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

OBJECTIVES

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.

KEY WORDS

fractions

numerator

denominator

proper fractions

factor

improper fractions

mixed number

simplification

like fractions

unlike fractions

least common denominator

decimal

decimal fractions

decimal point

mixed decimal fractions

ciphers

ratio

proportion

percent

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

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.

[FIGURE 4-1 OMITTED]
```TIPS ... To Insure Perfect Solutions

The parts of a fraction are the numerator (top number) and the
denominator (bottom number). Think of the letter d in
denominator as the d in down (in other words, the bottom
number).

Numerator/Denominator
```

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:
```5 Numerator

8 Denominator
```

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

8 Denominator
```

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:

7/4 = 1 3/4 mixed number

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

1 1/3, 3 3/4, and 8 2/3

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:
```12/28 (/ 4 greatest factor) = 3/7

16/24 (/ 8 greatest factor) = 2/3
```

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:

2/9 + 5/9 = 7/9 5/9 - 3/9 = 2/9

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

[FIGURE 4-2 OMITTED]
```TIPS ... To insure Perfect Solutions

numbers of the fractions). The denominators (the bottom
numbers of the fractions) remain unchanged.

When subtracting like fractions, subtract the numerators only
(the top numbers). The denominators (the bottom numbers)
remain unchanged.
```

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:

2/5 + 3/4 = 7/20

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:

2/5 = 8/20

+ 3/4 = 15/20

23/20 = 1 3/20 Sum

Subtract

3/4 = 15/20

- 2/5 = 8/20

Difference 7/20

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.

1/3 = 5/15

+ 1/5 = 3/15

8/15

Multiplying Fractions

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:

2/3 x 7/8 = 14/24 = 7/12

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:

17/1 x 3/4 = 51/4 = 12 3/4

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.

4 24/1/ x 5/6 1 = 20

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:

8/1 x 3 5/8 = 8/1 x 29/8 = 29

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?

5 / 1/3 = x

5 x 3/1 = x

5 / 1 x 3/1 = x

15 = x

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

Dividing Fractions

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.

Example B:

14/1 / 1/2 = 14/1 x 2/1 = 28

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.
```SUMMARY REVIEW 4-1

Find the sum in each of the following addition problems. Simplify all

1. 3/7
+ 3/7

2. 4/9
+ 2/9

3. 6/9
+ 2/9

4. 1/2
+ 1/8

5. 7/16
+ 2/4

6. 3/4
+ 1/8

7. 1 5/16
+ 3 3/4

8. 23/32
+ 15/16

9. 2 1/2
3 3/4
+ 6 1/8

10. 4/9
5/12
+ 5/18

11. 1/2
1/3
+ 1/5

12. 6 5/6
10 5/12
+ 13 2/3

13. 20 4/5
16 3/15
+ 18 3/10

14. 12 1/6
9 1/3
+ 7 1/9

15. 12 3/8
6 1/4
+ 10 1/2

SUMMARY REVIEW 4-2

Find the difference in each of the following subtraction problems.
Simplify all answers to lowest terms.

16. 3/7
- 1/7

17. 4/9
- 2/9

18. 7/16
3/16

19. 3/4
- 1/3

20. 3 3/4
- 1 3/16

21. 12 1/4
- 5 5/16

22. 14 7/24
- 6 5/16

23. 45 5/8
- 32 7/16

24. 18 7/16
- 13 1/4

25. 23 5/18
- 7 5/12

26. 42 23/32
- 23 15/16

27. 43 5/8
- 18 1/4

28. 21 3/16
- 16 1/8

29. 15 5/9
- 8 2/3

30. 25 5/12
- 12 3/16

SUMMARY REVIEW 4-3

Find the product in each of the following multiplication problems.

31. 5/8 x 1/2 = --

32. 7/8 x 2/4 = --

33. 5 1/2 x 3 1/2 = --

34. 1 3/4 x 4 5/8 = --

35. 36 x 5 3/4 = --

36. 45 x 7 1/2 = --

37. 4 3/4 x 5 1/2 = --

38. 18 x 9 5/9 = --

39. 2/3 x 12 1/3 = --

40. 24 1/3 x 7 2/3 = --

41. 22 5/16 x 4 = --

42. 8 2/9 x 3 1/2 = --

43. 10 3/4 x 3 1/2 = --

44. 22 x 5 1/4 = --

45. 3 5/8 x 1/2 = --

SUMMARY REVIEW 4-4

Find the quotient in each of the following division problems:

46. 15/16 / 3/4 = --

47. 15/16 / 2 = --

48. 3/4 / 2 = --

49. 7/16 / 3/16 = --

50. 1 1/2 / 6 = --

51. 2 5/8 / 7 = --

52. 1/4 / 10 = --

53. 3/4 / 2 = --

54. 5/16 / 3 = --

55. 2 1/4 / 1 1/2 = --

56. 7 5/8 / 2 1/4 = --

57. 9 3/8 / 3/4 = --

58. 10 2/3 / 1/2 = --

59. 7/8 / 1/4 = --

60. 12 3/4 / 2/3 = --
```

DECIMALS

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:

1/10 9/100 89/1,000 321/10,000
```TIPS ... To Insure Perfect Solutions

A little trick to help remember the decimal: keep in mind that a
decade equals 10 years. Therefore, a decimal is based on the
number 10.
```

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

1/10 = 0.1

9/100 = 0.09

89/1,000 = 0.089

321/10,000 = 0.0321

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.
```Thousands Hundreds Tens Units or Ones
1,000       100     10        1
```

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:
```  2.135 Addend

14.698 Sum
```

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

When subtracting decimal fractions:
```  9.825 Minuend
- 5.450 Subtrahend
4.375 Difference
```

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:
```  4.32 Multiplicand
x 0.06 Multiplier

0.2592 Product
```

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:
```  0.445 Multiplicand
x  0.16 Multiplier

2670
445

0.07120 Product
```

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.

[ILLUSTRATION OMITTED]
```TIPS ... To Insure Perfect Solutions

Dividend divided by Divisor equals Quotient

[ILLUSTRATION OMITTED]

TIPS ... To Insure Perfect Solutions

When dividing using the calculator, enter the dividend first.
```

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.

[ILLUSTRATION OMITTED]

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.

SUMMARY REVIEW 4-5

Change the following fractions to decimals.

1. 9/10 = --

2. 3/10 = --

3. 47/100 = --

4. 7/10 = --

5. 81/100 = --

6. 49/100 = --

7. 79/100 = --

8. 83/100 = --

9. 67/100 = --

10. 3/10 = --

Write the following decimals and mixed decimal fractions in words.

11. 0.6 --

12. 0.29 --

13. 0.85917 --

14. 0.002 --

15. 0.0578 --

16. 0.67187 --

17. 6.3 --

18. 7.42 --

19. 9.135 --

20. 3.41 --

Write each of the following numbers as decimals.

21. Five tenths --

22. Fourteen hundredths --

23. Sixteen hundredths --

24. Sixty-eight ten thousandths --

25. One hundred twenty-two thousandths --

26. Sixty-four hundred thousandths --

27. Three and five tenths --

28. Sixteen hundred thousandths --

29. One hundred thousandths --

30. Eight and nine hundredths --

Find the sum in each problem.

31. 0.45 + 0.062 + 8.169 + 0.046 = --

32. 0.58 + 0.675 + 6.225 + 9.323 =

33. 0.015 + 0.702 + 10.318 + 12.792 =

34. 0.056 + 0.015 + 0.711 + 6.25 + 16.37 = --

35. 0.046 + 0.002 + 643 + 6.1675 = --

Find the difference in each problem.

36. 9.765 - 0.046 = --

37. 10 - 0.123 = --

38. 1 - 0.685 = --

39. 0.0622 - 0.0421 = --

40. 139.371 - 123.218 = --

Find the product in each problem. Round the answers to the nearest ten-thousandth when necessary.

41. 412 x 0.52 = --

42. 0.822 x 1.52 = --

43. 0.0524 x 0.132 = --

44. 6.53 x 0.38 = --

45. 7.323 x 5.452 = --

Find the quotient in each problem. Give quotients three decimal places when necessary.

46. 0.49 / 7 = --

47. 16 / 4.8 = --

48. 945 / 500 = --

49. 0.0684 / 24 = --

50. 46.76 - 400 =

51. 4 / 0.25 = --

52. 2.65 / 1.5 = --

53. 6.5 / 2.45 = --

54. 45.5 / 3.5 = --

55. 54.25 / 4.8 = --

RATIOS

The term ratio is used in food service to express a comparison between two numbers. A ratio between two quantities is the number of times one contains the other. An example of using a ratio as a comparison is shown in the following problem:

Last week, the Sirloin Steak Restaurant sold five times as many steaks as pork chops. How many steaks were sold last week?

To determine how many steaks were sold, the amount of pork chops sold must be known. The restaurant sold 425 orders of pork chops.

Multiplying the amount of pork chops that were sold by the number 5 will solve this problem.

Mathematically, the problem will be set up this way: 425 x 5 = 2,125

Therefore, 2,125 steaks were sold last week at the Sirloin Steak Restaurant.

Another way to state this fact is by saying that steaks outsold pork chops by 5 to 1.
```TIPS ... To Insure Perfect Solutions

