# 4: Percentages and Decimals

• 4.1: Understand Percent
A percent is a ratio whose denominator is 100. Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per 100, so the denominator of the fraction is 100. To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal. To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent.
• 4.2: Solve General Applications of Percent
We will solve percent equations by using the methods we used to solve equations with fractions or decimals. Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we'll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.
• 4.3: Decimals (Part 1)
Decimals are another way of writing fractions whose denominators are powers of ten. To convert a decimal number to a fraction or mixed number, look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If not, the decimal converts to a mixed number. The numbers to right of the decimal point become the numerator while the place value corresponding to the final digit represent to the denominator. Finally, simplify the fraction if possible.
• 4.4: Decimals (Part 2)
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line. To round a decimal, locate the given place value and mark it with an arrow. Underline the digit to the right of the place value and determine if it is greater than or equal to 5. If it is, add one to the digit in the given place value. If not, don't change the digit. Finally, rewrite the number, removing all digits to the right of the given place value.
• 4.5: Decimal Operations (Part 1)
To add or subtract decimals, write the numbers vertically so the decimal points line up. Use zeros for place holders, as needed. Then, add or subtract the numbers as if they were whole numbers. Lastly, place the decimal in the answer under the decimal points in the given numbers. Multiplying decimals is like multiplying whole numbers—we just have to determine where to place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
• 4.6: Decimal Operations (Part 2)
Just as with multiplication, division of decimals is very much like dividing whole numbers. To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual with long division. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder. To divide decimals, we multiply both the numerator and denominator by the same power of 10 to make the denominator a whole number.

Most interest rates are quoted and advertised in terms of a percentage. But if you want to run calculations using those numbers, you’ll need to convert them to decimal format. The simplest way to do that is to divide the number by 100. ﻿ ﻿

Example: To convert 75% to decimal format, divide 75 by 100.

Search engines such as Google and Bing also make it easy to do quick calculations online, or you can also fire up your favorite calculator app if you prefer. To calculate with a search engine, type the expression you’re trying to solve into the search field. For example, type in “75/100.” ﻿ ﻿

## DECIMALS

Description: In Hotel Decimalfornia, kids play the role of an escort Snowy Owl, who has to take as many Screech Owl hotel guests to their rooms as possible before the morning comes. But these pesky Screech Owls make nothing easy. Rather than just telling the elevator escort what room they are in, they make him figure out an addition or subtraction decimal problem! Worse yet, this hotel’s rooms are numbered by decimals! The good news is the Snowy Owl escort receives a $5.00 tip for each Screech Owl he delivers successfully. If the escort brings the Screech Owl to the wrong room, however, he loses$5.00!

Type: Math Game - Decimals Focus

Mr. Nussbaum's Boardwalk Challenge - Online Game

Description: This super-fun online game requires students to correctly order whole numbers and decimals on a number line. If students accomplish this task, they are rewarded with tokens that can be redeemed for chances to play any of four boardwalk games: Ski-ball, Whack-a-Pirate, Air Hockey, and Roll the Ball. Tokens are saved as long as students play on the same computer. See instructional video for more information.

Type: Math Game - Decimals Focus

Decimals of the Caribbean - Online Game

Description: You play the role of a 17th century Caribbean Buccaneer who sails from port to port looking to steal from Spanish treasure ships. Read the decimal message that appears at the top of the screen. Then, shoot the boat that matches the decimal message (that has the numerical version of the message) with your decimal cannonball by clicking on it. You move from round to round by destroying all of the ships. After each round you successfully pass, you can obtain a special code that will take you to that round each time you play. Decimals of the Caribbean hits multiple Common Core standards as it can be played with or without decimals. Yes —- there is a version of DECIMALS of the Caribbean without decimals : )

Type: Math Game - Decimals Focus

Place Value Pirates - Online Game

Description: In Place Value Pirates, students must use their place value skills to dispatch of the wretched pirate Sir Francis Place Value and his horrible band of place value pirates! Simply read the prompt that appears at the top of the game and click or touch the pirate with the number that matches the prompt. For example, a prompt might read ""7" in the thousands place. Scan the pirates and dispatch of the one with the number that contains a "7" in the thousands place. The game consists of five rounds each round more challenging the last. For computer versions, users can earn codes after each round so they never have to start all over again. In addition, the game can be played with or without decimals.

CC Standards: 2.NBT.A.3, 4.NBT.A.1, 5.NBT.A.3, 2.NBT.A.1

Fun Adding Decimals Games - From ComputerMice

Description: Need to practice adding decimals? Fun Adding Decimals Games from Computer Mice is the perfect solution. You can practice adding three-digit numbers with decimals by playing any of 15 embedded games including target practice games, ninja baby games, spinning wheel games, and many more. Look throughout our games, math, and language arts section for more games from Computer Mice soon.

