Unit 1, Part 3: Operations with Whole Numbers
Learning objectives:
- Define Key Terms
- Add and Subtract whole numbers
- Multiply and divide numbers
- Convert time and manage time
- Solve algebraic and word problems
Key Terms:
- sum
- addition
- difference
- subtraction
- quotient
- product
- variable
- remainder
- inverse operation
You must be familiar with the terms in this section. For some word problems you will be asked to find the "quotient" for a problem, you need to be able to identify what that means. Take a moment to find the meaning for each of the key terms above.
1.3.A Basic Operations
In this section we will consider the four basic operations with whole numbers: addition, subtraction, multiplication, and division. It is assumed that you are familiar with computations, which are reviewed only briefly here. Our emphasis will be on how the operations are related to each other, as well as their application to real-world situations.
Addition represents the idea of finding a total count, the sum, or summing up, of values. Since we use only ten digits in our system (remember base 10), it is often necessary to use place value to “carry” digits.
Example 1: Find the sum: 458 + 375 + 296
Solution: Unless you are using a calculator, it is easier to organize this problem vertically:
458
375
+296
Starting in the ones place value, 8 + 5 + 6 = 19. Representing this sum as “carrying” into the tens place:
1
458
375
+296
9
Now working in the tens place value, 1 + 5 + 7 + 9 = 22. Representing this sum as “carrying” into the hundreds place:
21
458
375
+296
29
Finally working in the hundreds place value, 2 + 4 + 3 + 2 = 11. Representing this sum as “carrying” into the thousands place:
21
458
375
+296
1129
Thus 458 + 375 + 296 = 1129.
Example 2: The following table lists the monthly expenditures for food for the Jamison family in 2015:
| Month | Expense ($) |
| January | 386 |
| February | 335 |
| March | 287 |
| April | 264 |
| May | 279 |
| June | 315 |
| July | 298 |
| August | 214 |
| September | 269 |
| October | 314 |
| November | 368 |
| December | 416 |
Find the total expenditure in food for the Jamison family in 2015.
Solution: The total expenditure represents the sum of all twelve values in this table.
Representing this sum vertically:
67
386
335
287
264
279
315
298
214
269
314
368
+416
3745
The Jamison’s total expenditure in 2015 is $3,745
Subtraction of whole numbers is a natural result of an addition, called the inverse operation of addition. Given the addition statement 6 + 4 = 10, there are two associated subtraction statements:
10 – 6 = 4 and 10 – 4 = 6
Thus subtraction represents the idea of “undoing” addition. In general, if a + b = c, then c – a = b and c - b = a.
Example 3: Find the difference: 1426 – 548.
Solution: As with addition, we align the problem vertically and then “borrow” from the next higher place value, when necessary. Borrowing 1 ten = 10 ones, then subtracting 16 – 8 = 8:
Multiplication of whole numbers represents the idea of repeated addition. For example:
4 x 5 = 5 + 5 + 5 + 5 = 20
5 x 4 = 4 + 4 + 4 + 4 + 4 = 20
Note that this illustrates that multiplication is commutative, as was addition. Summarizing the properties of multiplication:
Note that the commutative and associative properties are similar to those of addition. The identity property for multiplication is similar to that of addition, except that 1 is the number used for the identity (rather than 0 as was used in addition). Notice the two new properties for multiplication involving the number 0. The multiplication property of 0 states that multiplication of anything by 0 always results in 0, and the zero factor property states that if a product results in 0, then one of the original numbers being multiplied (called factors) must be 0. Since we cannot re-write every multiplication problem as addition, we use memorized values from elementary school and carry as in addition.
Example 4: Compute the product: 256 • 47
Division of whole numbers represents the idea of repeated subtraction. For example:
Example 5: Compute the quotient: 6120 ÷ 45
Practice 1.3.A
Directions: Using the examples above, complete the following operations. Show your work. If you don't know how to use Word Document equation formatting, show your work on a separate sheet of paper and upload or turn in. Complete the following practice and submit all practice assignments for this section below.
Find the sum or the difference.
Find the product.
Find the quotient.
1.3.B. Time
Military or the 24-Hour clock is commonly used in the medical field. It is important that you are able to covert between standard and military time. Military time uses a 24-hour clock rather than the 12-hour (am/pm) clock we are familiar with.
