Decimal operations let us calculate with money, measurements, data, and scientific quantities that are not whole numbers. The decimal point marks place value, so every digit has a size based on its position. Small errors in decimal placement can change an answer by a factor of 10, 100, or more.
Learning clear rules for adding, subtracting, multiplying, and dividing decimals helps make calculations accurate and dependable.
For addition and subtraction, the main idea is to line up the decimal points so equal place values are combined. For multiplication, multiply as if the numbers were whole numbers, then place the decimal by counting the total decimal places in the factors. For division, the quotient decimal point is placed directly above the decimal point in the dividend, and dividing by a decimal requires shifting both numbers by the same power of 10.
These rules all come from preserving place value and keeping the value of the expression unchanged.
Key Facts
- Add decimals by lining up decimal points: 12.4 + 3.56 = 15.96.
- Subtract decimals by lining up decimal points and adding zeros if needed: 8.2 - 3.47 = 8.20 - 3.47 = 4.73.
- Multiply decimals by first ignoring decimal points, then count decimal places: 2.3 x 1.4 = 3.22.
- Number of decimal places in a product = decimal places in factor 1 + decimal places in factor 2.
- For decimal division, move the decimal in both divisor and dividend the same number of places: 5.6 ÷ 0.7 = 56 ÷ 7 = 8.
- Multiplying or dividing both numbers in a division problem by the same nonzero number keeps the quotient the same: a ÷ b = 10a ÷ 10b.
Vocabulary
- Decimal point
- The symbol that separates the whole number part from the fractional part of a decimal number.
- Place value
- The value of a digit based on its position, such as ones, tenths, hundredths, or thousandths.
- Dividend
- The number being divided in a division problem.
- Divisor
- The number that the dividend is divided by in a division problem.
- Quotient
- The result of a division problem.
Common Mistakes to Avoid
- Lining up the last digits instead of the decimal points in addition or subtraction is wrong because it combines different place values, such as tenths with hundredths.
- Forgetting placeholder zeros in subtraction is wrong because numbers like 8.2 and 8.20 have the same value, but the zero helps show the hundredths place during borrowing.
- Placing the decimal point in a product by lining it up with the factors is wrong because multiplication uses the total number of decimal places in all factors.
- Dividing by a decimal without moving the decimal in both numbers is wrong because changing only the divisor changes the value of the expression.
Practice Questions
- 1 Calculate 47.08 + 6.735.
- 2 Calculate 9.6 ÷ 0.24 by rewriting the problem with a whole-number divisor.
- 3 A student says 3.4 x 0.25 should have one decimal place because 3.4 has one decimal place. Explain the error and give the correct decimal-place rule.