Fun Subtracting Decimals Games - From ComputerMice

Description: Need to practice subtracting decimals? Fun Subtracting Decimals Games from Computer Mice is the perfect solution. You can practice subtracting three-digit numbers with decimals by playing any of 15 embedded games including target practice games, ninja baby games, spinning wheel games, and many more. Look throughout our games, math, and language arts section for more games from Computer Mice soon.

Death to Decimals and the Adventures of Fraction Man - Online Game

Description: This fun game requires students to role play as Fraction Man - a super hero who must save the world from horrible, parachuting decimal aliens by converting them to harmless fractions.

Type: Math Game - Decimals Focus

Death to Decimals - Online Game - Spanish Version

Description: This fun game requires students to role play as Fraction Man - a super hero who must save the world from horrible, parachuting decimal aliens by converting them to harmless fractions.

Half-court Rounding - Online Game

Description: Half-court Rounding is a game in which students try to score as many points as possible by rounding numbers to the nearest ten, hundred, or tenth. This is a perfect game across grade-levels, because students can choose to shoot free-throws (1 point – round to the nearest ten) jump shots (2 points – round to the nearest hundred) and three-pointers (round to the nearest tenth). Users can try any kind of shot within the game and have 90 seconds to score as many as points as possible and beat their opponent. If they answer incorrectly, the student misses the shot. Students can also choose a two-player game where they can play a friend or classmate.

CC Standards: 3.NBT.A.1, 4.OA.A.3, 4.NBT.A.3, 5.NBT.A.3

Description: Lunch Line is a fun (and funny) game in which students practice their fractions, decimals, and percentages ordering skills. Students must arrange the celebrities and historical figures in a lunch line based on the values floating on top of their heads from least to greatest. If students arrange all ten correctly, the lunch line will proceed smoothly to the cafeteria in a straight line and they’ll be able to print out a certificate showing the line leader. If figures are positioned incorrectly, the lunch line will stagger crookedly and inefficiently to the cafeteria, thereby angering the teacher.

Type: Math Game - Decimals Focus

Description: Ensure the lunch lady doesn't turn grumpy by helping her get all eight students through the lunch line successfully. For each student, add up the prices of their lunch items and enter the value into the cash register. You have three minutes before the next class comes rushing in! Very fun.

Decimals Workshop - Online

Description: This innovative program allows students to perform decimals calculations in addition, subtraction, multiplication, and division. The program is totally customizable and allows users to select the number of problems and the numbers of digits before or after the decimal in each problem. It also provides a drag and drop, decimal-friendly work space

CC Standards: 5.NBT.A.3, 5.NBT.B.7, 6.NS.B.3

Description: In Tipster, students player the role of restaurant manager who must calculate the tip amounts for his or her servers. This fun game involves calculating percentages of numbers and quality of service. Quality of service indicated by the customers determined percentage of total bill that constitutes tip. For example, the total bill at a table is $100.00, and the service was level was a "3," the customer pays 15% making the total bill$115. Very fun!

Computation Castle - Online Game

Description: Computation Castle is a fun game that requires students to use their fractions, measurement, place value, and exponents skills to reverse a spell placed that caused the royal family to be turned into various animals.

Description: This innovative game requires students to "purchase" as many of the world's most important landmarks as possible with ten billion dollars. Students must purchase by using the game's tools to convert dollars to the native currency. For example, to buy the Eiffel tower, players must convert dollars to euros. The game provides real-time currency exchange rates and has numerous twists and turns. Games can be saved! For more detailed video instructions, check out the Burnside Billion's instructional video.

Type: Math Game - Decimals Focus

Expanded Notation Workshop

Description: This fun workshop allows students to practice the concept of standard notation. It is completely customizable and students can choose from demonstration mode or play mode. You can include or exclude decimals.

Description: This activity requires students to order ten decimals ranging from 0.08 to 8.8. All values contain just zeros and eights.

Format: Printable Activity

Ordering Decimals from Least to Greatest - Online

Description: This activity requires students to drag and drop decimals from least to greatest. Works only on desktop computers.

Use as Assessment on Google Classroom.

Ordering Decimals from Least to Greatest (to the tenths) - Online

Description: This activity requires students to drag and drop decimals from least to greatest. Works only on desktop computers.

Ordering Decimals from Least to Greatest (to the hundredths) - Online

Description: This activity requires students to drag and drop decimals from least to greatest. Works only on desktop computers.

Use as Assessment on Google Classroom.

Ordering Decimals from Least to Greatest (to the tenths and hundredths) - Online

Description: This activity requires students to drag and drop decimals from least to greatest. Works only on desktop computers.