The 24-hour clock begins at midnight (which is 0000 hours). So, 1:00 AM is 0100 hours (it is read as “Zero, One-Hundred Hours”), 2:00 AM is 0200 hours, and so-on up until 11:00 PM which is 2300 hours (it is read as “Twenty-three Hundred Hours”).
Here's the entire list:
Midnight (12:00 AM) -- 0000 hrs
1:00 AM -- 0100 hrs
2:00 AM -- 0200 hrs
3:00 AM -- 0300 hrs
4:00 AM -- 0400 hrs
5:00 AM -- 0500 hrs
6:00 AM -- 0600 hrs
7:00 AM -- 0700 hrs
8:00 AM -- 0800 hrs
9:00 AM -- 0900 hrs
10:00 AM -- 1000 hrs
11:00 AM -- 1100 hrs
12:00 PM -- 1200 hrs
1:00 PM -- 1300 hrs
2:00 PM -- 1400 hrs
3:00 PM -- 1500 hrs
4:00 PM -- 1600 hrs
5:00 PM -- 1700 hrs
6:00 PM -- 1800 hrs
7:00 PM -- 1900 hrs
8:00 PM -- 2000 hrs
9:00 PM -- 2100 hrs
10:00 PM -- 2200 hrs
11:00 PM -- 2300 hrs
You should memorize this chart.
Practice 1.3.B
Directions: Convert the following from either military to standard or standard to military.
1. 1900
2. 5:30 pm
3. 1222
1.3.C. Word Problems
WORD PROBLEMS require practice in translating verbal language into algebraic language.
Example: All problems like the following lead eventually to an equation in that simple form.
While ordering the medical supplies Jane spends $420 on her current order. This was $140 less than twice what she spent last month. How much was the order last month?
Solution. Use this formula to solve this word problem: ax ± b = c. Every word problem has an unknown number. In this problem, it is the cost of last month’s supply order. Always let x represent the unknown number. That is, let x answer the question.
Let x represent the cost of last month’s supply order. The problem states that "This" -- that is, $420 -- was $140 less than two times x.
Here is the equation:
2x − 140 = 420 (Plug in numbers)
2x = 420 + 140 (Add)
x = 560 (Divide)
2
x = 280
TIME WORD PROBLEMS: Solving word problems involving addition and subtraction of time.
Now practice word problems with time.
Example: Brandon left work at 3:15 p.m. He walked to the library to read while he waited for his ride. It took 15 minutes to walk to the library. Brandon's friend picked him up at the library one hour after he arrived. What time did Brandon's friend pick him up?
Solution:
3:15 p.m. + 15 minutes = 3:30 p.m.
3:30 p.m. + 60 minutes (1 hour) = 4:30 p.m.
We know there is 60 minutes in each hour, so we can only add 30 minutes to 3:30, making it 4:00 pm. Then we need to add the additional 30 minutes, making it 4:30 p.m.
See Khan Academy on Solving Word Problems.
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More Khan Academy Word Problems
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And More Khan Academy Word Problems
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Practice 1.3.C.
Using the examples above, solve the following word problems.
1. Jill must keep track of her daily caloric intake. She is allowed to subtract calories every time she burns calories by exercising. For the entire day she ate three oranges at 60 calories each, 2 eggs at 35 calories each, a PB&J sandwich at 456 calories, 3 cookies at 100 calories each, a steak dinner at 1200 calories for the whole meal. She drank 5 cups of kool-aid at 70 calories each. She went for a one hour walk and burned 432 calories. What is Jill's intake for today.
2. 8 was multiplied by a particular number. This product was then added to 8, then divided by 3. The resulting quotient was 32. Give the initial number.
3. How many months, weeks, and days are there between April 13, 2016 and December 23, 2016?
4. Emily and Sarah live at opposite ends of the neighborhood. It takes each of them 20 minutes to walk to school. If they have to be at school at 8:15 a.m., what is the latest time they can leave home and still make it to school in time without running?
5. Marcus met Max at the movies at 7:00 p.m. The movie started at 7:10 and lasted 1 hour and 35 minutes. What time did the movie end?
Submit all practice assignments Here