Number & Operations in Base Ten

Decimal Place Value and Operations

Grade 5

✓By the end of this lesson students will be able to understand the value of digits in decimals to the thousandths place.
✓By the end of this lesson students will be able to explain patterns when multiplying or dividing decimals by powers of 10.
✓By the end of this lesson students will be able to fluently add and subtract decimals to the hundredths place.
✓By the end of this lesson students will be able to accurately multiply and divide decimals to the hundredths place.

Key concepts

Decimal Place Value

Decimal place value extends the base-ten system to numbers less than one. Each place to the right of the decimal point represents a fraction with a denominator that is a power of 10. For example, the first digit to the right of the decimal point is in the tenths place (1/10), the second digit is in the hundredths place (1/100), and the third digit is in the thousandths place (1/1000). A digit in one place represents 1/10 of what it represents in the place to its left.

Powers of 10

Powers of 10 are numbers like 10 (10^1), 100 (10^2), 1,000 (10^3), and so on. When you multiply a decimal by a power of 10, the decimal point moves to the right the same number of places as there are zeros in the power of 10 (or the exponent). When you divide a decimal by a power of 10, the decimal point moves to the left the same number of places as there are zeros in the power of 10 (or the exponent).

Adding and Subtracting Decimals

To add or subtract decimals, you must first line up the decimal points. This ensures that you are adding or subtracting digits with the same place value (tenths with tenths, hundredths with hundredths, etc.). You can add zeros to the end of decimals to make them have the same number of decimal places, which can help with alignment. Then, add or subtract as you would with whole numbers, bringing the decimal point straight down into your answer.

Multiplying Decimals

To multiply decimals, first multiply the numbers as if they were whole numbers, ignoring the decimal points. After you have your product, count the total number of decimal places in both of the original factors. The product will have the same total number of decimal places. Starting from the right of your whole-number product, count left and place the decimal point.

Dividing Decimals

To divide decimals, if the divisor (the number you are dividing by) is a decimal, you need to make it a whole number. Do this by moving the decimal point to the right until it is at the end of the number. You must then move the decimal point in the dividend (the number being divided) to the right the same number of places. If needed, add zeros to the dividend. Once the divisor is a whole number, divide as you would with whole numbers. Place the decimal point in the quotient (the answer) directly above the new decimal point in the dividend.

Key facts to remember

1The decimal point separates the whole number part from the fractional part of a number.
2Each place value to the right of the decimal point is 1/10 of the place value to its left.
3Multiplying by 10^n moves the decimal point 'n' places to the right.
4Dividing by 10^n moves the decimal point 'n' places to the left.
5When adding or subtracting decimals, always line up the decimal points.
6When multiplying decimals, count the total decimal places in the factors to determine the decimal places in the product.
7When dividing by a decimal, make the divisor a whole number by shifting decimal points in both the divisor and dividend.

Worked examples

Example 1

Calculate: a) 3.45 × 100 b) 87.2 ÷ 10

Ia) To multiply 3.45 by 100, we look at the power of 10. 100 has two zeros (or 10^2).

IIThis means we move the decimal point in 3.45 two places to the right.

IIIStarting with 3.45, moving the decimal one place gives 34.5.

IVMoving it another place gives 345.

Vb) To divide 87.2 by 10, we look at the power of 10. 10 has one zero (or 10^1).

VIThis means we move the decimal point in 87.2 one place to the left.

VIIStarting with 87.2, moving the decimal one place to the left gives 8.72.

Answer

a) 345 b) 8.72

Remember, multiplying by powers of 10 makes the number larger (decimal moves right), and dividing makes it smaller (decimal moves left).

Example 2

Solve: a) 12.3 + 5.87 b) 9.5 - 3.24

Ia) To add 12.3 and 5.87, first line up the decimal points. Add a zero to 12.3 to make it 12.30 for easier alignment.

II 12.30

III + 5.87

IV ------

V 18.17

VIAdd each column from right to left, carrying over when necessary, and bring the decimal point straight down.

VIIb) To subtract 3.24 from 9.5, first line up the decimal points. Add a zero to 9.5 to make it 9.50 for easier alignment.

VIII 9.50

9 - 3.24

10 ------

11 6.26

12Subtract each column from right to left, borrowing when necessary, and bring the decimal point straight down.

Answer

a) 18.17 b) 6.26

Always line up the decimal points before adding or subtracting decimals. Adding trailing zeros does not change the value of the number.

Example 3

Calculate: a) 4.2 × 0.6 b) 7.5 ÷ 0.5

Ia) To multiply 4.2 by 0.6, first multiply as if they were whole numbers: 42 × 6.

II 42 × 6 = 252.

III Now, count the total number of decimal places in the factors: 4.2 has one decimal place, and 0.6 has one decimal place. So, there are 1 + 1 = 2 total decimal places.

IV Starting from the right of 252, move the decimal point two places to the left: 2.52.

Vb) To divide 7.5 by 0.5, first make the divisor (0.5) a whole number by moving its decimal point one place to the right. It becomes 5.

VI Then, move the decimal point in the dividend (7.5) one place to the right. It becomes 75.

VII Now, divide 75 by 5.

VIII 75 ÷ 5 = 15.

9 Place the decimal point in the quotient directly above the new decimal point in the dividend (which is now at the end of 75, so the answer is a whole number).

Answer

a) 2.52 b) 15

When dividing by a decimal, always adjust both the divisor and dividend first to make the divisor a whole number.

Common mistakes

✗Not lining up the decimal points when adding or subtracting decimals.
✗Incorrectly moving the decimal point when multiplying or dividing by powers of 10 (e.g., moving left instead of right).
✗Forgetting to count all decimal places in the factors when multiplying decimals.
✗Not moving the decimal point in the dividend the same number of places as in the divisor when dividing decimals.
✗Confusing the place values (e.g., thinking hundredths is 1/10 instead of 1/100).

Exam tips

★Read the problem carefully to determine which operation to use (add, subtract, multiply, divide).
★For addition and subtraction, rewrite the problem vertically, carefully aligning the decimal points.
★For multiplication, estimate the answer first to check if your final decimal placement is reasonable.
★For division, show your work clearly, especially when adjusting the decimal points in the divisor and dividend.

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