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Chapter 3 Addition, subtraction, multiplication, and division.

OBJECTIVES

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

1. Find sums.

2. Find differences.

3. Check subtraction by adding.

4. Find products using the multiplication table.

5. Use a step-by-step procedure to find products.

6. Check the accuracy of the multiplication product.

7. Find quotients.

8. Find the remainder.

9. Check the accuracy of the division quotient.

KEY WORDS

addition

sum

subtraction

minuend

subtrahend

difference

trading (borrowing)

multiplication

multiplicand

multiplier

product

subproducts

division

dividend

divisor

quotient

remainder

In this chapter, the four most basic, and essential, math functions--addition, subtraction, multiplication, and division-will be covered. In these pages, you will not only revisit the foundation of all computation, but you will also see how these computations are integral to food service operations.
TIPS ... To Insure Perfect Solutions

We read words from left to right. We solve math problems
working from right to left.


ADDITION

Addition can be considered one of the most popular math functions because it means an increase is taking place. In any business venture, it is the desire of the operator that the increase shows up in the form of a nice profit.

Addition is the act of putting things together, or combining things or units that are alike, to obtain a total quantity. This total quantity is called the sum. If you have $6.00 and are given $10.00, you have a total of $16.00. This simple addition problem can be written two different ways. For example:

$6.00 + $10.00 = $16.00

Written with the numbers placed in a row or line, this is called the horizontal position. The horizontal position is seldom used in a problem involving large numbers because the way the numbers are positioned makes it difficult to calculate the answer.

This same problem can also be written in what is called the vertical or column position.

$ 6.00 +10.00/$16.00

[FIGURE 3-1 OMITTED]
TIPS ... To Insure Perfect Solutions

Addend plus Addend equals Sum

or

Addend
+ Addend

Sum


In both instances the plus sign, a symbol of operation, is used to indicate that the numbers are to be added. Figure 3-1 shows an example of how addition is used in food service.

As another example, when serving a sirloin steak dinner, if there are 25 steaks ready to be served, 18 cooking on the broiler, and 125 stored in the refrigerator, the food service establishment has a total of 168 steaks on hand. Since all of the units to be added are alike, it is unnecessary to write out what the units are. Therefore, the addition can be written in either of the following ways:
25 + 18 + 125 = 168 (or) 25
                         18
                      + 125

                        168


It is not necessary to use the plus sign when three or more numbers are added together in the vertical position because it cannot be confused with any other arithmetic operation. Remember that each digit must be placed in the correct column to give it the proper value.

Computing addition problems manually may be considered a task of the past because everyone uses calculators today. Nevertheless, it is still wise to understand the rules and proper steps for solving addition problems this way because situations may arise when a calculator is unavailable, or, worse yet, the battery may go dead while you are performing an important calculation. It is never wise to depend entirely on calculators or adding machines. Absorb the information and examples presented in this chapter so that you can add rapidly and accurately, should automation ever fail you.

Guidelines

There are several basic guidelines that can be followed in arithmetic problems to save time and improve accuracy.

Be Neat. If you are interested in a food service career, neatness is essential in both your physical appearance and personal hygiene. Neatness is also important in math. It does not require additional time to write neatly and carefully, placing each number in the proper column directly under the number above it, as shown below.
  2 1 5 8
      3 6
    5 2 6
    2 0 9
      8 5
+ 1 9 2 2

  4 9 3 6

TIPS ... To Insure Perfect Solutions

Always write down trades (carryovers).


Neatness is also important when trading (carrying) numbers over from one column to the next. The sum of the ones column in the preceding problem is 36 and not b. The 3 (actually 30) is traded (carried over) to the tens column. When the trade (carryover) 3 is written, it is placed neatly at the top of the tens column so that it is not overlooked when counting that column. The sum of the tens column is 23 (actually 230). The 2 is traded (carried over) to the hundreds column. Again, place the 2 neatly over the top of the hundreds column. Follow this procedure for each column in the problem as shown below:
(1)(2)(3)
 2  1  5 8
       3 6
     5 2 6
     2 0 9
       8 5
 + 1 9 2 2

   4 9 3 6


Check Your Work. All work must be checked, even when using a calculator, to be sure your addition is correct. Mistakes can be made even when calculating automatically. Even if you find, through checking, that your work is always correct, the practice should still be continued. The penalty for mathematical errors in the classroom is only a lower grade; the penalty for errors in a food service operation can result in a monetary loss for both you and your employer.

The common method of checking addition is to add the individual columns in reverse order. For example, if you originally added the columns from top to bottom, which is the usual practice, you can check your work by adding a second time, from bottom to top. Just reverse your procedure.

