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Chapter 1—Whole Numbers

1-9 Ordering Operations with Whole Numbers.

 When you complete the work for this section, you should be able to: Demonstrate the proper procedure for working problems that include both addition and subtraction. Demonstrate the proper procedure for working problems that include combinations of addition, subtraction, multiplication, and division. Demonstrate how to solve problems that include signs of grouping.

You have been doing a lot of work here with basic arithmetic problems. You are given two whole-number values and asked to add them, subtract them, multiply them, or divide them. Or you are given a group of numbers and asked to add them or to multiply them. This lesson introduces the procedures for combining more than two numbers and using more than one kind of arithmetic operation.

• You should already know how to do this:  2 + 4 = _____, but here you will learn how to do this: 2 + 4 – 3 = _____.
• You should already know how to do this: 3 x 6 x 8 = _____, but here you will learn how to do this: 3 + 6 x 8  ÷ 2 = _____.

You have been doing a lot of work here with adding numbers and subtracting numbers. Here you see now to combine addition and subtraction operations.

 Rule Always perform combinations of addition and subtraction from left to right. Example: 12 + 8 – 5 = _____ It is important that you always solve problems of this type from left to right. Why? Sometimes the answer is different, depending upon which way you go.

Example of Right and Wrong

12 – 4 + 6 – 1 = ______

Solving from left to right (the correct way):

12 – 4 + 6 – 1 = 8 + 6 – 1
8 + 6 – 1 = 14 – 1
14 – 1= 13

12 – 4 + 6 – 1 = 13 Solving from right to left (the incorrect way):

12 – 4 + 6 – 1 = 12 – 4 + 5
12 – 4 + 5 = 12 – 9
12 – 9 = 3

12 – 4 + 6 – 1 = 3 It can make a big difference whether you solve combinations of addition and subtraction from left to right (the correct way), or from right to left (the incorrect way).

Check out these examples of operations that include both addition and subtraction.

More Examples

Exercises

 Complete the addition and subtraction operations. Remember: Always work from left to right. Continue these exercises until you can solve them without errors.

Combining Multiplication and Division

You have been doing a lot of work here with multiplying and dividing numbers. Here you see now to combine multiplication and division operations.

 Rule Always perform combinations of multiplication and division from left to right. Example: 24 ÷ 6 x 2 = _____ You must always solve problems of this type from left to right. Otherwise, you will likely get a different (incorrect) result.

Example of Right and Wrong

4 x 12 ÷ 6 x 2 = _____

Solving from left to right (the correct way):

4 x 12 ÷ 6 x 2 = 48 ÷ 6 x 2
48 ÷ 6 x 2 = 8 x 2
8 x 2 = 16

4 x 12 ÷ 6 x 2 = 16 Solving from right to left (the incorrect way):

4 x 12 ÷ 6 x 2 = 4 x 12 ÷ 12
4 x 12 ÷ 12 = 4 x 1
4 x 1 = 4

4 x 12 ÷ 6 x 2 = 4 More Examples

Check out these examples of operations that include both multiplication and division.

Exercises

 Complete the multiplication and division operations. Remember: Always work from left to right. Continue these exercises until you can solve them without errors.

Combining Addition, Subtraction, Multiplication, and Division

You have been doing a lot of work here with combinations addition/subtraction and multiplication/division. Now it's time to combine three or four of these operations in the same problem.

 Rule Always perform combinations of multiplication and division before doing combination of addition and subtraction. Always perform the operations from left to right.

Example

4 x 3 + 8 ÷ 2 – 1 = _____

Do the multiplication and division first from left to right:

4 x 3 + 8 ÷ 2 – 1 = 12 + 8 ÷ 2 – 1
12 + 8 ÷ 2 – 1 = 12 + 4  – 1

Then do the addition and subtraction from left to right:

12 + 4 – 1 = 16 – 1
16 – 1= 15

4 x 3 + 8 ÷ 2 – 1 = 15

More Examples

Exercises

Complete the operations.

Remember:

Do multiplication and division first, and from left to right.
Then do the addition and subtraction—from left to right.

Continue these exercises until you can solve them without errors.

Introducing Signs of Grouping

Signs of grouping are often used for clarifying and simplifying expressions that have a mix of operations. They "group" operations into distinct sets of operations.

The first, and most common, sign of grouping is a set of parentheses ( ). Here is an example of a problem that includes signs of grouping:

2 x (8 – 4) + 1 = _____

When a problem includes signs of grouping, you must complete the operations within the group first. Then you handle any multiplication/division followed by addition/subtraction.

 Rule Operations enclosed in a sign of grouping are always completed first.

Example

Problem    2 x (8 – 4) + 1 = _____

Complete the operation in parentheses first:

2 x (8 – 4) + 1 = 2 x (4) + 1

When there is just one number inside the signs of grouping, you can just omit the signs. So:

2 x (4) + 1 = 2 x 4 + 1

There are no longer any signs of grouping, so you now use the usual rules for the order of operations:

2 x 4 + 1 = 8 + 1
8 + 1 = 9

Solution     2 x (8 – 4) + 1 = 9

 Note A sign of grouping can be omitted when it contains only a single number.

More Examples

Exercises

 Use these examples and exercises to strengthen your ability to simplify expressions that include signs of grouping as well as the usual arithmetic operations.

Signs of grouping may be nested signs of grouping placed within other signs of grouping. Here is an example of two sets of groupings within another.

Consider this example:

2 x [ 3 + 4 x ( 12 – 10 ) + 15 ]

• The inner group is enclosed in parentheses ( )
• The outer group is enclosed in brackets [ ] If there were yet another outer group that enclosed the bracket group, the sign of grouping would be a set of braces, or "curly brackets"  { }

When you are working out a problem that has nested signs of grouping, you should always clear the inside groups first ... no matter what kinds of operations they contain.  In our example here, the first thing to do is the (12 - 10)  that is the innermost group.

2 x [ 3 + 4 x ( 12 – 10 ) + 15 ] = 2 x [ 3 + 4 x 2 + 15 ]

The inner sign of grouping is now gone.

Then work the operations enclosed in brackets:

Do multiplication first    2 x [ 3 + 4 x 2 + 15 ] = 2 x [ 3 + 8+ 15 ]

Then the addition    2 x [ 3 + 8+ 15 ] = 2 x 26

Now the outer sign of grouping is gone, too.

Finish the operation 2 x 26 = 52

So 2 x [ 3 + 4 x ( 12 – 10 ) + 15 ] = 52

 Rule When working with nested signs of grouping, always clear the innermost groups first.

Examples

Examples & Exercises

 Use these examples and exercises to check your understanding of nested signs of grouping.

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