Learning Objectives
  • Understand decimal place value and compare decimal numbers.
  • Perform the four operations with decimals.
  • Convert between terminating decimals, recurring decimals and fractions.
  • Use decimals correctly in measurements and money.
Key Terms
Decimal place
A position to the right of the decimal point, such as tenths or hundredths.
Terminating decimal
A decimal that ends after a finite number of digits.
Recurring decimal
A decimal in which a digit or block of digits repeats forever.
Place value
The value of a digit determined by its position.
Decimal point
The symbol separating the whole-number part from the fractional part.
Decimal Place Value

The first position after the decimal point represents tenths, followed by hundredths, thousandths and so on. Zeros may be important place holders. For example, 4.07 is four and seven hundredths, not four and seven tenths.

Number Units Tenths Hundredths Thousandths
5.482 5 4 8 2
0.306 0 3 0 6
12.050 2 0 5 0
Comparing And Ordering Decimals

Align decimal points and compare corresponding place values from left to right. Adding trailing zeros does not change a decimal, so 0.7=0.70=0.700. This helps compare numbers such as 0.68 and 0.7: write 0.68 and 0.70, so 0.70 is greater.

Worked Example: Ordering

Question: Arrange 3.405, 3.45, 3.054 and 3.5 in ascending order.

  1. Write equal numbers of decimal places: 3.405, 3.450, 3.054, 3.500.
  2. Compare tenths first, then hundredths and thousandths.

Answer: 3.054<3.405<3.45<3.5.

Addition And Subtraction

Place decimal points vertically above one another. Fill missing places with zeros if helpful. The decimal point in the answer stays in the same aligned column.

Worked Example: Decimal Operations

Question: Calculate 18.7-6.958+2.04.

  1. Write 18.700-6.958=11.742.
  2. Then 11.742+2.040=13.782.

Answer: 13.782

Multiplication

Multiply as if the numbers were integers, then place the decimal point by counting the total number of decimal places in both factors. Estimation should be used to check the position of the decimal point.

Worked Example: Multiplication

Question: Calculate 3.48\times0.62.

  1. Ignore decimal points: 348\times62=21576.
  2. There are four decimal places altogether.
  3. Place the decimal point: 2.1576.
  4. Check: 3.5\times0.6\approx2.1, so the answer is reasonable.

Answer: 2.1576

Division

When dividing by a decimal, multiply both dividend and divisor by the same power of 10 until the divisor is a whole number. This does not change the quotient.

Worked Example: Division

Question: Calculate 7.56\div0.24.

  1. Multiply both numbers by 100: 756\div24.
  2. Calculate 756\div24=31.5.

Answer: 31.5

Terminating And Recurring Decimals

A fraction in simplest form has a terminating decimal exactly when its denominator contains no prime factors other than 2 and 5. Other denominators usually produce recurring decimals. Recurring notation places a dot or bar above the repeating digit or block.

Worked Example: Recurring Decimal To Fraction

Question: Convert 0.\overline{27} to a fraction.

  1. Let x=0.272727\ldots.
  2. Multiply by 100: 100x=27.272727\ldots.
  3. Subtract: 100x-x=27.
  4. So 99x=27 and x=\frac{27}{99}=\frac{3}{11}.

Answer: \frac{3}{11}

Worked Example: One Non-Recurring Digit

Question: Convert 0.1\overline{6} to a fraction.

  1. Let x=0.16666\ldots.
  2. Then 10x=1.6666\ldots and 100x=16.6666\ldots.
  3. Subtract: 100x-10x=15.
  4. So 90x=15 and x=\frac{1}{6}.

Answer: \frac{1}{6}

Decimals In Context

In money, the number of decimal places is normally fixed by the currency, so 4.8 dollars should be written as $4.80. In measurement, units matter: 2.35 m is not the same as 2 m 35 mm. Convert units before calculating.

8th Edition Chapter Map
  • Decimals and fractions, including recurring decimals
  • Operations involving decimals
  • Conversions of metric units for length, mass and volume/capacity
Converting Recurring Decimals To Fractions

A terminating decimal ends after a finite number of places. A recurring decimal repeats a digit or block forever. The algebraic method eliminates the repeated part by multiplying by a suitable power of 10 and subtracting.