Increase Your Accuracy and Speed. Addition is often simplified if numbers are combined and then added. For example, in the problem 7 + 3 + 8 + 2 + 5 + 3 = 28, the addition is greatly simplified by combining 7 and 3 into 10, and 8 and 2 into 10, which adds up to twenty. Adding the remaining numbers (5 and 3) gives 8, which makes a total of 28.

Eliminate unnecessary steps when adding. One method of increasing your speed and accuracy is that instead of thinking 7 plus 3 equals 10, automatically see the 7 and 3 combination as 10. When adding the problem in the previous paragraph, do not think that 10 plus 10 equals 20, plus 5 equals 25, plus 3 equals 28. Think 10, 20, 28.

If these guidelines are followed, they will help you to ensure the accuracy and speed required for any type of addition problem, especially those related to food service. Some of these guidelines, such as neatness and checking your work, apply to all arithmetic operations.

SUMMARY REVIEW 3-1

Find the sum of each of the following addition problems. Use the methods and guidelines suggested in this chapter. To improve your math skills, calculate these problems manually and then check your work.
1. 6
 + 7

2. 4
   7
   6
 + 3
 ___
3. 3 + 5 + 8 + 17 = --

4. 24 + 19 + 12 + 28 = --

5. 259
 + 148

6. 338
   225
 + 648

7. $56.17
    49.54
    26.38
  + 18.67

8. 312
   422
   345
   239
 + 751

9. $366.26
    441.31
    374.43
    223.23
   +154.73

10. 8
   28
  335
 2765
  222
  589
   17
  259
+ 126

11. $43.16
     42.19
     41.20
     31.42
     29.73
     25.43
     27.63
     18.64
   + 12.25

12. $555.25
     216.11
     140.18
     713.14
     726.12
     289.82
     326.22
     129.10
   + 222.12


13. If a person's guest check includes an omelette @ $6.99, French toast @ $5.99, and coffee @ $1.25, how much is the total check? --

14. When preparing a fruit salad bowl, the following items were used: oranges $1.98, apples $0.79, grapes $0.92, bananas $1.69, strawberries $2.68, peaches $1.47, and pineapple $0.98. What was the total cost of the fruit salad bowl? --

15. The restaurant had 203 orders of chicken in the freezer, 126 in the walk-in refrigerator, and 109 in the reach-in refrigerator. How many orders of chicken did they have on hand? --

SUBTRACTION

Subtraction means to take away. It is the removal of one number of things from another number of things (see Figure 3-2). The word "subtract" is very seldom used except in its mathematical sense. The popular word used in the business world is "deduct," which also means to take away. For example, instead of saying "he subtracted a discount of $2.00 from my bill," the statement would be "he deducted a discount of $2.00 from my bill."

[FIGURE 3-2 OMITTED]

If you have $12.00 and spend $8.25, the subtraction problem is written as follows:
$12.00
- 8.25

$ 3.75


The minus sign (-) must always be used so that the problem is not confused with another mathematical operation.

Each of the factors in subtraction has a name. The original number before subtraction, or before anything is removed, is called the minuend. In the example above, the minuend is $12.00. The number removed from the minuend is called the subtrahend. Finally, the amount left over or remaining after the problem is completed is called the difference.
 1585 Minuend
- 742 Subtrahend

  843 Difference


Trading (Borrowing)

Subtraction frequently requires trading (borrowing). When trading, add ten to the ones column of the minuend. At the same time, diminish the number in the tens column by one. Although the minuend is usually a larger number than the subtrahend, a particular digit in the minuend may be less than the digit beneath it in the subtrahend, so trading (borrowing) is required. Example: 1,723 - 688 = 1,038. When this problem is set up in the vertical position, it looks like this:

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

Minuend minus Subtrahend equals Difference

or

  Minuend
- Subtrahend

Difference

TIPS ... To Insure Perfect Solutions

Always write down trades (carryovers).


The minuend (1,723) is dearly a larger number than the subtrahend (688). However, the digit 8 in the ones column of the subtrahend is larger than the digit 3 in the ones column of the minuend. Since 8 cannot be subtracted from 3, it becomes necessary to trade (borrow) from the tens column.

