people who start in their thirties should save 25% of their pretax dollars for retirement

(c) Forties?

Hint symbol
Comment

(c) and people who start in their forties need to put away 35%.

[ Discouraged? Good news: Saving a little is better than not saving at all. And if you work in a job you enjoy, you probably won’t mind working until later in life, which means your savings have a longer time to grow and you won’t rely on them as early. ] Starting later than mid 40s is even more of a challenge as these recommended percentages increase further.

According to this advice, how much should the following people, who are just starting to create their first retirement savings, put into their respective retirement accounts each year? (All yearly incomes given are pretax.)

(i) A 36-year-old groundskeeper earning $25,000 a year.

Solution symbol
Answer

(i) The groundskeeper is in his or her thirties, and thus should save 25 percent prefix multiplication of dollar times 25 comma 000 equals 0.25 multiplication dollar times 25 comma 000 equals dollar times 6250 per year. (This would be equation left hand side dollar times 6250 divided by 12 almost equals right hand side dollar times 521 or about $520 a month.)

(ii) A 43-year-old nurse earning $70,000 a year.

Solution symbol
Answer

(ii) The nurse is in his or her forties, and thus should save 35 percent prefix multiplication of dollar times 70 comma 000 equals 0.35 multiplication dollar times 70 comma 000 equals dollar times 24 comma 500 per year. (This would be about $2040 a month.)

(iii) A 27-year-old entrepreneur earning $46,000 a year.

Solution symbol
Answer

(iii) The entrepreneur is in his or her twenties, and thus should save 15 percent prefix multiplication of dollar times 46 comma 000 equals 0.15 multiplication dollar times 46 comma 000 equals dollar times 6900 per year. (This would be about $575 a month.)

Looking at these percentages, you see why it is a good idea to start paying into a retirement plan or account as soon as you can.

8.1.16 Sales Tax

Calculator icon The calculator can be accessed on the left-hand side bar under Toolkit.

Percentages are used frequently in business transactions when calculating extra charges such as sales tax, a tip left in a restaurant, working out discounts on sale items, or specifying interest on a loan or savings account. As you know, sales tax and discounts are given in percent. To calculate the total price of a purchase, you can work out the extra charge (or discount) and then add it to (or subtract it from) the original price.

As an example, the sales tax in Oklahoma, as of 2011, was 4.5%. To work out the total price on a $148 lounge chair purchased there, you can calculate 4.5% of $148, then add that amount to the ticket price. If you just want a rough idea what the item will cost at check out, an estimation will give you an idea.

Let's make an estimate, first. 4.5% is close to 5%, and $148 is approximately $150. You can find 10% of a number very easily by dividing the amount by 10, and 5% is half of 10%, so taking half of the amount from the previous step will do the trick. This makes the added tax about equation left hand side dollar times 15 divided by two equals right hand side dollar times 7.50 and the entire purchase approximately dollar 150 postfix plus dollar 7.50 equals dollar 157.50.

You also know that you rounded both figures up—the original price, as well as the sales tax—so if you have at least this amount to pay with, it will be more than enough to make this lounge chair yours.

Using the calculator and working exactly, 4.5% of $148 is 0.045 postfix multiplication dollar 148 equals dollar 6.66.

This is reasonably close to the estimate we found above for the sales tax, which was $7.50.

The total price of for this lounge chair including Oklahoma sales tax is dollar 148 postfix plus dollar 6.66 equals dollar 154.66.

Note: You can also calculate equation left hand side 1.045 multiplication dollar times 148 equals right hand side dollar times 154.66 if you don’t want to do the calculation in two steps. The 1.045 then stands for 100 percent prefix plus of 4.5 percent equals 104.5 percent, because you are paying 100% of the price for the chair, plus 4.5% of the price additional in tax. Calculating 104.5% of the original cost, therefore, gives the final price.

8.1.17 Calculating Sales Tax

Notepad iconActivity: Expensive Sales Tax

(a) Sabrina was visiting California, and bought a pair of sneakers with a ticket price of $69. The California sales tax rate at the time was 8.25%. First, without using a calculator, give an estimate of what the shoes cost in all. Afterward, find the exact price of her purchase.