Ordering Decimals to the hundredths and higher (to the tenths and hundredths) - Online

Description: This activity requires students to drag and drop decimals from least to greatest. Works only on desktop computers.

Decimal Tenths on a Number Line - Online

Description: This activity requires students to analyze number lines where the decimal numbers fall. It gives immediate feedback.

Use as Assessment on Google Classroom.

Rounding Decimals to the Nearest Tenth - Online

Description: This activity requires students to round decimals to the nearest tenth. It gives immediate feedback.

Use as Assessment on Google Classroom.

Rounding Decimals to the Nearest Hundredth - Online

Description: This activity requires students to round decimals to the nearest hundredth. It gives immediate feedback.

Use as Assessment on Google Classroom.

Understanding Decimals as Written Words - Online

Description: This activity requires students to convert decimal words to decimal numbers.

Expanded Notation and Decimals - Online

Description: This activity requires students to configure decimal numbers from standard notation. It gives immediate feedback.

Place Value and Decimals - Online

Description: This activity requires students to identify the values of digits in numbers with decimals. Immediate feedback is provided.

Adding Numbers With Decimals to the Tenths - Online

Description: This activity requires students to add two numbers that include tenths. It gives immediate feedback.

Adding Numbers With Decimals to the Hundredths - Online

Description: This activity requires students to add two numbers that include hundredths. It gives immediate feedback.

Adding Numbers With Decimals to the Tenths and Hundredths - Online

Description: This activity requires students to add two numbers that include either tenths or hundredths. It gives immediate feedback.

Subtracting Numbers With Decimals to the Tenths - Online

Description: This activity requires students to subtract two numbers that include tenths. It gives immediate feedback.

Subtracting Numbers With Decimals to the Hundredths - Online

Description: This activity requires students to subtract two numbers that include hundredths. It gives immediate feedback.

Subtracting Numbers With Decimals to the Tenths and Hundredths - Online

Description: This activity requires students to subtract two numbers that include either tenths or hundredths. It gives immediate feedback.

Multiplying Decimals by Powers of Ten - Online

Description: This activity requires students to multiply decimals by powers of ten. It gives immediate feedback.

Multiplying a Whole Number by a Decimal (to the tenth) - Online

Description: This activity requires students to multiply a whole number by a tenth. For example, 8 x 0.4. Immediate feedback is given.

Multiplying Decimals to the Tenths - Online

Description: This activity requires students to multiply decimals to the tenths. For example, 4.3 x 2.7. Immediate feedback is given.

Multiplying Decimals to the Hundredths - Online

Description: This activity requires students to multiply decimals to the hundredths. For example, 2.35 x 4.72. Immediate feedback is given.

Multiplying Decimals to the Tenths and Hundredths - Online

Description: This activity requires students to multiply decimals to the tenths and hundredths. For example, 7.56 x 3.3. Immediate feedback is given.

Dividing Decimals by Powers of Ten - Online

Description: This activity requires students to use mental math to divide decimal numbers by powers of ten (e.g. 19.7/10)

Dividing Decimals to the Tenths - Online

Description: This activity requires students to divide decimals by a whole number. For example, 7.8/2. Immediate feedback is given.

Comparing Equations with Fractions and Decimals Using <, >, and = - Online

Description: This activity requires students to compare equations that include fractions and decimals using inequalities.

Baseball Card Math - Jackie Robinson - Operations with Decimals

Description: Baseball Card Math is an activity in which students must make calculations and conclusions based on a player's statistics (listed on the back of a baseball card). This particular example reinforces operations with decimals.

Format: Printable Activity

Use as Assessment on Google Classroom.

Baseball Card Math - Giancarlo Stanton - Decimals and Percentages

Description: This activity reinforces the calculation of decimals and percentages with the 2016 and 2017 statistics of Giancarlo Stanton of the New York Yankees. The activity provides explanations of all statistics and vocabulary.

Format: Printable Activity

Use as Assessment on Google Classroom.

Baseball Card Math - Bryce Harper - Decimals, Ratios, and Percentages

Description: This activity reinforces the calculation of decimals and percentages with the 2016 and 2017 statistics of Bryce Harper of the Washington Nationals. The activity provides explanations of all statistics and vocabulary.

Format: Printable Activity

Use as Assessment on Google Classroom.

Death to Decimals Practice Version 2 - Converting Decimals to Fractions - Online

Description: This activity will help students learn to play Death to Decimals and reinforces converting decimals to fractions.

Death to Decimals Practice - Converting Decimals to Fractions - Online

Description: This activity will help students learn to play Death to Decimals and reinforces converting decimals to fractions.