To indicate that a ten has been traded (borrowed), cross out the 2 in the tens column and write 1 above it. If this step is neglected, you may sometimes forget that you traded (borrowed).
  1 13
 1 723
-  688

     5


Add the traded (borrowed) ten to the 3 in the ones column, which increases the 3 to 13. Then subtract 13 - 8 = 5. The 5 is written beneath the bar in the ones column.
  113
 1723
- 688

    5


In the tens column, 8 cannot be subtracted from 1 (actually 80 from 10), so it is necessary to trade (borrow) a hundred from the hundreds column. This is done by crossing out the numeral 7 in the hundreds column and writing 6 above it.
   6 11 13
 1 7 2 3
-  6 8 8

      5


Return to the tens column, and subtract 8 from 11 (actually 80 from 110) to get 3 (actually 30). Write the 3 in the tens column beneath the bar.
 6 11 13
 1 7 2 3
- 6 8 8

     35


Moving to the hundreds column, 6 can be subtracted from 6 (which is actually 600 from 600). Even though nothing will remain, a zero is used to hold a place. Trading (borrowing), therefore, is unnecessary in this column.
  6 11 13
 1 7 2 3
-  6 8 8

   0 3 5


The problem is completed by bringing down the 1 that remains in the thousands column. The completed problem appears below:
   6 11 13
 1 7  2  3
-  6  8  8

   1 0 3 5


Checking Subtraction

Checking any math problem is always a wise step to take, even when using a calculator, because errors are so easily made. The common way of checking a subtraction answer is to add together the subtrahend and the difference. The sum of these two numbers should equal the minuend.
Subtraction:

  3 6 4 2 Minuend
- 2 1 3 2 Subtrahend

  1 5 1 0 Difference

Check:

  2 1 3 2 Subtrahend
+ 1 5 1 0 Difference

3 6 4 2 Minuend


SUMMARY REVIEW 3-2

Find the difference in the following subtraction problems. Calculate these problems manually and then check your work.
1. 955 - 314 = --

2. 688 - 520 = --

3. 4925 - 1649 = --

4. 743
 - 526

5. 828
 - 593

6. 5197
 - 2058

7. $24.43
  - 18.20

8. $76.32
  - 56.54

9. $132.77
   - 59.99

10. $7,333.64
   - 6,132.45
    _________

11. $221,004.03
    - 79,625.07


12. The bill for the wedding party came to $1,585.21. They were given a discount of $121.00 because the guaranteed number of people attended. What was the total bill? --

13. The restaurant thought they had 76 live lobsters on hand. Taking a quick inventory, it was discovered that 29 were dead. How many live lobsters did the restaurant have left? --

14. A restaurant had 240 chickens in the freezer. They used 31 for a special party. How many did they have left? --

15. A restaurant purchased 268 pounds of sirloin. Forty-eight pounds were lost in boning and trimming. How many pounds were left? --

MULTIPLICATION

Multiplication is another math operation where an increase takes place. Methods that bring forth an increase, such as addition, seem to be most popular, especially if the increase is spelled "profit" or carries a dollar sign. Multiplication can be thought of as a shortcut for a certain type of addition problem. In multiplication, a whole number is added to itself a specified amount of times. For example, 4 x 2 = 8 is another way of expressing 4 + 4 = 8. The number 4 is added to itself two times.

Since a relationship exists between addition and multiplication, it does not matter which operation is used in simple problems such as the one above. However, when the problem consists of problems involving large numbers (such as 4531 X 6580 = ?), addition is an impractical way of solving it. This is when multiplication becomes useful as a shortcut for addition. Working out the preceding problem is quite lengthy using multiplication, as shown in the following example. Now just imagine what would be involved if done by addition:
   6580
 x 4531

   6580

  19740
 32900
26320

29813980

TIPS ... To Insure Perfect Solutions
Multiplier times Multiplicand equals Product

or

Multiplier
x Multiplicand

Product


Each number involved in the multiplication process has a name. The number that is added to itself (4 in the example, 4 x 2 = 8) is called the multiplicand (which means "going to be multiplied"). The number representing the amount of times the multiplicand is to be added to itself is called the multiplier (number 2 in the example). The result of multiplying the multiplicand by the multiplier is called the product. The product in our example is 8.

The following example gives the names and functions of the various numbers involved in the multiplication operation.
  362 Multiplicand
x  32 Multiplier

  724 Subproduct
10860 Subproduct

11584 Product

Function                                Name

Number to be added to itself            Multiplicand
Number of times to be added to itself   Multiplier
Product of the ones column              Subproduct
Product of the tens column              Subproduct
Final result (answer)                   Product


In this example, subproducts are shown. Subproducts (sub meaning under, below, or before; product being the result of multiplying) occur whenever the multiplier consists of two or more digits. In this case, the multiplier is 32. The first subproduct is the result of the product 362 x 2 = 724. The second subproduct (10,860) is the result of multiplying 362 x 30. The zero at the end of this subproduct is not necessary, because 4 + 0 = 4. It does not affect the outcome of the problem, and is only shown here to illustrate that the product of multiplying 362 x 30 is 10,860 and not 1,086. It also helps keep all of the digits in their proper columns (ones in the ones column, tens in the tens column, and so forth), as mentioned earlier in relation to subtraction.

Once all of the subproducts are determined, they are added together to obtain the final total or product (in this example, 11,584). The multiplication sign (also called the times sign) is always used in a multiplication problem to distinguish it from any other type of arithmetic operation.

The Multiplication Table

It has now been demonstrated to you through examples that multiplication is a shortcut for certain types of addition problems. A shortcut method is only valuable if it can be used efficiently and accurately. The key to using multiplication efficiently is the multiplication table (see Figure 3-3, giving products up to 12 x 12). Accuracy depends on your efforts and how well you have developed your multiplication skills.

Practice the multiplication table until it is memorized. To test how well you have memorized this table, write each problem on one side of an index card and the product on the other side. You should be able to look at a problem and know its answer within five seconds, without looking at the other side.
TIPS ... To Insure Perfect Solutions

Remember the old saying: practice makes perfect.


Another method of presenting the multiplication table is shown in Figure 3-4. This unique table gives the products of numbers up to 25 x 25 = 625. It is relatively simple to use. For example, to find the product of 8 x 9, locate the number 8 in the vertical (up and down) column to the far left. Then move your finger to the right until the 9 is located in the horizontal (left to right) column at the top of the table. The number 72 is in the place where the 8 column and the 9 column intersect. Therefore, 72 is the product of 8 x 9. Look one place below the 72 and find the number 81. This is the product of 9 x 9. Drop down another place to find that 10 x 9 = 90.

Simplifying Multiplication by a Step-By-Step Procedure. This multiplication example is intended to illustrate the step-by-step procedures involved in finding the product. As mentioned earlier, multiplication is a variation of, and has a very close association with, addition. In this example, it will be shown that the product of the problem is the result of adding together the subproducts of each step of the problem.
TIPS ... To Insure Perfect Solutions

Placing a card or sheet of paper across the table horizontally is
helpful in locating the products of the various numbers.


Example: 924
        x 65

Ones column:

Step 1. 5 x 4 = 20
Step 2. 5 x 20 = 100
Step 3. 5 x 900 = 4,500

Subproduct of ones column 4,620

Tens column:

Step 4. 60 x 4 = 240
Step 5. 60 x 20 = 1,200
Step 6. 60 x 900 = 54,000

Subproduct of tens column 55,440

Add subproducts:

Step 7. 4,620 + 55,440 = 60,060 (Product)


This example shows the steps in finding the product of 924 X 65. Generally when the problem is worked, the unneeded zeros are eliminated, but carryover numbers are used, as shown in the following example:
    9 2 4 Multiplicand
   x  6 5 Multiplier

  4 6 2 0 Subproduct of ones column
5 5 4 4   Subproduct of tens column

6 0 0 6 0 Product


2 is the carryover number for the ones column.

1 is the carryover number for the tens column.

Notice that the subproduct of the ones column in both methods of working the problem is 4,620. The same is true for the subproduct of the tens column, 55,440. The zero is left off the subproduct in the second method because its only purpose is to hold a place. As long as the other figures are in their proper places, the zero is unnecessary.

Checking the Product

The accepted and common method of checking the accuracy of a multiplication product is to invert, or turn over, the multiplicand with the multiplier and work the problem from a reverse position.
Original Multiplication

   3 4 8   Multiplicand
   x 5 4   Multiplier

 1 3 9 2 Subproduct
1 7 4 0   Subproduct

1 8 7 9 2 Product

Problem Checked

      5 4 Multiplier
  x 3 4 8 Multiplicand

    4 3 2 Subproduct
  2 1 6   Subproduct
1 6 2     Subproduct

1 8 7 9 2 Product


If the problem is worked accurately in both instances, the products will be the same. If two different products are obtained, invert the problem back to its original form and try again with a little better effort. Today, checking can be done on a calculator. However, it is a good practice to work problems through manually and then double-check the work on a calculator. Remember that doing problems manually sharpens your math skills, and that times will arise when you must function without automation.

Guidelines

A few guidelines are offered here to help provide speed and accuracy to multiplication work.

Be Neat. Neatness is very important, as pointed out earlier in this chapter. The customary method of multiplying eliminates end zeroes (zeroes that appear at the end of a number), and you must therefore be very careful in writing each number in its proper place.

Be Careful with Carryover Numbers. Remember that carryover numbers are added to the product of the two numbers being multiplied. The carryover numbers are not multiplied, as pointed out in the following example. (Note: It is unnecessary to write the carryover numbers as shown in the following example.)
(4)(5)

  7 5 8
   x  7

5 3 0 6


The first numbers to multiply are 7 x 8 = 56. Write the b in the ones column beneath the bar and carry the 5 over to the tens column. Next, 7 x 5 = 35. To this we add the carryover number 5. So, 35 + 5 = 40. Write the 0 in the tens column beneath the bar and carry the 4 to the hundreds column. The next step in proper order is 7 x 7 = 49. To this, add the carryover number 4. This becomes 49 + 4 = 53. Write the 3 in the hundreds column beneath the bar and the 5 in the thousands column beneath the bar. 7 x 758 = 5,306.
TIPS ... To Insure Perfect Solutions

When the multiplier consists of two or more numerals, be careful
to note the carryover numbers.


To Quickly Determine the Product When Multiplying by 10, 100, 1000, and so forth Add the Correct Number of Zeroes to the Multiplicand. For example: 10 x 222 = 2,220. Since there is one zero in 10, add one zero to 222 to obtain the product. When multiplying by 100, two zeroes would be added. 100 x 222 = 22,200. When multiplying by 1000, three zeroes are added.

Use Units in the Product When They Are Used in the Multiplicand. A unit was explained in a previous lesson as a single quantity of like things. Most multiplication problems in the world of work involve some sort of designated units. If a chicken processing plant has 45 chickens each in 15 separated pens, how many chickens does it have on hand? (45 x 15 = 675) The product is not simply 675, but 675 chickens. This type of unit does not have to be written next to the multiplicand when working the problem, but remember what type of units are being multiplied so the result will be a certain number of those designated units.

SUMMARY REVIEW 3-3

Find the answer to the following multiplication problems. If units are indicated, write the unit in the answer. Do these problems manually and then check your work. Round to the nearest cent.
1. 156
   x 3

2. 5682
   x 76

3. 3562
   x 29

4. 372
 x 121

5. 9763
 x 1847

6. $355.46
      x 32

7. $5,351.44
        x 53

8. $8,421.55
      x 1.69


9. A side of beef weighs 323 pounds and costs $2.73 per pound. How much does the side cost? --

10. If a foresaddle of veal weighs 46 pounds and costs $7.60 per pound, what is the total cost? --

11. Jim's catering truck gets 13 miles to every gallon of gas. How many miles can it travel on 128 gallons of gas? --

12. When preparing meat loaf, it is required that four pounds of ground beef go into each loaf. How many pounds of ground beef must be ordered when preparing 65 loaves? --

13. Ribs of beef weigh 24 pounds and cost $2.97 per pound. What is the total cost of the ribs? --

14. A wedding reception is catered for 575 people. The caterer charges $25.75 per person. What is the total bill? --

15. A cook earns $125.00 a day. How much is earned in a year if the cook works 328 days? --

DIVISION

Division is the act of separating a whole quantity into parts. It is a sharing of that whole part, a process of dividing one number by another. Division is basically the method of finding out how many times one number is contained in another number. It is, in a sense, a reverse of multiplication, as shown below:
Multiplication 5 x 8 = 40
Division       40 / 8 = 5
               40 / 5 = 8


Look at division from another viewpoint. Assume that your employer promises to pay you $8.00 per hour. After putting in eight difficult hours, you receive a check for only $56.00 (before deductions, of course). Just looking at the total, you know something is wrong. A simple division problem will show you that $56 / 8 = $7. You were only being paid $7.00 an hour, or you were paid only for seven hours. Using division can thus help you find out that a mistake was made.

Division is frequently used in the food service business because foods are constantly being divided. A strip sirloin is divided into steaks, vegetables into portions, cakes into servings, and so forth (see Figure 3-5, which shows another example). An example illustrating the division of food can be seen by observing how a solution is found to the following problem:

A No. 10 can of applesauce contains about 105 ounces. How many guests can be served if each guest receives a 3-ounce portion?

105 / 3 = 35

Thirty-five guests can be served from the 105-ounce can of applesauce. Using division has helped the cook to know how many cans must be opened when serving a certain number of people.

In division, the number to be divided is called the dividend. In the problem 80 / 4 = 20, the number 80 is the dividend. The name for the number by which the dividend is divided (number 4, in this example) is the divisor. The result of dividing the dividend by the divisor is called the quotient. The quotient in this example is the number 20.
TIPS ... To Insure Perfect Solutions

Dividend divided by Divisor equals Quotient

Quotient
Divisor) Dividend


[FIGURE 3-5 OMITTED]
TIPS ... To insure Perfect Solutions

When dividing using the calculator, enter the dividend first.


In situations where the divisor is not contained in the dividend an equal or exact number of times, the figure left over or remaining is called the remainder. For example: 83 / 4 = 20 with 3 left over, because 4 is too large to be contained in 3. In this problem, 3 is the remainder.

There are several division signs that can be used when working division problems. Usually the sign selected depends on how long or difficult the problem may be. Up to this point in our examples of the division operation, division has been indicated by a bar with a dot above and below (/). This symbol is mainly used when the problem is simple or when first stating a division problem that is to be solved. Simple problems are also written using the fraction bar. For example, 84 / 7 = 12 can also be written 84/7 = 12.

For what is known as a long division problem, the accepted sign to use has the appearance of a closed parenthesis sign with a straight line coming out of the top:

[ILLUSTRATION OMITTED]

Step-By-Step Division

When reviewing basic mathematical operations, division is usually discussed last because it involves both multiplication and subtraction in its operation (including carryover numbers and trading). For example: 8,295 / 15 = ?

Since this problem would be considered a lengthy one, the long division form is used.

[ILLUSTRATION OMITTED]

A long division problem is started from the left, rather than from the right, as in addition, subtraction, and multiplication. The first step is to estimate how many times 15 is contained in 82 (actually 8,200). It is known that 5 x 15 = 75, so your first estimate would be 5. The 5 is written above the division bar in the hundreds place.

[ILLUSTRATION OMITTED]

To make sure that 5 is the correct figure, it is multiplied by 15. The product is 75. The 7 is written in the thousands column, under the 8, and the 5 in the hundreds column, under the 2. Then 75 is subtracted from 82. If the difference is less than 15, the estimate of 5 in the hundreds place of the quotient is correct. 82 - 75 = 7. Since 7 is less than 15, the estimate is correct.

[ILLUSTRATION OMITTED]

The next step is to bring the 9 in the tens place of the dividend down to the right of the 7, giving 79. We estimate that 79 contains 15 only 5 times. 79 - 75 = 4, which is smaller than 15. (Note that sometimes the estimate is too low or too high. When this happens, the estimate must be increased or decreased accordingly.)

So far our problem has advanced to this point:

[ILLUSTRATION OMITTED]

The next step is to bring down the remaining figure 5 from the ones column and place it to the right of the 4 (left over from subtracting 75 from 79). Again, estimate how many times 45 contains 15. It is easy to see that 3 should be the estimated figure, making the problem come out even. The completed problem is shown as follows:

[ILLUSTRATION OMITTED]

The Remainder

As explained earlier, the figure in a division problem that may be left over when the dividend does not contain the divisor exactly is called the remainder. For example:

[ILLUSTRATION OMITTED]

At this point, there are no more numerals to bring down from the dividend. There are no 33s contained in the number 18. Therefore, the leftover 18 is called the remainder. Therefore, the solution would be 65 R18. The remainder can also be written in fractional or decimal form, as shown below:

2,163 / 33 = 65 18/33 (a fraction) or 65.545 (a decimal)

Checking Division

The common method of checking division is to multiply the quotient by the divisor. The product of multiplying the divisor and quotient together should be the same as the dividend.

[ILLUSTRATION OMITTED]
To check the division:

  1 3 2 Quotient
x   2 2 Divisor

  2 6 4
2 6 4

2 9 0 4 Dividend


When the quotient includes a remainder, multiply the quotient by the divisor as shown in the previous example, then add the remainder to the product.

[ILLUSTRATION OMITTED]

To check the division with a remainder present:
  2 7 5 Quotient
x   3 1 Divisor
  2 7 5
8 2 5

8 5 2 5 Product
+   1 9 Remainder

8 5 4 4 Dividend


SUMMARY REVIEW 3-4

Find the quotient for the following division problems. Carry answers three places to the right of the decimal point. Calculate these problems manually and then check all work.

1. 45 / 15 = --

2. 96 / 3 = --

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

7. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

8. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

9. A 480-pound side of beef is purchased for $950.40. What is the cost per pound? --

10. In one week, the Prime Time Catering Company's delivery truck traveled 336 miles. They used 24 gallons of gasoline. How many miles did they average per gallon? --

11. The chef at the Starlight Restaurant earns $43,200 per year. If this chef works 48 weeks per year, how much does she earn in one week? --

12. The preparation cook at the Blue Star Restaurant works five days a week and earns $356.00 per week. How much does he earn in one day? --

13. A restaurant orders 392 pounds of pork loins. Each loin weighs 14 pounds. How many loins are contained in the shipment? --

14. The Deluxe Catering Company's delivery truck averages 13 miles per gallon of gasoline. During 30 days of operation, the truck travels 2,028 miles. How much gasoline is used? --

15. One hundred twenty-eight ounces of orange juice are contained in one gallon. How many 8-ounce glasses of juice can be served? --
Chef Sez ...

Math was never one of my favorite subjects
in high school. I kept thinking, "Why do I
need to know math?" Today, as executive
chef at Fairview Southdale Hospital, I use
math every minute of the day!

My budget for food is $96,000 a month
based on 31 days, and $93,000 for a
30-day month. This food budget is used for
all patient meals, cafe service, catering, and
foods used for floor stock. Daily, we
average 225 patient meals, 150 "meals on
wheels," and 1,800 guests in our cafe. My
per-meal cost for a patient is $1.33, with
another $1.25 in snacks and nourishments
per day.

The challenge my staff and I face is this:
as we increase revenue in our cafe, the
amount of money I spend on my monthly
food purchases should not increase. I must
find ways to control my costs without
spending any more money on food. This is
accomplished through forecasting and
knowing our patient census. We take this
information to our production system and
calculate our food production process. This
allows me to control the amount of food to
order and prepare, resulting in minimal
leftovers and waste. My goal is to operate
the cafe at about 40% food cost, with
catering at 35% externally and internally at
cost.

Every day, I am using all four functions of
math--adding, subtracting, multiplying,
and dividing--to keep my costs in line!


Bob Newell, Certified Executive Chef (CEC)

Executive Chef

Sodexho

Fairview Southdale Hospital

Edina, Minnesota

Fairview Southdale Hospital, located in a suburb of Minneapolis, Minnesota, is a 390-bed facility. The hospital has a staff of over 1,000 physicians, 2,200 health providers, and more than 500 volunteers. Mr. Newell has had a varied career as a chef for a variety of hospitals, as well as for private industry like the Dow Corning Corporation, Alma College, and hotels. He has also operated his own catering business.
Figure 3-3

Multiplication table of numbers from 1 through 12

1 x 1 = 1
1 x 2 = 2
1 x 3 = 3
1 x 4 = 4
1 x 5 = 5
1 x 6 = 6
1 x 7 = 7
1 x 8 = 8
1 x 9 = 9
1 x 10 = 10
1 x 11 = 11
1 x 12 = 12

2 x 1 = 2
2 x 2 = 4
2 x 3 = 6
2 x 4 = 8
2 x 5 = 10
2 x 6 = 12
2 x 7 = 14
2 x 8 = 16
2 x 9 = 18
2 x 10 = 20
2 x 11 = 22
2 x 12 = 24

3 x 1 = 3
3 x 2 = 6
3 x 3 = 9
3 x 4 = 12
3 x 5 = 15
3 x 6 = 18
3 x 7 = 21
3 x 8 = 24
3 x 9 = 27
3 x 10 = 30
3 x 11 = 33
3 x 12 = 36

4 x 1 = 4
4 x 2 = 8
4 x 3 = 12
4 x 4 = 16
4 x 5 = 20
4 x 6 = 24
4 x 7 = 28
4 x 8 = 32
4 x 9 = 36
4 x 10 = 40
4 x 11 = 44
4 x 12 = 48

5 x 1 = 5
5 x 2 = 10
5 x 3 = 15
5 x 4 = 20
5 x 5 = 25
5 x 6 = 30
5 x 7 = 35
5 x 8 = 40
5 x 9 = 45
5 x 10 = 50
5 x 11 = 55
5 x 12 = 60

6 x 1 = 6
6 x 2 = 12
6 x 3 = 18
6 x 4 = 24
6 x 5 = 30
6 x 6 = 36
6 x 7 = 42
6 x 8 = 48
6 x 9 = 54
6 x 10 = 60
6 x 11 = 66
6 x 12 = 72

7 x 1 = 7
7 x 2 = 14
7 x 3 = 21
7 x 4 = 28
7 x 5 = 35
7 x 6 = 42
7 x 7 = 49
7 x 8 = 56
7 x 9 = 63
7 x 10 = 70
7 x 11 = 77
7 x 12 = 84

8 x 1 = 8
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
8 x 5 = 40
8 x 6 = 48
8 x 7 = 56
8 x 8 = 64
8 x 9 = 72
8 x 10 = 80
8 x 11 = 88
8 x 12 = 96

9 x 1 = 9
9 x 2 = 18
9 x 3 = 27
9 x 4 = 36
9 x 5 = 45
9 x 6 = 54
9 x 7 = 63
9 x 8 = 72
9 x 9 = 81
9 x 10 = 90
9 x 11 = 99
9 x 12 = 108

10 x 1 = 10
10 x 2 = 20
10 x 3 = 30
10 x 4 = 40
10 x 5 = 50
10 x 6 = 60
10 x 7 = 70
10 x 8 = 80
10 x 9 = 90
10 x 10 = 100
10 x 11 = 110
10 x 12 = 120

11 x 1 = 11
11 x 2 = 22
11 x 3 = 33
11 x 4 = 44
11 x 5 = 55
11 x 6 = 66
11 x 7 = 77
11 x 8 = 88
11 x 9 = 99
11 x 10 = 110
11 x 11 = 121
11 x 12 = 132

12 x 1 = 12
12 x 2 = 24
12 x 3 = 36
12 x 4 = 48
12 x 5 = 60
12 x 6 = 72
12 x 7 = 84
12 x 8 = 96
12 x 9 = 108
12 x 10 = 120
12 x 11 = 132
12 x 12 = 144

Figure 3-4 Multiplication table of numbers from 1 through 25

 1     2     3     4     5     6     7     8     9    10
 2     4     6     8    10    12    14    16    18    20
 3     6     9    12    15    18    21    24    27    30
 4     8    12    16    20    24    28    32    36    40
 5    10    15    20    25    30    35    40    45    50
 6    12    18    24    30    36    42    48    54    60
 7    14    21    28    35    42    49    56    63    70
 8    16    24    32    40    48    56    64    72    80
 9    18    27    36    45    54    63    72    81    90
10    20    30    40    50    60    70    80    90   100
11    22    33    44    55    66    77    88    99   110
12    24    36    48    60    72    84    96   108   120
13    26    39    52    65    78    91   104   117   130
14    28    42    56    70    84    98   112   126   140
15    30    45    60    75    90   105   120   135   150
16    32    48   &4     80    96   112   128   144   160
17    34    51   &8     85   102   119   136   153   170
18    36    54    72    90   108   126   144   162   180
19    38    57    76    95   114   133   152   171   190
20    40    60    80   100   120   140   160   180   200
21    42    63    84   105   126   147   168   189   210
22    44    66    88   110   132   154   176   198   220
23    46    69    92   115   138   161   184   207   230
24    48    72    96   120   144   168   192   216   240
25    50    75   100   125   150   175   200   225   250
 1     2     3     4     5     6     7     8     9    10

 1    11    12    13    14    15    16    17    18    19
 2    22    24    26    28    30    32    34    36    38
 3    33    36    39    42    45    48    51    54    57
 4    44    48    52    56    60    64    68    72    76
 5    55    60    65    70    75    80    85    90    95
 6    66    72    78    84    90    96   102   108   114
 7    77    84    91    98   105   112   119   126   133
 8    88    96   104   112   120   128   136   144   152
 9    99   108   117   126   135   144   153   162   171
10   110   120   130   140   150   160   170   180   190
11   121   132   143   154   165   176   187   198   209
12   132   144   156   168   180   192   204   216   228
13   143   156   169   182   195   208   221   234   247
14   154   168   182   196   210   224   238   252   266
15   165   180   195   210   225   240   255   270   285
16   176   192   208   224   240   256   272   288   304
17   187   204   221   238   255   272   289   306   323
18   198   216   234   252   270   288   306   324   342
19   209   228   247   266   285   304   323   342   361
20   220   240   260   280   300   320   340   360   380
21   231   252   273   294   315   336   357   378   399
22   242   264   286   308   330   352   374   396   418
23   253   276   299   322   345   368   391   414   437
24   264   288   312   336   360   384   408   432   456
25   275   300   325   350   375   400   425   450   475
 1    11    12    13    14    15    16    17    18    19

 1    20    21    22    23    24    25
 2    40    42    44    46    48    50
 3    60    63    66    69    72    75
 4    80    84    88    92    96   100
 5   100   105   110   115   120   125
 6   120   126   132   138   144   150
 7   140   147   154   161   168   175
 8   160   168   176   184   192   200
 9   180   189   198   207   216   225
10   200   210   220   230   240   250
11   220   231   242   253   264   275
12   240   252   264   276   288   300
13   260   273   286   299   312   325
14   280   294   308   322   336   350
15   300   315   330   345   360   375
16   320   336   352   368   384   400
17   340   357   374   391   408   425
18   360   378   396   414   432   450
19   380   399   418   437   456   475
20   400   420   440   460   480   500
21   420   441   462   483   504   525
22   440   462   484   506   528   550
23   460   483   506   529   552   575
24   480   504   528   552   576   600
25   500   525   550   575   600   625
 1    20    21    22    23    24    25
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No portion of this article can be reproduced without the express written permission from the copyright holder.
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Title Annotation:PART II Review of Basic Math Fundamentals
Publication:Math Principles for Food Service Occupations
Geographic Code:1USA
Date:Jan 1, 2007
Words:7169
Previous Article:Chapter 2 Numbers, symbols of operations, and the mill.
Next Article:Chapter 4 Fractions, decimals, ratios, and percents.
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