and estimate what the correct answer should be. In the previous
problem, if the answer came out to be 425 or less, the answer
would not be reasonable. If a ratio is set up the wrong way, the
```

SUMMARY REVIEW 4-6

Determine the answers to the following questions using ratios.

1. The Cabernet Cafe sold six times as much Cabernet Sauvignon as Zinfandel. How many bottles of Cabernet Sauvignon were sold if they sold 5 bottles of Zinfandel? --

2. Lanci's sold eight times as much osso bucco as veal marsala. How many orders of osso bucco were sold if they sold 33 orders of veal marsala? --

3. The Bears Restaurant sells 15 times as many orders of chateaubriand as chicken breasts. How many orders of chateaubriand were sold if 12 orders of chicken breasts were sold? --

4. The Casola Dining Room sold half as many orders of appetizers as desserts. How many appetizers were sold if the restaurant sold 250 desserts? --

5. Scotti's sold 25 times as many orders of sausage and cheese pizzas than plain cheese pizzas. How many sausage and cheese pizzas were sold if they sold 280 cheese pizzas? --

PROPORTIONS

Mathematical or word problems that include ratios may be solved by using proportions. A proportion is defined as a relation in size, number, amount, or degree of one thing compared to another. For example, when preparing baked rice, the ratio is 2 to 1, that is, 2 parts liquid to 1 part rice. If cooking 1 quart of raw rice, the formula calls for 2 quarts water or stock. The order in which the numbers are placed when expressing a ratio is important. If you say that the ratio when cooking barley is 4 to 1, meaning 4 parts water to 1 part barley, it is not the same as saying 1 to 4, which would reverse the ratio and mean 1 part water to every 4 parts of barley (which would cause the barley to scorch and burn).

How to Use Proportions to Solve Problems

The problem reads: How much water is needed to cook 8 quarts of barley using a ratio of 4 to 1 liquid to barley?
```Step 1

Set up a written equation

Right now, you don't know how much water is needed to cook 8 quarts
of barley. Since the amount of water is an unknown, that unknown
quantity will be named x. X is the amount of water to be determined;
it does not mean to multiply.

Facts you must know to do a proportion

The water is represented by an x, since that is the answer to be
determined. "Is to" in a mathematical equation is represented by a
full colon (:) The numbers on the outside of the equation are called
extremes The numbers on the inside of the equation are called means

Step 2

Set up a mathematical equation

x : 8 = 4 : 1

Step 3

Multiply the means together

8 times 4 = 32, and put the answer on the right of the = sign
Multiply the extremes together
x times 1 = 1x, and put the answer on the left of the = sign

Step 4
The mathematical equation looks like this:
1 x = 32

Step 5

Divide both sides of the equation by the number next to the x
x /1 = x (the 1 cancels out the 1)
32/1 = 32