Place Value Pirates Practice - Decimals to the thousandths and ten thousandths - Online

Description: This activity will help students get used to playing Place Value Pirates and identifying decimals to the tenths, hundredths, thousandths, and ten thousandths.

Place Value Pirates Practice - Decimals to the hundredths and thousandths - Online

Description: This activity will help students get used to playing Place Value Pirates and identifying decimals to the tenths hundredths, and thousandths.

Place Value Pirates Practice - Decimals to the tenths and hundredths - Online

Description: This activity will help students get used to playing Place Value Pirates and identifying decimals to the tenths and hundredths.

Format: Printable Activity

Decimals of the Caribbean Practice - Numbers as Written Words With Decimals - Online

Description: This activity will help students learn to play Decimals of the Caribbean and will help them identify numbers with decimals as written words.

Half-court Rounding Practice - Online

Description: This online activity requires students to round the basketballs to the nearest tens and hundreds. It gives immediate feedback.

## Conversions!

#### From Percent to Decimal

To convert from percent to decimal divide by 100 and remove the % sign.

An easy way to divide by 100 is to move the decimal point 2 places to the left:

Don't forget to remove the % sign!

#### From Decimal to Percent

An easy way to multiply by 100 is to move the decimal point 2 places to the right:

Don't forget to add the % sign!

#### From Fraction to Decimal

To convert a fraction to a decimal divide the top number by the bottom number:

### Example: Convert 25 to a decimal

Divide 2 by 5: 2 ÷ 5 = 0.4

### Example: To convert 0.75 to a fraction

0.75 × 1001 × 100

#### From Fraction to Percentage

To convert a fraction to a percentage divide the top number by the bottom number, then multiply the result by 100%

### Example: Convert 38 to a percentage

First divide 3 by 8: 3 ÷ 8 = 0.375

Then multiply by 100%: 0.375 × 100% = 37.5%

#### From Percentage to Fraction

To convert a percentage to a fraction, first convert to a decimal (divide by 100), then use the steps for converting decimal to fractions (like above).

## Decimals

A decimal shows the numerical amount of a fraction or percentage. The fraction ¾ as a decimal is 0.75 (75/100). The percentage 50% as a decimal is 0.50. It's essential to know how to convert fractions and percentages to decimals so you can do the math needed to figure out problems. At the store, you will most likely see decimals when looking at the prices of products. While learning about decimals, it's also helpful to practice rounding. Marketing experts say that a person is more likely to buy something if the price is just a penny shy of the next whole number. For example, you will see many products that cost $1.99 or$4.99, as opposed to $2 and$5. Even though it's practically the same amount, our brains are more likely to buy it at $1.99 than at$2. But when you're aware of this and understand decimals and how to round them, you'll be able to see that $1.99 is practically$2.

## Percents to Fractions

When converting percents to fractions, apply the definition of percent and then reduce.

Example 15: Convert to a fraction: 28 % .

Applying the definition of percent is equivalent to removing the percent sign and multiplying by 1 100 .

Example 16: Convert to a fraction: 66 2 3 % .

Solution: First, convert to an improper fraction and then apply the definition of percent.

Try this! Convert to a fraction: 3 7 31 % .

### Video Solution

Example 17: Using the given pie chart, calculate the total number of students that were 21 years old or younger if the total US community college enrollment in 2009 was 11.7 million.

Solution: From the pie chart we can determine that 47% of the total 11.7 million students were 21 years old or younger.

Source: American Association of Community Colleges.

Convert 47% to a decimal and multiply as indicated by the key word “of.”

Answer: In 2009, approximately 5.5 million students enrolled in US community colleges were 21 years old or younger.

### Key Takeaways

• To convert a decimal to a mixed number, add the appropriate fractional part indicated by the digits to the right of the decimal point to the whole part indicated by the digits to the left of the decimal point and reduce if necessary.
• To convert a mixed number to a decimal, convert the fractional part of the mixed number to a decimal using long division and then add it to the whole number part.
• To add or subtract decimals, align them vertically with the decimal point and add corresponding place values.
• To multiply decimals, multiply as usual for whole numbers and count the number of decimal places of each factor. The number of decimal places in the product will be the sum of the decimal places found in each of the factors.
• To divide decimals, move the decimal in both the divisor and dividend until the divisor is a whole number and then divide as usual.
• When rounding off decimals, look to the digit to the right of the specified place value. If the digit to the right is 4 or less, round down by leaving the specified digit unchanged and dropping all subsequent digits. If the digit to the right is 5 or more, round up by increasing the specified digit by one and dropping all subsequent digits.
• A percent represents a number as part of 100: N % = N 100 .
• To convert a percent to a decimal, apply the definition of percent and write that number divided by 100. This is equivalent to moving the decimal two places to the left.
• To convert a percent to a fraction, apply the definition of percent and then reduce.
• To convert a decimal or fraction to a percent, multiply by 1 in the form of 100%. This is equivalent to moving the decimal two places to the right and adding a percent sign.
• Pie charts are circular graphs where each sector is proportional to the size of the part out of the whole. The sum of the percentages must total 100%.

