Whole Numbers and Their Place Value
Whole numbers are the counting numbers and zero: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…Whole numbers are also sometimes called digits.
Each digit has a specific value. The position, or place, of a digit in a number written in standard form determines the value the digit represents.
The number 543,210 has a 5 in the hundred thousands place, a 4 in the ten thousands place, a 3 in the thousands place, a 2 in the hundreds place, a 1 in the tens place, and a 0 in the ones place.
The expanded form of a number is the sum (addition) of its various place values: 500,000 + 40,000 + 3,000 + 200 + 10 + 0.
Rounding Whole Numbers
You can estimate by using rounded numbers. Numbers can be rounded to different place values. For example, to round to the nearest ten means to find the closest number having all zeros to the right of the tens place.
When the digit 5, 6, 7, 8, or 9 appears to the right of the place you are rounding to, round up. When the digit 0, 1, 2, 3, or 4 appears to the right of the place you are rounding to, stay the same.
Choosing an Operation
Often a problem will tell you exactly which operation you should do. However, sometimes you will have to translate the words in a word problem into the operations. Look for these clues when you have to choose the operations.
You add (+) when you are asked to
§ find a sum
§ find a total
§ combine amounts
§ all together
You subtract (–) when you are asked to
§ find a difference
§ take away an amount
§ compare quantities
§ how many more than
§ how many fewer than
§ how much less than
§ how much is left over
You multiply (×) when you are asked to
§ find a product
§ times
§ each
§ add the same number over and over
You divide (÷) when you are asked to
§ find a quotient
§ per
§ split an amount into equal parts
Examples
- Add the product of 6 and 3 to the sum of 10 and 4.
To solve this problem, begin by translating the words into math symbols. You know from the list of clues that the word product indicates multiplication. So you will need to multiply 6 and 3. You also know that sum indicates addition. You could write the problem like this: 6 × 3 + 10 + 4.
Now, follow the order of operations (PEMDAS). Multiply 6 and 3: 18 + 10 + 4.
Add in order from left to right: 28 + 4 = 32. The answer is 32.
- Missy and Amy went to a movie at the theater. They shared a large popcorn. Each girl paid for her own drink. The movie cost $6.25. The popcorn cost $4.50. Each drink cost $2. How much did each girl pay?
Begin by translating the words into math symbols. The cost of the popcorn should be divided between the two girls. So, each girl paid the following: $6.25 + ($4.50 ÷ 2) + $2.
Now, follow the order of operations. Do operations in parentheses first: $6.25 + ($2.25) + $2. Add in order from left to right: $8.50 + $2 = $10.50.
Each girl paid $10.50 for the movie and refreshments.
Order of Operations
The basic operations of real numbers include addition, subtraction, multiplication, division, and exponentiation. Often, in expressions, there are grouping symbols—usually shown as parentheses—which are used to make a mathematical statement clear. In math, there is a predefined order in which you perform operations. This is an agreed-upon order that must be used. For the five basic operations, the order is:
§ First, perform all operations enclosed in parentheses i.e. (5 + 2).
§ Second, evaluate all exponents. i.e. (42)
§ Third, perform any multiplication and division, in order, working from left to right.
§ Finally, evaluate any addition and subtraction, in order, working from left to right.
Examples
- 8 + 15 × 3
There are no parentheses or exponents, so evaluate multiplication first: 15 × 3 = 45. Now perform the addition: 8 + 45 = 53.
- 7 + 24 ÷ 6 × 10
There are no parentheses or exponents, so evaluate multiplication and division from left to right. First, do the division: 24 ÷ 6 = 4. Next, perform multiplication: 4 × 10 = 40. Finally, perform addition: 7 + 40 = 47.
- (36 + 64) ÷ (18 – 20)
First, evaluate the parentheses, from left to right: (36 + 64) = 100 and (18 – 20) = –2. Now, do the division: 100 ÷ –2 = –50.
Whole Number Addition and Subtraction
Example
20 + 529 + 24 =
First, align the numbers you want to add on the ones column. Because it is necessary to work from right to left, begin to add starting with the ones column. The ones column equals 13, so write the 3 in the ones column and regroup or carry the 1 to the tens column. Now, add the tens column, including the regrouped 1. Finally, add the hundreds column. Because there is only one value, write the 5 in the answer.
Example
Find the difference between 36 and 75.
Start with the ones column. Because 5 is less than the number being subtracted (6), regroup or borrow a 10 from the tens column, leaving 6 tens. Add the regrouped amount to the ones column. Now, determine 15 – 6 in the ones column. Regrouping 1 ten from the tens column left 6 tens. Subtract 3 from 6, and write the result in the tens column of your answer.
Whole Number Multiplication and Division
In multiplication, you combine the same amount multiple times. In some cases, multiplication can be used instead of addition. For example, instead of adding 60 four times, 60 + 60 + 60 + 60, you could simply multiply 60 by 4. If a problem asks you to find the product of two or more numbers, you should multiply.
Example
Find the product of 12 and 16.
Line up the place value as you write the problem in columns. Multiply the ones of the top number by the ones of the bottom number: 2 × 6 = 12. Write the 2 in the ones place in the first partial product. Regroup the 10.
Multiply the tens place in the top number by 6: 6 × 1 = 6. Then add the regrouped amount: 6 + 1 = 7. Write the 7 in the tens column of the partial product.