Therefore

x = 32, or 32 quarts of water is needed to end up with 8 quarts of
barley.
```

SUMMARY REVIEW 4-7

Solve the following problems using the proportions given.

1. How much water is needed to bake 3/4 gallon of rice using a ratio of 2 to 1 liquid to rice? --

2. How much water is required to soak 1/2 gallon of navy beans using a ratio of 3 to 1 water to beans? --

3. How much water should be used to cook 1 pint of barley using a ratio of 4 to 1 water to barley? --

4. How much water is needed to prepare orange juice using 1 1/2 pints of orange concentrate and a ratio of 3 to 1 water to concentrate? --

5. How much water is needed to prepare lemonade using 1 3/4 pints of frozen lemonade and a ratio of 4 to 1 water to lemonade? --

6. How much chicken stock is needed to bake 1 1/4 quarts of rice using a ratio of 2 to 1 stock to rice? --

7. How much water will it take to simmer 11 quarts of barley using a ratio of 4 to 1 water to barley? --

8. How much water will it take to soak 3/4 gallon of red beans using a ratio of 3 to 1 water to beans? --

9. How much water is needed to prepare orange juice using 1 3/4 quarts of orange concentrate and a ratio of 4 to 1 water to concentrate? --

10. How much water is required when making fruit punch if 1 1/4 pints of frozen punch concentrate is used and the ratio is 5 to 1 water to concentrate? --

PERCENTS

Percent plays a big part in food service language. It is used to express a rate when dealing with important business matters such as food costs, labor costs, profits, and even that undesirable figure called loss. When management meets with food service employees and wishes to emphasize points that need improving, the message conveyed is usually done by expressing a percentage. For example: last month our food cost was 45%. We must bring this figure under control.

Percent (%) means "of each hundred." Thus, 5% means 5 out of every 100. This same 5% can also be written 0.05 in decimal form. In fraction form, it is 5/100. The percent sign (%) is, in reality, a unique way of writing 100.

Percents are a special tool used to express a rate of each hundred. If 50% of the customers in a restaurant select a seafood entree, a rate of a whole is being expressed. The whole is represented by 100% and all of the customers entering the restaurant would represent the whole, or 100%. Another example of percent in food service is shown in Figure 4-3.

To find what percent one number is of another number, divide the number that represents the part by the number that represents the whole. For example: a hindquarter of beef weighs 240 pounds. The round cut weighs 52 pounds. What percent of the hindquarter is the round cut?

The 240-pound hindquarter represents the whole. The 52-pound round cut represents only a fractional part of the hindquarter. As a fraction, it is written 52/240. Simplified or reduced to lowest terms, it is 13/60. To express a common fraction as a percent, the numerator is divided by the denominator. The division must be carried out two places (hundredth place) for the percent. Carrying the division three places behind the decimal gives the tenth of a percent, which makes the percent more accurate. The finished problem looks like this:

[ILLUSTRATION OMITTED]

[FIGURE 4-3 OMITTED]

Percent means hundredths, so the decimal is moved two places to the right when the percent sign is used.

This same percent could be obtained by converting the fraction 13/60 into a percent by dividing the denominator into the numerator, as shown in the following example:

[ILLUSTRATION OMITTED]

When a percent is expressed as a common fraction, the given percent is the numerator and 100 is the denominator. For example:

40% = 40/1900 or 2/5;

20% = 20/100 or 1/5.

When a percent is changed to a decimal fraction, the percent sign is removed and the decimal point is moved two places to the left. For example:

24.5% = 0.245

75.7% = 0.757

When a decimal fraction is expressed as a percent, move the decimal point two places to the right and place the percent sign to the right of the last figure. For example:

0.234 = 23.4%

0.826 = 82.6%

The preceding percents are read as twenty-three and four-tenths percent, and eighty-two and six-tenths percent.

Remember the following points when dealing with percents:

* When a number is compared with a number larger than itself, the result is always less than 100%. For example:

72 is 80% of 90 since 72/90 = 4/5 = 80%.

* When a number is compared with itself, the result is always 100%. For example:

72 is 100% of 72 since 72/72 = 1 = 100%.

* When taking a percent of a whole number, the method of operation is to multiply. For example:

If a restaurant takes in \$9,462 in one week, but only 23% of that amount is profit, what is the profit?
```\$ 9,462
x  0.23

28386
18924

\$2,176.26 Profit
```

Finding Percents of Meat Cuts

Since the use of percents in expressing a rate is a common practice in the food service business, this exercise is included for two reasons. First, to show the student how the sides of beef, veal, pork, and, lamb are blocked out into wholesale or primal cuts, and, second, to provide another opportunity to practice and understand percents.

A side is half of the complete carcass. A saddle, used in reference to the lamb carcass, is the front or hind half of the complete carcass that is cut between the twelfth and thirteenth ribs of lamb. The side or saddle is blocked out into wholesale or primal cuts at the meat processing plant.

Shown below is a blocked out side of beef and how the percentage of each wholesale cut is found. For the side of beef in Figure 4-4, find the percentage of each wholesale cut.

To find a percentage, divide the whole into the part. In this case, the whole is represented by the total weight of the side of beef, 444 pounds. The part is represented by the weight of each individual wholesale cut. For example, the percentage of chuck is found as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 4-4 OMITTED]

The percentages of the other eight beef cuts are found by following the same procedure. Carry three places to the right of the decimal. When the exercise is completed and the percentages are totaled, the result should be 99% plus a figure representing a tenth of a percent. In the following example, the figure expressing the sum is 99.7%. A 100% result will be unlikely, since the individual percents will almost never figure out evenly.
```25.2% Chuck
4.5% Shank
7.2% Brisket
6.7% Short plate
11.2% Rib
10.3% Sirloin
5.4% Flank
7.6% Rump
21.6% Round