### Topic Exercises

Perform the operations. Round dollar amounts to the nearest hundredth.

31. A gymnast scores 8.8 on the vault, 9.3 on the uneven bars, 9.1 on the balance beam, and 9.8 on the floor exercise. What is her overall average?

32. To calculate a batting average, divide the player’s number of hits by the total number of at-bats and round off the result to three decimal places. If a player has 62 hits in 195 at-bats, then what is his batting average?

Part B: Percents to Decimals

Convert each percent to its decimal equivalent.

51. Convert one-half of one percent to a decimal.

52. Convert three-quarter percent to a decimal.

54. What is 50% of one hundred?

58. What is 9 1 2 % of $1,200? 59. If the bill at a restaurant comes to$32.50, what is the amount of a 15% tip?

60. Calculate the total cost, including a 20% tip, of a meal totaling $37.50. 61. If an item costs$45.25, then what is the total after adding 8.25% for tax?

62. If an item costs $36.95, then what is the total after adding 9¼% tax? 63. A retail outlet is offering 15% off the original$29.99 price of branded sweaters. What is the price after the discount?

64. A solar technology distribution company expects a 12% increase in first quarter sales as a result of a recently implemented rebate program. If the first quarter sales last year totaled $350,000, then what are the sales projections for the first quarter of this year? 65. If a local mayor of a town with a population of 40,000 people enjoys a 72% favorable rating in the polls, then how many people view the mayor unfavorably? 66. If a person earning$3,200 per month spends 32% of his monthly income on housing, then how much does he spend on housing each month?

## Converting Fractions into Decimals and Percentages

Hello and welcome to this video about Converting Fractions to Decimals and Percents! In this video we will explore:

• How to visually convert fractions to decimals and percents and
• How to numerically convert fractions to decimals and percents

Before we get started, let’s review a couple of key concepts that we will use to help the math make sense.

Consider the fraction (frac<10><10>). One way to think about fractions is to think of them as division.

In other words, ten-tenths is one whole.

(frac<10><10>) can be written equivalently as (frac<100><100>). The fraction bar can be said as “per”, so this expression can be said as “100 per 100”. The word “percent” literally means “per 100”, so “100 per 100” means 100 percent.

Therefore, when the same number is divided by itself, the result as a decimal is 1 and as a percent is 100%.

But what happens when our fraction is less than 1? Let’s take a look:

Consider the fraction (frac<1><4>). Visually, this is what’s happening:

We can see in the diagram that (frac<1><4>) of the whole is (frac<25><100>). (frac<25><100>) means “25 per 100”, so it equals 25%.

Now let’s figure out how to convert this into a decimal.

We’re going to take our fraction (frac<1><4>), which is the same as saying 1 divided by 4.

Dividing this way doesn’t look like it will work. But using our knowledge of place value, we can make it work:

First, rewrite the 1 as 1.0. Instead of 1, the dividend is now ten tenths.

Second, 4 ones will go into ten-tenths two-tenths times.

Fifth, we’re gonna rewrite the original dividend as 100-hundredths and bring the new 0 down.

Sixth, 4 ones goes into 20-hundredths 5-hundredths times.

Seventh, we’re gonna multiply (4 imes 0.05=0.20). Then we’ll subtract to get 0.

Let’s see this work with a non-unit fraction, like (frac<3><16>).

Here’s what the division looks like. The sequence of adding a decimal and dividing repeats as often as necessary until either the remainder is 0 or the decimal begins to repeat.

So, (frac<3><16>=0.1875) and (0.1875=frac<1,875><10,000>=frac<18.75><100>). Therefore, 0.1875 is the same as 18.75% because it’s 18.75 per, 100.

How about a repeating decimal? Everything is the same and the process can be stopped when it is clear that the decimal repeats. Here’s a quick example: (frac<1><3>).

The process of subtracting 9 units from 10 units repeats, causing the decimal to repeat.

Lastly, consider a fraction that is greater than 1, such as (frac<5><2>).

Visually, here’s what we have:

We can see that the shaded quantity is (frac<1><2>+frac<1><2>+frac<1><2>+frac<1><2>+frac<1><2>=frac<5><2>). Each (frac<1><2>=50), so we can say equivalently (frac<250><100>), which, as a decimal, is 2.5 (meaning (2frac<1><2>) wholes, which the diagram shows). We can also equivalently say 250 per 100, or 250%.