Calculator icon The calculator can be accessed on the left-hand side bar under Toolkit.

Hint symbol
Comment

It does not matter if you figure the sales tax as an estimated 10% (rounded up very roughly) or 8% (rounded down to the closest percent). Working with 10% clearly is much easier. The $69 rounds to $70, and since 10% of $70 is $7 (just divide 70 by 10), the total comes out to less than $77.

Now, use the exact numbers to find the actual purchase price Sabrina paid. Pay close attention to your decimal places, as you are converting a one-digit percentage to a decimal.

Solution symbol
Answer

(a) 8.25% of $69 is equation left hand side 0.0825 multiplication dollar times 69 almost equals right hand side dollar times 5.69 (rounded to the closest cent) The total price was dollar 69 postfix plus dollar 5.69 equals dollar 74.69.

You may also have calculated it like this:equation left hand side 1.0825 multiplication dollar times 69 almost equals right hand side dollar times 74.69.

(b) [ Sabrina was fortunate. If you travel to England now, 20% VAT is added to all purchases. ] When Sabrina traveled to Europe in 2010, she was amazed to see that England has a 17.5% sales tax, which is called VAT (value added tax). In London, she bought a pair of boots that were £26 before tax. After calculating a rough estimate of sales tax and final purchase amount, can you find the exact sales tax in your head using mental strategies from chapter 2? Then find the exact purchase amount.

Hint symbol
Comment

Make an estimate first. 20% of £30 is approximately two postfix multiplication pound three equals pound six. Since we rounded up, we expect the VAT to be less than £6.

For the mental approach, think about breaking the VAT of 17.5% down into 17.5 percent equals 10 percent prefix plus of five percent prefix plus of 2.5 percent. Do you see that each of these partial percentages is half of the last? You also know how to find 10% of an amount. Use these strategies to do your mental math.

Solution symbol
Answer

(b) 10% of £26 is £2.60.

5% of £26 is £1.30 (half of £2.60).

2.5% of £26 is £0.65 (half of £1.30).

Therefore, 17.5% of £26 is pound 2.60 postfix plus pound 1.30 postfix plus pound 0.65 equals pound 4.55, which is less than £6, and thus reasonable when compared to our estimate.

The final price is the original cost plus the VAT is pound 26 postfix plus pound 4.55 equals pound 30.55.

The corresponding amount in U.S. dollars depends on the current exchange rate. In 2010, the exchange rate was about $1.60 per British pound. This would make the boots approximately £equation left hand side 30.55 multiplication 1.6 equals right hand side dollar times 48.88. We’ll look at more conversions like this in Unit 9.

8.1.18 Leaving a Tip

If you go out to eat, you know that it is customary to leave a tip (gratuity) for the server based on the amount of your bill. 15% is considered typical, although many people agree that if you enjoyed the service, a tip of 20% is appropriate. [ Did you know that most U.S. wait staff are paid less than minimum wage, and that the main portion of their income comes from tips? They are also required to pay income tax on the gratuity they receive. ]

Video clip iconHow do you figure out how much to give as a tip? This video helps you to learn how to calculate a standard 15% tip quickly:

Interactive feature not available in single page view (see it in standard view).

Watch the video and then review your knowledge in the next activity.

Activity iconActivity: Leaving a Tip

Imagine you are at a restaurant, and you pay your bill with your debit card. You decide to tip about 15% on the $41.68 bill (which already included sales tax).

How much tip will you add, and what is the total cost of the meal with tip?

Hint symbol
Comment

You have many methods now to arrive at 15% of about $41.68. Can you do it with the method from the video? Don’t hesitate to make a rounded estimate to get started.

Solution symbol
Answer

If you decide to round the $41.68 differently from this sample solution, you may arrive at a slightly different overall amount. But, since you are not required to leave exactly 15% in tip, other results can be correct. To avoid mistakes, make sure the answer is reasonable. It is your decision whether you calculate the tip based on the entire bill amount or just on the cost of the food and not on the sales tax.