99.7% Total
```

SUMMARY REVIEW 4-8

1. 3/8 = --

2. 5/9 = --

3. 3/10 = --

4. 5/12 = --

5. 3/16 = --

6. 5/8 = --

7. 5/6 = --

8. 4/9 = --

9. 2/3 = --

10. 3/4 = --

Express the following percents as common fractions, whole numbers, or mixed numbers. Reduce to the lowest common denominator.

11. 3% = --

12. 40% = --

13. 75% = --

14. 60% = --

15. 100% = --

16. 38% = --

17. 68% = --

18. 200% = --

19. 65% = --

20. 140% = --

Solve the following problems. Round answers to the nearest hundredth.

21. 5% of 85 = --

22. 30% of 678 = --

23. 42% of 500 = --

24. 14.5% of 92 = --

25. 39% of \$38.20 = --

26. 75% of \$750.00 = --

27. 43% of \$468.00 = --

28. 65% of \$2,480.00 = --

29. 26% of \$4,680.00 = --

30. 79% of \$2,285.00 = --

31. A 526-pound side of beef is ordered. The chuck cut weighs 76 pounds and the round cut weighs 58 pounds. What percent of the side is the chuck? -- What percent is the round? --

32. A party for 325 people is booked. The cost of the party is \$8,125. If they are given an 8% discount on their total bill, what is the cost of the party? --

33. A party for 120 people is booked. The cost of the party is \$3,000. If they are given a 12% discount on their total bill, what is the cost of the party? --

34. If a restaurant takes in \$25,680 in one week and 26% of that amount is profit, how much is profit? --

35. Mrs. Hill purchased three new coffee urns at \$1,586 each. For buying in quantity, she is given a 3o discount on the total bill. What is the amount she paid? --

36. If a 48-pound beef round is roasted and 9 pounds are lost through shrinkage, what percent of the round is lost through shrinkage? --

37. The food cost percentage for the month is 38%. If \$26,485 was taken in that month, how much of that amount went for the cost of food? --

38. If a restaurant's gross receipts for one week total \$12,000, of which \$8,000 is profit, what percent of the gross receipts is profit? --

39. If a restaurant's gross receipts for one day total \$18,500, of which \$5,600 are expenses, what percent of the gross receipts are expenses? --

40. If a 45-pound round of beef is roasted and 9 pounds are lost through shrinkage, what percent of the round is lost through shrinkage? --

41. If a restaurant's gross receipts for one week total \$22,080 but only 24% of that amount is profit, how much are expenses? --

42. If a 52-pound round is roasted and 9 pounds are lost through shrinkage, what percent of the round roast is lost through shrinkage? --

43. If a restaurant's gross receipts for one week is \$42,682 and 26% of that amount is for labor, 37% for food cost, and 15% for miscellaneous items, how much is the profit? --

Find the percentage of beef and pork wholesale cuts in the following problems.

44. Find the percentage of each cut of beef that makes up the side of beef shown in the figure.

[ILLUSTRATION OMITTED]
```-- % Chuck         Total weight of the side --
-- % Shank
-- % Brisket
-- % Short Plate
-- % Rib
-- % Loin
-- % Flank
-- % Rump
-- % Round
```

45. Find the percentage of each cut of pork that makes up the side of pork shown in the figure.

[ILLUSTRATION OMITTED]
```-- % Jowl         Total weight of the side --
-- % Boston Butt
-- % Clear Plate
-- % Ham
-- % Picnic
-- % Hock
-- % Feet
-- % Loin
-- % Bacon
-- % Spare Ribs
-- % Fat Back

Chef Sez ...

"When developing a new chef's knife, the
percentage of each raw material going into
that item must be precisely calculated. If
the metal alloys chromium, molybdenum,
steel are not present in the correct
amounts, the knife blade could be too soft,
too hard, or too brittle. If the percentages
were incorrect, we'd end up with a poor-quality
knife that would be unacceptable
for culinary professionals to use."

Canada Cutlery Inc. is a supplier of
professional quality cutlery and kitchen
tools. They have been marketing
professional chefs knives and tools since
1954. Their CCI Superior Culinary Master'
professional knives and tools are
manufactured in state-of-the-art facilities to
not only meet, but to exceed, the demands
of the professional chef. Only the highest
quality, European, high-carbon, stainless
surgical steel is used.
```

Peter Huebner

President and Owner of Canada Cutlery

```Table 4-1 Relationship between fractions and decimals

Fraction   Decimal Equivalent

1/8             .125

1/5              .2

1/4             .25

1/3          .33333 ...

2/5              .4

1/2              .5

2/3          .66666 ...

3/4             .75

7/8             .875
```