## Chapter 4 Fractions, decimals, ratios, and percents.

At the completion of this chapter, the student should be able to:

1. Simplify (or express) a fraction in lower terms without changing the value of the fraction.

2. Add, subtract, multiply, and divide fractions.

3. Change fractions to decimals.

4. Write decimals and mix decimal fractions in words.

5. Write numbers as decimals.

6. Find sums, differences, products, and quotients in decimal problems.

7. Solve problems by using given ratios.

8. Write common fractions as percents.

9. Write percents as common fractions, whole numbers, or mixed numbers.

10. Find the percents of meat cuts.

Now that you have learned about the roles that addition, subtraction, IM multiplication, and division play in food service operations, it is time to learn about the equally important roles played by fractions, decimals, ratios, and percents.

Fractions are sometimes used in a restaurant operation. They may play an important part when converting standard recipes, dealing with the contents of a scoop or dipper, and dividing certain items into serving portions, but compared to most other math operations, the use of fractions is limited. This does not mean, however, that the knowledge of fractions is unimportant. Situations will occur in your workplace and everyday life where knowledge of this subject will be required, so review this section just as intensely as the others.

A fraction indicates one or more equal parts of a unit. For example, a cake is usually divided into eight equal pieces. (See Figure 4-1.) If this is done, the following statements are true about the parts or slices of the cake:

One part is 1/8 of the cake.

Three parts are 3/8 of the cake.

Seven parts are 7/8 of the cake.

Eight parts are 8/8 of the cake or the whole cake.

Another example of this same teaching tool is the division of a 9-inch pie into slices. A 9-inch pie is usually cut into seven equal servings. In this case, the fractional parts would be a little different but the same theory shown for the sliced cake would hold true:

One part is 1/7 of the pie.

Three parts are 3/7 of the pie.

Six parts are 6/7 of the pie.

Seven parts are 7/7 of the pie or the whole pie.

Since fractions indicate the division of a whole unit into equal parts, the numeral placed above the division or fraction bar indicates the number of fractional units taken and is called the numerator. The numeral below the bar represents the number of equal parts into which the unit is divided and is called the denominator. Thus, if a cantaloupe is cut into eight equal wedges, but only five of those wedges are used on a fruit plate, the wedges used are represented by the fraction 5/8.

A common fraction is written with a whole number above the division bar and a whole number below the bar. For example:

A proper fraction is a fraction whose numerator is smaller than its denominator. For example:

This type of fraction is in its lowest possible terms when the numerator and denominator contain no common factor. A factor refers to two or more numerals that, when multiplied together, yield a given product. For example: 3 and 4 are factors of 12. The fraction 5/8 is in its lowest possible terms because there is no common number by which both can be divided. (See "The Simplification of Fractions" later in this chapter.)

An improper fraction is a fraction whose numerator is larger than its denominator, and whose value is greater than a whole unit. If, for instance, 1 3/4 hams is expressed as an improper fraction, it is expressed as 7/4 since the one whole ham would be 4/4, and the extra 3/4 makes it 7/4. Such fractions can be expressed as a mixed number by dividing the numerator by the denominator, as shown below:

A mixed number is a whole number mixed with a fractional part. For example:

The Simplification of Fractions

Simplification is a method used to express a fraction in lower terms without changing the value of the fraction. This is achieved by dividing the numerator and denominator of a fraction by the greatest factor (number) common to both. For example:

The value of these fractions is unchanged, but they have been simplified or reduced to their lowest possible terms.

A mixed number is usually expressed as an improper fraction when it is to be multiplied by another mixed number, a whole number, or a fraction. The first step is to express the mixed number as an improper fraction. This is done by multiplying the whole number by the denominator of the fraction, and then adding the numerator to the result. The sum is written over the denominator of the fraction. For example:

1 3/4 x 4 1/4 = 7/4 x 17/4 = 119/16 = 7 7/16

In this example, the whole number (1) is multiplied by the denominator of the fraction (4). To this result (4), the numerator (3) is added. The sum (7) is written over the denominator (4), creating the improper fraction 7/4. The same procedure is followed in expressing the mixed number 4 1/4 as the improper fraction 17/4. When the two mixed numbers are expressed as improper fractions, the product is found by multiplying the two numerators together and the two denominators together, resulting in the improper fraction 119/16, and simplifying (reducing) it to the lowest terms, 7 7/16.

Fractions are used most often to increase and decrease recipe ingredients. Ingredients such as herbs and spices generally appear in a recipe in fractional quantities. The addition and subtraction of fractions are used most often when adjusting recipes. However, all operations dealing with fractions will be required at some point on the job or in everyday activity. One example of the use of fractions in food service is illustrated in Figure 4-2.