Worked Example: One Repeating Digit

Let x=0.\overline{7}. Then 10x=7.\overline{7}. Subtracting gives 9x=7, so x=\frac79.

Worked Example: A Repeating Block After A Non-Repeating Digit

Convert 0.1\overline{36} to a fraction.

  1. Let x=0.1363636\ldots.
  2. 10x=1.363636\ldots and 1000x=136.363636\ldots.
  3. Subtract: 990x=135.
  4. x=\frac{135}{990}=\frac{3}{22}.
Decimal Operations And Place-Value Control

When adding or subtracting, align decimal points. When multiplying, first multiply as whole numbers, then place the decimal point so the answer has the combined number of decimal places. When dividing by a decimal, multiply both dividend and divisor by the same power of 10 until the divisor is an integer.

Worked Example: Decimal Division

18.564\div0.24=1856.4\div24=77.35. Multiplying both numbers by 100 changes the appearance but not the quotient.

Metric Unit Conversions

Length, mass and capacity use place-value relationships. A conversion to a smaller unit produces a larger numerical value; a conversion to a larger unit produces a smaller numerical value. Keep squared and cubed units for area and volume for Chapter 12; this chapter focuses on ordinary measures.

Measure Key relationships
Length 1\text{ km}=1000\text{ m}, 1\text{ m}=100\text{ cm}=1000\text{ mm}
Mass 1\text{ tonne}=1000\text{ kg}, 1\text{ kg}=1000\text{ g}
Capacity 1\text{ litre}=1000\text{ ml}
Worked Example: Mixed Units

Convert 3.47 km to metres: 3.47\times1000=3470 m. Convert 6850 g to kilograms: 6850\div1000=6.85 kg.

Reasonableness And Calculator Displays

Estimate before accepting a calculator result. For 39.8\times0.204, the estimate 40\times0.2=8 shows that a display of 81.192 has a misplaced decimal point. Preserve enough working digits during a calculation and round only at the end unless instructed otherwise.

Extended Practice

A. Convert 0.\overline{27} to a fraction.

100x-x=27, so 99x=27 and x=\frac3{11}.

B. Calculate 4.536\times0.07.

4.536\times0.07=0.31752.

C. A bottle holds 1.75 litres. How many 125 ml cups can be filled completely?

1.75\text{ L}=1750\text{ ml}, and 1750\div125=14 cups.

Examination Guidance
  • Estimate before or after a decimal calculation to detect a misplaced decimal point.
  • Use a bar or dots clearly when writing recurring decimals.
  • Keep exact recurring fractions during working rather than using a rounded calculator display.
  • For money, give the answer to the correct number of decimal places and include the currency symbol or unit.
Common Mistakes
  • Aligning final digits instead of decimal points in addition and subtraction.
  • Moving the decimal point in only the divisor during division.
  • Treating 0.45 as larger than 0.5 because 45 is larger than 5.
  • Using a rounded decimal when the question asks for an exact fraction.
Knowledge Check

1. Arrange 0.506, 0.56, 0.065 and 0.5 in ascending order.

Answer

0.065<0.5<0.506<0.56.

2. Calculate 4.08\times0.35.

Answer

1.428.

3. Calculate 6.72\div0.16.

Answer

42.

4. Convert 0.\overline{45} to a fraction.

Answer

\frac{5}{11}.

5. Explain why \frac{7}{40} terminates.

Answer

In simplest form the denominator is 40=2^3\times5, containing only factors 2 and 5.

Mixed Review With Full Solutions

1. Write 0.2\overline{45} as a fraction.

Let x=0.2454545\ldots. Then 10x=2.454545\ldots and 1000x=245.454545\ldots. Subtracting gives 990x=243, so x=\frac{27}{110}.

2. Calculate (6.08-2.735)\div0.15.

6.08-2.735=3.345. Then 3.345\div0.15=22.3.

3. A wire is 3.75 m long and is cut into pieces of length 12.5 cm. How many complete pieces are obtained?

3.75\text{ m}=375\text{ cm}. Then 375\div12.5=30 pieces.

4. Put 0.62,\frac58,0.6\overline3,\frac{19}{30} in increasing order.

Values are 0.62, 0.625, 0.6333… and 0.6333…. Therefore 0.62<\frac58<0.6\overline3=\frac{19}{30}.

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