$41.68 can be rounded up to $42.00. Dividing by 10 gives you 10% of $42.00, which is $4.20. Multiplying by 2 creates 20% being $8.40. Did you catch what he did next? He is taking the average of the sum of these two dollar amounts. He can do this because the average of 10% and 20% is equation sequence open 10 plus 20 close percent divided by two equals 30 percent divided by two equals 15 percent.

Now do it with the money: equation sequence dollar times 4.20 plus dollar times 8.40 divided by two equals dollar times 12.60 divided by two equals dollar times 6.30 .

Lastly, dollar 42.00 postfix plus dollar 6.30 equals dollar 48.30.

The standard tip (15%) is about $6.30, and the total amount would be about $48.30. In such a situation, you probably will consider to round up to $49.00 or even $50.00 if the service was good (the latter coming close to a more generous 20% in gratuity).

If the averaging arithmetic is a little tricky, you can round the sums before doing it. For example, round $4.20 to $4 and $8.40 to $8. It’s easier to see that the average of those is dollar equation left hand side open four plus eight close divided by two equals right hand side dollar six, which is still reasonable for a tip.

8.1.19 Online Shopping

Activity symbolActivity: Online Shopping

A website that you sometimes shop on offers free shipping and no sales tax. This week, they are also having a promotion which will make your entire order 20% off.

You place an order for items that without the discount would cost $45.50. How much will you pay?

Hint symbol
Comment

How much is the 20% discount on the $45.50 order? You can use it to determine the final cost.

Solution symbol
Answer

How about figuring out the discount mentally?

10% of $45.50 is $4.55. Double that to 20%, which is equation left hand side two multiplication dollar times 4.55 equals right hand side dollar times 9.10.

You could also calculate the discount directly, maybe with a calculator: 20% of $45.50 is equation left hand side 0.2 multiplication dollar times 45.50 equals right hand side dollar times 9.10.

The items will cost dollar 45.50 postfix minus dollar 9.10 equals dollar 36.40.

Alternatively: 100 percent negative 20 percent equals 80 percent. You pay 80% of the original price, which is equation left hand side 0.80 multiplication dollar times 45.50 equals right hand side dollar times 36.40.

8.1.20 Going Down, Going Up

This activity involves working out increases and decreases together.

Activity symbolActivity: Going Down, Going Up

Let’s say you live and shop in Maryland, which in 2010 had a 6% sales tax. You have set your eyes on a new dresser for your room, which is on sale at 30% off. Have you ever asked yourself if it makes a difference if the discount is applied first, and then sales tax, or if it is done in the opposite order: Sales tax applied first, and then the discount?

Hint symbol
Comment

Try a numerical example first to get a feel for the problem. For example, what happens if the price of the dresser was $100 (before the discount and sales tax have been applied)?

Solution symbol
Answer

If the item costs $100, reducing the price first gives equation left hand side dollar times 100 minus dollar times 30 equals right hand side dollar times 70.

The sales tax is 6% of $70, which is $4.20.

So, the final bill is dollar 70 postfix plus dollar 4.20 equals dollar 74.20.

Now, let's try it the other way.

Adding the sales tax first gives equation left hand side 1.06 multiplication dollar times 100 equals right hand side dollar times 106.

The discount is 30% of $106, which is equation left hand side 0.3 multiplication dollar times 106 equals right hand side dollar times 31.80.

So, the total bill is dollar 106 postfix minus dollar 31.80 equals dollar 74.20.

It does not seem to matter whether the discount or the sales tax is applied first. But can you be sure? Doing more numerical examples would confirm it. However, these could just be lucky picks. Let’s analyze what we can tell using these particular discount/sales tax rates.

If the discount is 30%, the price after the discount will be 100 percent negative 30 percent equals 70 percent of the original cost. You can find the discounted base price by multiplying the cost by 0.7.

If the discount is applied first, then the sales tax, it’s 106% of 70% of the original price, and can be expressed as 1.06 multiplication 0.7 postfix multiplication original price.

If the sales tax is applied first, then the discount, it’s 70% of 106% of the original price, and can be expressed as 0.7 multiplication 1.06 postfix multiplication original price.

Since it does not matter in which order the multiplication operations are carried out, the answer will be the same using either method. This will work with any type of discount and sales tax.

8.1.21 Working Backwards

In some situations, you may be given the total after the discount or tax has been applied and asked to find the original amount. For example, suppose an online store charges an extra 5% of the cost of the purchases for postage and packing. You have a gift voucher for $30. What is the maximum you can spend at the store, so that the total including the postage is less than $30?

As you can see in the diagram above, 105% is equivalent to $30.

Therefore, 1% is equivalent to equation left hand side dollar times 30 division 105 almost equals right hand side dollar times 0.2857, and multiplying this quotient by 100 creates 100%, which results in equation left hand side 100 multiplication dollar 30 divided by 105 almost equals right hand side dollar 28.57 (rounded down to the nearest cent).

The maximum amount that can be spent then is $28.57. You can check your answer by working out the 5% postage charge on $28.57 to see that the grand total will be $30.

Activity symbolActivity: Working Backwards

The price of a tennis racquet, including a 4% sales tax, is $45.24. What was the price before sales tax?

Hint symbol
Comment

The $45.24 includes sales tax. What percentage does this total represent? Find the dollar amount that represents 1%, and use that figure to arrive at 100%.

Solution symbol
Answer

104% of the cost is $45.24.

1% is dollar 45.24 division 104 and 100% is equation left hand side 100 multiplication dollar 45.24 divided by 104 equals right hand side dollar 43.50.

Thus, the price of the tennis racquet before sales tax was applied was $43.50.

8.2 Ratios

The media reports: “One in three people rejects technology such as computers and mobile phones.”

Alternatively, you could say “For every one person who rejects technology, there are two people who embrace it.” Mathematically, we say, “The ratio of people who reject technology to those who embrace technology is one to two.” This ratio could be written as one divided by two, or in colon notation as 1:2. Ratios provide another way to convey information.

Let’s consider the following example. Baseball is America’s pastime. In 2008, the L.A. Dodgers won the division title in the National League West. They won 84 games and lost 78. What is the ratio of wins to losses for the L.A. Dodgers?

The ratio of wins to losses can be set up as a fraction, placing the wins in the numerator (top of fraction) and the losses in the denominator (bottom of fraction). We have 84 wins over 78 losses, or 84 divided by 78. Although most major leaguers would just report the ratio as is, let’s use our fraction knowledge to reduce it: equation sequence 84 times 42 divided by 78 times 39 equals 42 times 14 divided by 39 times 13 equals 14 divided by 13. Thus, the ratio of wins to losses is 14 to 13. Go, Dodgers!

Key Points

A ratio is the quotient (division) of two values and can be expressed in fraction or colon notation.

8.2.1 Salad Dressing Recipe

Activity symbolActivity: Salad Dressing Recipe

You have a recipe for salad dressing that serves 8  but want to know what proportion the oil and vinegar are in so that you can make it for any number of servings.

The recipe is:

  • 50 ml olive oil
  • 15 ml white wine vinegar

Work out the ratio of oil to vinegar.

Hint symbol
Comment

To find the ratio of two values, you should use division.

Solution symbol
Answer

For the oil to vinegar ratio, you should set up the calculation volume of oil divided by volume of vinegar.

Ratio of oil to vinegar equals times 50 ml divided by 15 ml times equals 3.3333 left parenthesis to four d full stop p full stop right parenthesis

So there is 3.333 times the volume of oil in the recipe compared to vinegar.

You will have noticed that this produces an answer where the 3 to the right of the decimal place is repeated.  In fact this continues to infinity and is known as a recurring number.  In this case 0.3 recurring is the same as 1 third.  Try dividing 1 by 3 on your calculator to check this out.

In this example, you calculated the ratio of one quantity to a second quantity by dividing the first number by the second. You can use this process to calculate other ratios, too.

8.2.2 To Buy or Not to Buy?

Video clip iconA very important ratio is the debt to equity (or liability to asset) ratio. For example, it is used as a guide to determine how much money a business or individual should be lent by banks. Check out this short video, which provides more details.

Interactive feature not available in single page view (see it in standard view).

Activity symbolActivity: To Buy or Not to Buy?

You really would like to purchase an eight-person hot tub. Before you apply for a $15,000 loan, you should determine what your debt-to-equity ratio would be before and after making such a purchase.

The combined amount in your checking and savings account is $35,300. Currently, you have credit card debt of $12,000 and still owe $11,200 on your school loans.

Calculate your debt-to-equity ratio before and after the purchase.

Hint symbol
Comment

Have you identified the total amount of debt before the purchase? How about after purchasing the hot tub?

Solution symbol
Answer

Before:

Your total debt (liability) is dollar 12 comma 000 postfix plus dollar 11 comma 200 equals dollar 23 comma 200. Your equity (asset) is $35,300.

The debt-to-equity ratio without the purchase is dollar times 23 comma 200 divided by dollar times 35 comma 300 almost equals 0.66.

After: [ Credit card debt is usually at a very high interest rate. You should seriously consider paying that off before making another large purchase! ]

The total debt would be dollar 12 comma 000 postfix plus dollar 11 comma 200 postfix plus dollar 15 comma 000 equals dollar 38 comma 200. The equity would remain $35,300. So, the debt-to-equity ratio after purchasing the hot tub would be dollar times 38 comma 200 divided by dollar times 35 comma 300 almost equals 1.08.

Prior to the purchase, your debt to equity ratio is less than 1, which means you have more than you owe.

However, if you decide to take out a loan to buy the hot tub, your debt to equity ratio becomes larger than 1, which means you owe more than you have.

In other words, it’s probably not a wise decision to take out a loan to purchase this commodity!

8.2.3 Ratios in Recipes

[ Feeling a little shaky with ratios written in colon notation? Check out this interactive website. ] So far, you have worked with ratios as fractions. Remember that ratios are also given in colon notation. For example, a recipe may call for two cups of raisins for every three cups of oatmeal. This can be written as the ratio 2:3. If you decided to make more or less of recipe, you need to preserve the ratio. In other words, there would need to be two parts raisins for every three parts oatmeal. Let’s look at an example.

A recipe for shortbread requires 12 ounces of all purpose flour, 4 ounces of sugar and 8 ounces of butter.

This means that the ratio of flour to sugar to butter is 12:4:8. Ratios can be canceled down like fractions. Dividing all parts of the ratio by 4, it can be expressed more simply as 3:1:2 or three parts flour, one part sugar and two parts butter.

Suppose you wish to make some shortbread following this recipe using 12 ounces of butter, how much flour and sugar will you need?

Start with the butter, 12 ounces is equivalent to two parts.

So one part is the same as 12 ounces prefix division of two equals six ounces.

Hence, three parts will be the same as three multiplication six ounces equals 18 ounces.

So, 18 ounces of flour and 6 ounces of sugar will be needed.

8.2.4 Ratios in Food and Drink

Activity symbolActivity: Using Ratios in Food and Drink

(a) A fruit drink has to be diluted by mixing 1 ounce of the concentrate with 5 ounces of water. How much water should be added to 4 ounces of concentrate? How much drink will this make?

Hint symbol
Comment

Have you considered drawing a sketch of a glass with ounces marked along its side?

Solution symbol
Answer

(a) The ratio of concentrate to water is 1:5, so five times as much water as concentrate is required. If 4 ounces of concentrate is used, five multiplication four ounces equals 20 ounces of water is needed. If 4 ounces of concentrate and 20 ounces of water are used, the amount of drink will be four ounces prefix plus of 20 ounces equals 24 ounces.

(b) The ratio of white flour to wholemeal flour in a bread recipe is 3:7. If 6 ounces of white flour is used, how much wholemeal flour is needed? Can you explain why this is called a “70% wholemeal loaf”?

Hint symbol
Comment

Because the ratio is 3:7, this means 6 ounces of white flour is equivalent to 3 parts. How much is 1 part of white flour?

Solution symbol
Answer

[ Alternatively, you could use the corresponding fraction three divided by seven and find an equivalent fraction where the numerator is 6 (because there are 6 ounces of white flour). ]

(b) The resulting denominator would be your answer.

If 3 parts are equivalent to 6 ounces, one part is equivalent to six ounces prefix division of three equals two ounces. 7 parts are equivalent to seven multiplication two ounces equals 14 ounces. Thus, 14 ounces of wholemeal flour are needed.

There are 10 parts altogether, and 7 parts are wholemeal, so the fraction of the flour that is wholemeal is seven divided by 10 or 70%; hence the name.

Notice that there are different ways of tackling these problems and sketches might help you to understand the situation better.

8.3 Extensions and Further Exploration

In this section, we will try to extend our new knowledge through investigation of some of the mathematical content that was discussed throughout the unit. You might find some of these activities to be quite challenging. If you get stuck, feel free to discuss them with a friend. Don’t panic; just keep going.

The goal of this section is to expose you to more types of math, and to help you realize just how often you already solve math problems in daily life. Remember, if you believe that you don’t have the time to spend further exploring these topics, this is a section that you could treat as optional.

Now, get ready to learn the secrets of art and nature, and even bake some delicious cookies!

8.3.1 Percent Increase/Decrease

Let’s take another look at percentages. In the following exploration, you will use the calculator to find a percent increase and a percent decrease.

Calculator icon The calculator can be accessed on the left-hand side bar under Toolkit.

Activity symbolActivity: Island Population

The population of an island was 5,678 in the year 2010, with a prediction that it will rise by 3.1% over the next ten years (from 2010 to 2020). What will the population be in 2020?

There are two ways you can look at this calculation. For the first method, you find the increase and add it to the original population, and for the second, you find the new population as a percentage of the original one.

Method 1 involves finding 3.1% of the 2010 population of 5,678, then adding it to 5,678 to find the population in the year 2020.

(a) Find 3.1% of 5,678, and then find the predicted population of the island in the year 2020.

Hint symbol
Comment

You learned how to find percentages of numbers earlier.

Solution symbol
Answer

(a) There are two methods for finding percentages of numbers, shown on the calculator screens here:

Sum showing 3.1 over 100 in parenthesis multiplied by 5678 equals 17 and 9 five hundredths equals 176.018
Sum showing 3.1 percent multiplied by 5678 equals 176.018

3.1% of 5,678 is 176. You need to round to a whole number because you can only have a whole number of people!

The predicted population of the island in the year 2020 is five comma 678 plus 176 equals five comma 854.

Method 2 is a little quicker. If the population rises by 3.1%, then the new population is 100 percent prefix plus of 3.1 percent, or 103.1% of the 2010 population. The population in 2020 is therefore 103.1% of 5,678.

(b) Use this method to find the new population.

Solution symbol
Answer

(b) Using the calculator to find 103.1% of 5,678, you get:

Calculator screen showing 103.1 over 100 in parenthesis multiplied by 5678 = 5854 and nine five hundredths = 5854

Or:

103.1 percent multiplied by 5678 equals 5854.018

In the first screenshot, the calculator cuts off the end of the answer, but the full answer is shown in the white screen below it.

This method gives the same answer: The population of the island in 2020 is predicted to be 5,854.

Now you try a percent calculation, a percent decrease this time.

(c) The population on a neighboring island is falling. In the year 2010, it was 820, with a drop of 24% predicted over the next ten years (from 2010 to 2020). What will the population be in the year 2020? Try both methods shown above and make sure that they give the same answer.

Hint symbol
Comment

For method 1, find 24% of 820, and then subtract it from 820.

For method 2, the new population is 100 percent negative 24 percent of the 2010 population.

Solution symbol
Answer

(c) Finding 24% of 820 on your calculator gives:

Two separate sums, one on left showing 24 over 100 in parenthesis multiplied by 820 equals 196 and four fifths equals 196.8 the sum on the right showing 24 percent multiplied by 820 equals 196.8

So, the population drops by 197, and the predicted population in the year 2020 is then 820 minus 197 equals 623.

For method 2, the new population is 100 percent negative 24 percent equals 76 percent of the 2010 population. Using the calculator to find 76% of 820:

Two separate sums, one on left showing 76 over 100 in parenthesis multiplied by 820 equals 623 and one fifth equals 623.2 and the one on the right showing 76 percent multiplied by 820 equals 623.2

The predicted population in the year 2020 is 623.

To recap:

To find the new amount after a percent increase, either:

  • find the increase and add it to the original amount, or
  • write the new amount as 100% plus the % increase, and find this percentage.

To find the new amount after a percent decrease, either:

  • find the decrease and subtract it from the original amount, or
  • write the new amount as 100% minus the % decrease, and find this percentage.

Now you can use your calculator for future percentage calculations.

8.3.2 Percent Decrease

Whether it’s coupons you receive in the mail or a raise in your salary, you know that percentages are used to describe increases or decreases.

For example, suppose last week a pair of sneakers cost $125, and this week the same pair costs $100. The price decreased by $25, which is referred to as the actual decrease.

To express the discount as a percentage, first write the actual decrease as a fraction of the original cost, then change the fraction into a percentage.

So the percent decrease is:

Image showing 25 over 125 multiplied by 100 percent equals 1 fift multiplied by 100 percent equals 0.2 multiplied by 100 percent equals 20 percent. Arrow are pointing at 25 over 125 saying actual decrease, arrow point at 100 percent saying original cost (or amount) and arrow point at 20 percent saying percent decrease

In other words, the price of the sneakers dropped by 20%. The process for calculating percent increases and decreases can be summarized as follows: percentage increase or decrease equation left hand side equals right hand side actual increase or decrease divided by original amount multiplication 100.

8.3.3 Ups and Downs

Notepad iconActivity: Ups and Downs

(a) Gas prices rose from $3.25 to $3.51 a gallon since last week. What is the percent increase over the week?

Hint symbol
Comment

What was the actual increase in price? What was the initial cost of gas?

Solutions icon
Answer

(a) The actual increase is dollar 3.51 postfix minus dollar 3.25 equals dollar 0.26.

So, the percent increase in gas prices is multiline equation row 1 dollar times 0.26 divided by dollar times 3.25 prefix multiplication of 100 percent equals eight percent row 2 up left arrow row 3 cap o times r times i times g times i times n times a times l cap c times o times s times t.

(b) The 1981 Encyclopedia Britannica said that the Amazon Rain Forest occupied 2,700,000 square miles. Three decades later, Wolfram Alpha reported its total area to be 2.1 million square miles. Assuming each report was correct at the time of publication, find the percent decrease from 1981 to 2011.

Hint symbol
Comment

What was the actual decrease in area? What was the original area occupied by the Amazon Rain Forest?

Solutions icon
Answer

(b) The actual decrease is approximately two comma 700 comma 000 times mi super two minus two comma 100 comma 000 times mi super two equals 600 comma 000 times mi super two.

Thus, the percent decrease in the amount of area covered by the Amazon Rain Forest is:

600 comma 000 mi super two divided by two comma 700 comma 000 mi super two equals 0.222 full stop full stop full stop almost equals 22 percent

(c) You were just promoted at work and are getting a 3% raise. Congratulations! Your new salary, $56,780, reflects this increase. What was your salary before the raise?

Hint symbol
Comment

In this exercise, you already know the percent increase; we need to find the original or base salary. If $56,780 includes your raise, then it represents 100 percent prefix plus of three percent equals 103 percent of your original salary. Should your original salary be less than or greater than your new salary?

Solutions icon
Answer

(c) We know that $56,780 is 103% of the original salary. This directly translates to dollar 56 comma 780 equals 103 percent multiplication original salary. To discover your original salary, we need to do the opposite of multiplication by 103%, which would be division by 103%. Thus, your original salary was: dollar times 56 comma 780 division 103 percent multirelation equals dollar times 56 comma 780 divided by 1.03 almost equals dollar times 55 comma 126.

(d) You just purchased a gas grill that was 35% off. The sale price was $312. What was the original cost of the grill?

Hint symbol
Comment

This activity is similar to part (d) as we already know the percent decrease. Since it is a decrease, the sales price $312 represents 100 percent negative 35 percent equals 65 percent of the original price. Should the original cost of the grill be less than or greater than the sale price?

Solutions icon
Answer

(d) We know that $312 is 65% of the original price, which is the same as writing dollar 312 equals 65 percent multiplication original price. Once again, to discover the original price, we do the opposite of multiplication by 65%, which is division by 65%. Thus, the original cost of the grill is dollar times 312 division 65 percent equation sequence equals dollar times 312 divided by 0.65 equals dollar times 480.

8.3.4 Fact or Fiction?

Although percentages are used quite often in everyday life, they can be misleading and even a little tricky. In the next activity, you are asked to decide whether some statements are true or false, and to explain why.

Activity: Fact or Fiction?

For each of the following statements, say whether or not it is correct. Try working out some numerical examples to test your conjectures. Can you justify your results?

(a) If the price of a piece of jewelry is increased by 20% and then decreased by 20%, the item returns to its original price.

Hint icon
Comment

Pick any easy amount, like $100, to work with. Increase it by 20%, then take that new amount and decrease it by 20%.

Solutions icon

a. 

True


b. 

False


The correct answer is b.

Answer

(a) Let’s try it using $100. If we apply a 20% increase, we will have 120% of the starting amount. 120% of $100 is 1.20 postfix multiplication dollar 100 equals dollar 120. If we then apply a 20% decrease to this new amount, we will have 80% left. 80% of $120 is 0.8 postfix multiplication dollar 120 equals dollar 96. So, the item does not return to its original price. The statement is false.

This happens for any price. To apply an increase of 20%, you multiply the price by 1.2, and then to apply a decrease of 20%, you multiply by 0.8. This is the same as multiplying by 1.2 multiplication 0.8, which is 0.96. The new price will always be only 96% of the original price.

This also happens for any percentage. If, for instance, you apply an increase of 40% and then decrease it by 40%, the result is the same as multiplying by 1.4 and then by 0.6. With a 40% price increase and decrease, the end result will always be 84% of the original price.

(b) If the price of a pair of jeans is decreased by 20%, and two pairs are purchased, then the total paid is a 40% decrease from the original price.

Hint icon
Comment

Select an easy number to work with and see what happens!

Solutions icon

a. 

True


b. 

False


The correct answer is b.

Answer

(b) Increasing a number by a certain percent and then decreasing it by the same percent will not result in the original number.

Suppose the pair of jeans costs $100. If we apply a 20% reduction, the new price will only be 80% of the original cost. 80% of $100 is 0.8 postfix multiplication dollar 100 equals dollar 80. So, if you buy two pairs at $80 each, the total is $160 instead of $200, a reduction of $40.

The overall percent decrease is therefore 40 divided by 200 multiplication 100 percent equals 20 percent. Thus, the statement is not correct.

To help convince you that this is the case, a diagram might help. You can see that the discount on the two items together is still only 20% of the total price.

8.3.5 Percentage Points

Percentage Points

When comparing percentages, the difference between the percentages is described in terms of “percentage points.” Subtracting percentages gives percentage points.

[ A mortgage point is 1% of the amount you borrow for the purchase of your house. It is an additional fee that the bank may collect in return for offering a reduction on the annual percentage rate (APR) of the mortgage—in other words, paying this fee entitles you to a cheaper loan term. ]

Note that this is not the same as the percent increase or decrease. This distinction is something to be aware of in media reports, particularly when interest rates or buying mortgage rate points are discussed.

For example: If 72% of students starting at a community college were placed in a noncredit math course one year, but only 63% the next, the college had a decrease of 9 percentage points in students not being ready for credit math courses.

The next activity illustrates the difference between percentage points and percent change.

Notepad iconActivity: Student Loans

You took out a private student loan for college fees. You see a headline reading, “Interest rates jump 2%.” Should you be worried?

Hint icon
Comment

Most people read this headline to mean that the interest rate on the loan increases by 2 percentage points. What would that mean for you?

Is there another way to think about the headline that isn’t as sensational?

Solutions icon
Answer

The headline is ambiguous. It could mean “Interest rates increase by 2 percentage points”—for example, from 10% to 12%. This is a big deal. Your interest rate just went up 20%: open .12 minus .10 close divided by .10 multiplication 100 percent equals 20 percent.

On the other hand, the headline could mean “Interest rates increase by 2% based on your previous interest rate.” In our example, that would mean rates go up from 10% to