Decimals

Decimals

What is the value of the digit 5 in the decimal below? How about the value of the digit 2

541.923

Similar to whole numbers, decimals have specific place values for each digit. Below is a standard decimal displayed in a place value chart:

 

HundredsTensOnesTenthsHundredthsThousandthsTen Thousandths
543.9876

In any decimal, the first digit after the decimal point represents the tenths place.

The digit 9 is in the tenths place.  The value of 9 is 0.9

The digit 8 is in the hundredths place.   The value of 8 in the hundredths place is  0.08.

The digit 7 is in the thousandths place.  The value of 7 in the thousandths place is 0.007.

The digit 6 is in the ten thousandths place.  The value of 6 is 0.0006.

The digit 3 is in the ones place.  The value of 3 is 3.

The digit 4 is in the tens place.  The value of 4 is 40.

The digit 5 is in the hundreds place.  The value of 5 is 500.

 

Decimals Exercises

205.7431

Answer the following questions:

1. What is the place value of the number 2?

2. What is the value of the number 5?

3. What is the value of the number 7?

4. What is the place value of the number 0?

5. What is the place value of the number 1?

 

Answers

  • Place value of the number 2: Hundreds
  • Value of the number 5: 5
  • Value of the number 7: 0.7
  • Place value of the number 0: Tens
  • Place value of the number 1: Ten-thousandths

Rounding Whole Numbers

Example 1.  Round off 298,245 to the nearest hundreds

In rounding whole numbers to the nearest hundreds, first, identify the hundreds digit, which is 2 in this example. Then, examine the digit immediately to its right, which is 4. If this digit is greater than or equal to 5 we round the number up by increasing the original digit by 1.  If its less than 5 then the original number will not be changed.  In this example, since we have 4, then 2 remains unchanged.  Finally, replace all the digits to the right of the original number with zeros. Since 4 is less than 5, the rounded value becomes 298,200.

 

Example 2.  Round off 346,532 to the nearest ten thousands

The ten thousands digit is 4.  The number immediately to its right is 6.  Since 6 is greater than 5, we round up 4 and make it 5.  Thus, the rounded value is 350,000

 

Example 3.  Round off 819,473 to the nearest hundred thousands.

The hundred thousands digit is 8. The number immediately to its right is 1 which is lower than 5.  Thus, the rounded value is 800,000

Rounding Decimals

Example 1. Round 25.4831 to the nearest tenth.

In rounding decimals to the nearest tenth, first, identify the tenth digit, which in this case is 4.  Then, examine the digit immediately to its right which is 8.  Since 8 is is greater than 5, add 1 to 4 making it 5, and drop all the digits to the right. Thus, the answer is 25.5

 

Example 2. Round 86.5763 to the nearest hundredth.

In rounding decimals to the nearest hundredth, first, identify the hundredth digit, which in this case is 7.  Then, examine the digit immediately to its right which is 6.  Since 6 is is greater than 5, add 1 to 7 making it 8, and drop all the digits to the right. Thus, the answer is 86.58

 

Example 3. Round 42.4951 to the nearest tenth.

In rounding decimals to the nearest tenth, first, identify the tenth digit, which in this case is 4.  Then, examine the digit immediately to its right which is 9.  Since 9 is is greater than 5, add 1 to 4 making it 5, and drop all the digits to the right. Thus, the answer is 42.5

Rounding Off Exercises

Round off the following numbers:

1. Round off 28.45 to the nearest tens.

2. Round off 425.96 to the nearest hundreds.

3. Round off 29.341 to the nearest tenths.

4. Round off 191.4563 to the nearest hundredths.

5. Round off 26.3951 to the nearest thousandths.

Answers

1. Round off 28.45 to the nearest tens: 30

2. Round off 425.96 to the nearest hundreds: 400

3. Round off 29.341 to the nearest tenths: 29.3

4. Round off 191.4563 to the nearest hundredths: 191.46

5. Round off 26.3951 to the nearest thousandths: 26.395

Rules for Comparing Decimals

1.  Start by comparing the whole number parts first.

Example 1.  2.5831 __ 1.996

In this example, the whole number 2 is greater than 1.  So, even if the decimal in the tenths place is higher for 1.996, still, 2.5831 is higher.