Before fractions can be added or subtracted, they must have the same denominator. Like fractions are fractions that have the same denominator. To add or subtract like fractions, add or subtract the numerators and write the result over the common denominator. Examples of adding and subtracting like fractions are shown below:

Note how simple it is to add and subtract like fractions. The next step, dealing with unlike fractions, becomes a little more difficult.

To determine the least common denominator, multiply the two denominators by each other. For example, 3/4 plus 5/7 equals what? We multiply 4 by 7, which equals 28. So, the least common denominator is 28. There are times when multiplying the two denominators does not result in the least common denominator. For example, 7/12 - 1/2 has a denominator of 24 but that is not the least common denominator. First, determine whether the 12 could be the least common denominator. Divide the 2 into the 24, and you discover that it is! A little practice and detective work in this area will help determine the least common denominator.

Unlike fractions have different denominators. They are more difficult because only like things can be added or subtracted. Therefore, to add or subtract fractions that have unlike denominators, the fractions must first be expressed so the denominators are the same. To find this common denominator, multiply the two denominators together (5 x 4 = 20). The product will, of course, be common to both. For example:

When a number is found that is a multiple of both denominators, the fractions are then expressed in terms of the common denominator, so 2/5 is 8/20 and 3/4 is 15/20. These fractions have now become like fractions that can be added or subtracted without too much difficulty, as shown below:

In adding and subtracting unlike fractions, the common denominator may be any number that is a multiple of the original denominators. However, always use the least common denominator to simplify the work The least common denominator is the smallest number that is a multiple of both denominators. For example: if 1/3 and 2/5 are to be added, the least common denominator is 15, since it is the smallest multiple of both 3 and 5.

Multiplying fractions is considered the simplest operation with fractions. When multiplying two fractions, multiply the two numerators and place the results over the result obtained by multiplying the two denominators. For example:

Note: 14/24 expressed in lowest terms is 7/12.

If multiplying a whole number by a fraction, multiply the whole number by the numerator of the fraction, place the result over the denominator of the fraction, and divide the new numerator by the denominator. For example:

Sometimes it is possible to simplify the problem before multiplying. In the example below, b is a factor of 24 because 24 contains b exactly 4 times. This step is commonly called canceling.

If the numerator and denominator can be divided evenly by the same number, simplify to lowest terms. For example:

32/48 Numerator and denominator can be divided 2/3 evenly by the common factor 16, resulting in:

If multiplying by one or two mixed numbers, express the mixed number or numbers as improper fractions and proceed to multiply as with two fractions. For example:

2 1/3 x 4 3/5 = 7/3 x 23/5 = 161/15 = 10 11/15

A note of caution: when dividing by a fraction (which is less than 1), the answer is greater than the dividend. For example, a serving of a hamburger is 1/3 pound. How many hamburgers can be obtained from five pounds of ground beef?

From five pounds of ground beef, 15 hamburgers (weighing 1/3 pound each) can be obtained.

Dividing fractions is perhaps the most difficult operation because it involves the process of inverting (turning over) the divisor. Always be careful to invert the correct fraction. Mistakes can be easily made when inverting takes place. After inverting the divisor, proceed to operate the same as you would when multiplying fractions. Example A:

5/8 / 1/2 = 5/8 x 2/1 = 5/4 x 1/1 = 5/4 = 1 1/4

Step 1: The divisor 1/2 is inverted to 2/1.

Step 2: Cancel a factor of 2 from the 8 and 2.

Step 3: Multiply 5/4 x 1/1 to get 5/4.

Step 4: The result 4 is an improper fraction and must be reduced to a mixed number, which would be 1 1/4.

Step 1: The divisor 1/2 is inverted to 2/1.

Step 2: Multiply 14/1 x 2/1 = 28.

Example B results in a whole number so, of course, reducing is not necessary.

A decimal is based on the number 10. The decimal system refers to counting by tens and powers of 10. The term decimal refers to decimal fractions. Decimal fractions are those fractions that are expressed with denominators of 10 or powers of 10. For example:

Instead of writing a fraction, a point (x) called a decimal point is used to indicate a decimal fraction. For example:

Individuals should know and memorize the relationship of decimals to common fractions that is used in the food service industry. If a food service employee is working at a delicatessen and a guest asks for 1/2 pound of cheese, the employee must know that when the scale reads 0.5, that represents 1/2 pound, based on the decimal system. Table 4-1 illustrates common fractions that are used in the food service industry and their decimal equivalents. It is helpful for a food service professional to memorize this information.