Now multiply by the tens place of 16. Write a placeholder 0 in the ones place in the second partial product, because you're really multiplying the top number by 10. Then multiply the top number by 1: 1 × 2 = 2. Write 2 in the partial product next to the zero. Multiply 1 by the top number in the tens place: 1 × 1 = 1. Your total second partial product is 120.
Add 72 and 120. Your answer is 192.
In division, the answer is called the quotient. The number you are dividing by is called the divisor and the number being divided is the dividend. The operation of division is finding how many equal parts an amount can be divided into.
Example
Find the quotient of 72 divided by 3 (72 ÷ 3) (dividend ÷ divisor).
Set up a long division problem with 3 as the divisor and 72 as the dividend:
What times 3 equals 7, or a whole number closest to 7? 3 × 2 = 6, so this is your best choice. Write a 2 over the 7 in the dividend because 3 is being divided into 7. 7 – 6 = 1, which is the remainder. Bring down the 2.
What times 3 equals 12? 3 × 4 = 12. Write a 4 over the 2 in the dividend because 3 is being divided into 12.
The quotient of 72 divided by 3 is 24.
You Try:
Solve
1) Find the difference of 518 and 329. 2) Find the sum of 183 and 8957.
3) Find the product of 78 and 851. 4) Find the quotient of 25776 and 24.
5) 324,879 + 9,776 6) 5023 – 489 7) 395 x 1874 8) 12564 ÷ 18
Name the underlined place value then round.
9) 36,745 10) 12,894 11) 7,405 12) 3,895,675
Decimals, Place Value and Reading/Writing Decimal Numbers
To read a decimal, read as a whole number. Then name the place value of the last digit. We read and write 0.53 as fifty-three hundredths. To read a decimal that has a whole number part, read the whole number part, read the decimal point as “and”, then read the decimal part as a whole number and then name the place value of the last digit. We read and write 23.705 as twenty-three and seven hundred five thousandths.
Comparing and Ordering Decimals
To compare two decimal numbers, begin at the left. Compare the digits in each place from left to right. Use “<” for less than, “>” for greater than and “=” for equal to.
Examples:
Compare 2.6 and 2.3.
2.6 The ones digits are the same. Compare the tenths.
2.3
6 > 3, so 2.6 > 2.3
Compare 0.035 and 0.35.
0.035 The ones digits are the same. Compare the tenths.
0.35
0 < 3, so 0.035 < 0.35
Compare 0.4 and 0.47.
0.40 The ones and tenths digits are the same. Compare the hundredths.
0.47
0 < 7, so 0.4 < 0.47
Rounding Decimals
27.17469 rounded to the nearest whole number is 27
36.74691 rounded to the nearest whole number is 37
12.34690 rounded to the nearest tenth is 12.3
89.46917 rounded to the nearest tenth is 89.5
50.02139 rounded to the nearest hundredth is 50.02
72.63539 rounded to the nearest hundredth is 72.64
46.83531 rounded to the nearest thousandth is 46.835
9.63967 rounded to the nearest thousandth is 9.640
Rules for rounding decimals:
1. Retain the correct number of decimal places (e.g. 3 for thousandths, 0 for whole numbers)
2. If the next decimal place value is 5 or more, increase the value in the last retained decimal place by 1.
Decimal Operations
Adding and Subtracting Decimals
Addition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. The easiest way to do that is to line up the decimal points.
In this section we will provide a few examples to remind you of the procedure for adding and subtracting decimals. Look at the examples below, and then read through the detailed examples.
Example
· Here is an example of adding 12.35 and 5.287. Notice how the decimal points are lined up. /· Here is an example of subtracting 2.28 from 12.993. Notice how the decimal points are lined up. /
Detailed Example of Addition
Add the following numbers 1.19 and .16
The answer to this is: 1.19 + .16 = 1.35.
· First line the numbers up in a column, lining up the decimal points. /· Add down the columns, starting at the right. Notice that 9 + 6 = 15, so we need to carry a 1 to the tenths column. /
· Continue to add down the columns, moving from right to left. /
Detailed Example of Subtraction
Subtract 1.387 from 12.17.
The answer to this is: 12.17 - 1.387 = 10.783
· Subtract down the columns, starting at the right. Notice that in the thousandths column 7 > 0. We must "borrow" from the hundredths column. /
· When we move to the hundredths column, we notice that 8 > 6. We must "borrow" from the tenths column. /
· Continue to subtract down the columns, moving from right to left. Again, we need to borrow from the ones place to be able to subtract the tenths. /
Multiplication of Decimals
When multiplying numbers with decimals, we first multiply them as if they were whole numbers. Then, the placement of the number of decimal places in the result is equal to the sum of the number of decimal places of the numbers being multiplied.
For example, if we multiply 2.3 times 4.5, each number has one digit to the right of the decimal, so each has one decimal place. When they are multiplied, the result will have two digits to the right of the decimal, or two decimal places. Now let's look at a detailed example of multiplying two number with decimals. /Detailed Example of Multiplication
Multiply 12.3 by 3.54.
The answer is 43.542
· Multiply the second digit from the right of the bottom number by the top. Notice how this second number is offset below the first. Do this for each digit of the bottom number, moving from right to left. /
· Continue doing this for each digit of the bottom number, moving from right to left. Remember to offset each number as you multiply. /
· Now add the numbers up going down the columns. /
· To determine how many digits are to the right of the decimal point in the result, we count the decimal places in the two numbers being multiplied and add these together. /
Division of Decimals
Division with decimals is easier to understand if the divisor (the dividend is divided by the divisor) is a whole number.