2. Compare each digit in the decimal as you would with whole numbers.

Example 1.  41.3281 __ 41.5021

When the whole numbers are equal in value, we check the decimals.  Based on the example, 5 which is the the digit in the tenths place is higher than 3.  So, 41.5021 is higher. 

Example 2: 5.6324 __ 5.6291

Based on the example, 3 which is the the digit in the hundredths place is higher than 2.  So, 5.6324 is higher. 

3. If needed, add zeros to the decimals to make the comparison easier, just like comparing whole numbers.

Example 1: 4.7 __ 4.730

To make the comparison easier add zeros to 4.7 making it 4.700.  Based on this example, 4.730 is higher

Comparing Decimals Exercises

Greater than or Less than

1. 78.64 __ 74.99

2. 82.62 __ 96.421

3. 2.5421 __ 2.8801

4. 7.421 __ 7.08

5. 11.621 __ 11.6

Arrange the following decimals from greatest to least

1. 2.34      2.04     23.04     .234

2. 7.07      70.7     7.007    70

Arrange the following decimals from Least to Greatest

1. 3.01      .031     30.1   301

2. 99.01      90.1      91.0     9.01

 

Answers

 

Addition and Subtraction of Decimals

Rule 1. Align the Decimal Points:

Start by aligning the decimal points of each number in a column. Write the numbers so that the decimal points line up vertically. If any positions in the equation are empty, such as the hundreds place in the first number or the thousandths place in the second, treat those places as zeros.

Rule 2. Perform the Addition or Subtraction:

Copy the decimal point directly below the aligned decimal points in the result. Then, begin adding or subtracting from the rightmost digit, moving leftward. Treat the operation as if you were working with whole numbers. Simply follow the rule on addition and subtraction of whole numbers.

Rule 3. Simplifying the Result

Once you’ve completed the addition or subtraction, you can remove any unnecessary zeros. However, you should only remove zeros from the leftmost or rightmost positions if they are zero. This will leave you with a number that begins and ends with a nonzero digit

Multiplication of Decimals

Rule 1. First, multiply the given problem as if they are whole numbers. 

Rule 2. Second, from right to left, count the number of decimal places from each factor then add them altogether.

Rule 3.  Third, place the decimal on your final answer.

Division of Decimals

Converting the Divisor to a Whole Number

When dividing decimals, the first step is to make the divisor a whole number. To do this, count the number of decimal places in the divisor, and then move the decimal point in both the divisor and the dividend to the right by the same number of places.

Performing Long Division

Once the divisor is a whole number, you can proceed with the division as you would with whole numbers. The decimal point in the dividend can be temporarily ignored while performing the division.

Placing the Decimal Point in the Quotient

After obtaining the quotient, place the decimal point in the quotient so that it aligns with the decimal point in the original dividend. If the dividend has no decimal point, there is no need to include one in the quotient.

Order of Operations

The Order of Operations is a standardized rule in mathematics that determines the sequence in which mathematical operations should be performed.

What is GEMDAS?

Here’s an explanation of GEMDAS in bullet points:

  • G: Groupings

    • Perform operations inside grouping symbols first, such as parentheses ( ), brackets [ ], or braces { }.

  • E: Exponents

    • Next, solve any exponents or powers.

  • MD: Multiplication and Division

    • After exponents, perform multiplication and division from left to right, as they appear in the expression.

  • AS: Addition and Subtraction

    • Finally, perform addition and subtraction from left to right, as they appear in the expression.

Note: Multiplication and Division are of equal rank, so you process them from left to right. The same goes for Addition and Subtraction.

Example 1. (25 ÷ 5 + 3) × 3 + (56 ÷ 8 × 2)

8 x 3 + 14

24 + 14

38

Example 2. (36 ÷ 6 x 3) + (8 x 4 + 8)

(6 x 3 ) + (32 + 8)

18 + 40

58

Example 3. ( 96 ÷ 1.2 x 4) ÷ (5 x 4 ÷ 8)

(80 x 4) ÷ (20 ÷ 8)

320 ÷ 2.5

128

Application

 

1. (48 + 12.4 × 3) − ( 8 × 2.2 − 7)

2. (7 × 4.3 ÷ 2) + (18 − 6 × 0.4)

3. (54 − 8) × (3 + 7.6) ÷ (12 × 2)

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