Numbers go in both directions from the decimal point. The place value of the numbers to the left starts with the units or ones column, and each column (moving left) is an increasing multiple of 10.

To the right of the decimal point, each column is one-tenth of the number in the column immediately to its left. For example, one-tenth of one is 1/10. Thus, the decimals to the right of the decimal point are 0.1, 0.01, 0.001, 0.0001, and so on. These numbers stated as decimal fractions are 1/10, 1/100, 1/1,000, and 1/10,000.

Decimal fractions differ from common fractions because they have 10 or a power of 10 for a denominator, whereas common fractions can have any number for the denominator. To simplify writing a decimal fraction, the decimal point is used. For example, to express the decimal fraction 725/1,000 as its equivalent using a decimal point:

1. Convert the decimal fraction to a decimal first by writing the numerator (725).

2. Count the number of zeroes in the denominator and place the decimal point according to the number of zeroes. There must always be as many decimal places as there are zeroes in the denominator (0.725).

Often, when writing a decimal fraction as a decimal, it is necessary to add zeroes to the left of the numerator before placing the decimal point to indicate the value of the denominator. For example: 725/10,000 = 0.0725, which should be read as seven hundred twenty-five ten thousandths.

When a number is made up of a whole number and a decimal fraction, it is referred to as a mixed decimal fraction. To write a mixed decimal fraction, the whole number is written to the left of the decimal point and the fractional part to the right of the decimal point. For example: 7 135/1,000 = 7.135. The decimal point is read as "and," so to read this mixed decimal fraction, the whole number is read first, then the decimal point as "and." Next, read the fraction as a whole number and state the denominator. Following this procedure, 7.135 is read "seven and one-hundred thirty-five thousandths."

To add or subtract decimal fractions, keep all whole numbers in their proper column and all decimal fractions in their proper column. Remember that the decimal point separates whole numbers from fractional parts. It is therefore very important that decimal points are directly in line with one another. For example, in adding decimal fractions:

Note that the decimal point in the sum goes under the decimal point of the other numbers.

When subtracting decimal fractions:

Note that the decimal point in the difference goes under the decimal point in the minuend and subtrahend.

To multiply decimal fractions, follow the same procedure as when multiplying whole numbers to find the product. To locate the decimal point in the product, count the number of decimal places in both the multiplicand and the multiplier. The number of decimal places counted in the product is equal to the sum of those in the multiplicand and multiplier. For example:

There are four decimal places in the multiplicand and multiplier. Therefore, four decimal places are counted from right to left in the product.

In many cases, the total number of decimal places in the multiplicand and multiplier exceeds the number of numerals that appear in the product. In such cases, ciphers (zeroes) are added to the left of the digits in the product to complete the decimal places needed:

Note that a cipher is added to the product to complete the five decimals places required.

To divide decimal fractions, proceed as if the numbers were whole numbers and place the decimal point as follows:

1. When dividing by whole numbers, place the decimal point in the answer directly above the decimal point in the dividend.

2. When dividing a whole number or mixed decimal by a mixed decimal or decimal fraction, change the divisor and dividend so the divisor becomes a whole number. This is accomplished by multiplying both the dividend and divisor by the same power of 10. The divisor and dividend can be multiplied by the same power of 10 without changing the value of the division.

In the preceding example, the divisor 0.25 is made into the whole number 25 by multiplying by 100, moving the decimal point two places to the right. Since the dividend must also be multiplied by 100, the decimal point in the dividend is also moved two places to the right, so 3 becomes 300. The decimal point in the quotient is always placed directly over the decimal point in the dividend. The answer is 12, a whole number. Note that when moving a decimal point, an arrow is used to show where the decimal point is to be moved.

• Write decimals as percents for example write 0.29, 0.05, or 1.07 as percents. Includes percentages that are more than 100%.
• Write decimals as percents for example, write 0.391 as a percent involves tenth of a percent for example, write 29% or 283% as decimals
• Convert percents to decimals and vice versa for example write 56% as a decimal or write 1.83 as a percent

Worksheets about percentages (for example, find what percentage 78 is of 123)

Free lessons on percent topics:

## Comparing Fractions, Decimals and Percents

When you want to compare a fraction to a decimal or a percent, it is usually easiest to convert to a decimal number first, and then compare the decimal numbers.

We first convert (frac<7><13>) to a decimal by dividing to get 0.5385. Now notice that

When we preform a hypothesis test in statistics, We have to compare a number called the p-value to another number called the level of significance. Suppose that the p-value is calculated as 0.0641 and the level of significance is 5%. Compare these two numbers.

We first convert the level of significance, 5%, to a decimal number. Recall that to convert a percent to a decimal, we move the decimal over two places to the right. This gives us 0.05. Now we can compare the two decimals: