Chapter focus
Decimal notation extends place value to parts smaller than one. Multiplication and division with decimals become easier when we use place value, equivalent fractions and estimation.
These original worked notes support the named Ganita Prakash Part II chapter. Check the assigned textbook for its complete lesson and exercise sequence.
1. Decimal place value
To the left of the decimal point are ones, tens, hundreds, and so on. To the right are tenths, hundredths, thousandths, and so on. For 3.407, the value is 3 + 4/10 + 0/100 + 7/1000.
Trailing zeroes do not change a value: 2.5 = 2.50. Compare decimals by aligning places: 0.68 = 0.680 and 0.7 = 0.700, so 0.7 > 0.68.
2. Multiply decimal numbers
Multiply as whole numbers, then count the total decimal places in the factors. The product must contain that many decimal places.
Worked example: 2.4 × 0.35
- Ignore the decimal points: 24 × 35 = 840.
- There are three decimal places in the factors altogether.
- Place the decimal three places from the right: 0.840 = 0.84.
- Estimate: 2.4 × about 0.4 is about 1, so 0.84 is reasonable.
3. Divide decimals
To divide by a decimal, multiply both dividend and divisor by the same power of 10 until the divisor is a whole number. The quotient stays unchanged because the numerator and denominator are scaled equally.
Worked example: 12.6 ÷ 0.3
Multiply both numbers by 10: 126 ÷ 3 = 42. Check: 42 × 0.3 = 12.6.
When dividing by 10, 100 or 1,000, each digit shifts one, two or three place values to the right in the quotient (equivalently, the decimal point moves left in the number being divided): 4.8 ÷ 10 = 0.48.
4. Terminating and repeating decimals
Division can terminate, as 3/8 = 0.375, or repeat, as 1/3 = 0.333… . Repeating digits continue in a fixed pattern. A decimal approximation should be labelled as approximate when it is rounded.
Use a remainder to tell whether a division will continue: if a remainder repeats during long division, the decimal digits repeat from that point.
Common errors to avoid
- Do not simply count decimal places in the divisor when setting up division; scale both numbers equally first.
- In multiplication, count decimal places across both factors together.
- Use a zero before a decimal smaller than one: 0.4 is clearer than .4.
- Estimate to catch a misplaced decimal point.
Practice: calculate and check
1. Multiply 0.6 × 0.08.
Answer: 6 × 8 = 48; three decimal places give 0.048.
2. Calculate 7.2 ÷ 0.6.
Answer: 72 ÷ 6 = 12.
3. Calculate 0.125 × 8.
Answer: 1.000 = 1.
4. Divide 5.04 by 0.7.
Answer: 50.4 ÷ 7 = 7.2.
5. Is 10 ÷ 3 terminating or repeating? Give the decimal form.
Answer: Repeating: 3.333… = 3.3̅.
Chapter summary
Decimal calculations follow place value. Multiply by treating the factors as whole numbers and restoring the combined decimal places; divide by scaling both numbers equally to make the divisor whole. Estimate and use the inverse operation to check.