Decimals

Key Concepts

• Decimals are numbers that show parts of a whole using place value.

  • Example: 3.45
    • 3 is in the ones place
    • 4 is in the tenths place
    • 5 is in the hundredths place

• The decimal point separates the whole number part from the fractional part.

  • Example: In 12.7, 12 is the whole number and 7 means 7 tenths.

• Each place value on the right of the decimal point becomes 10 times smaller as you move right.

  • Tenths: $\frac{1}{10}$
  • Hundredths: $\frac{1}{100}$
  • Thousandths: $\frac{1}{1000}$

Multiplying decimals

• When multiplying a decimal by a whole number:

  • Multiply as usual.
  • Then place the decimal point correctly in the answer.

• When multiplying a decimal by 10, 100, or 1000:

  • The digits move to the left in value, so the decimal point appears to move to the right.
  • Multiply by 10: move 1 place right
  • Multiply by 100: move 2 places right
  • Multiply by 1000: move 3 places right
  • Example:
    • $4.56 \times 10 = 45.6$
    • $4.56 \times 100 = 456$

• When multiplying a decimal by another decimal:

  1. Ignore the decimal points first.
  2. Multiply the numbers as whole numbers.
  3. Count the total number of decimal places in both factors.
  4. Put the decimal point in the product so that it has the same total number of decimal places.
  • Example:
    • $2.4 \times 1.3$
    • Multiply $24 \times 13 = 312$
    • There are 2 decimal places in total
    • Answer: 3.12

Dividing decimals

• When dividing a decimal by a whole number:

  • Divide as usual.
  • Put the decimal point in the quotient directly above the decimal point in the dividend.
  • Example:
    • $8.4 \div 4 = 2.1$

• When dividing by 10, 100, or 1000:

  • The digits move to the right in value, so the decimal point appears to move to the left.
  • Divide by 10: move 1 place left
  • Divide by 100: move 2 places left
  • Divide by 1000: move 3 places left
  • Example:
    • $56.7 \div 10 = 5.67$
    • $56.7 \div 100 = 0.567$

• When dividing a decimal by a decimal:

  1. Change the divisor into a whole number.
  2. Multiply both the dividend and divisor by the same power of 10.
  3. Then divide.
  • Example:
    • $6.24 \div 0.3$
    • Multiply both by 10
    • $62.4 \div 3 = 20.8$

• Always check whether the answer is reasonable.

  • Example: $6 \div 0.5 = 12$, so the answer can be greater than 6.

Conversion between fractions and decimals

• A fraction shows equal parts of a whole.

• A decimal is another way to write some fractions.

• Fractions with denominators of 10, 100, 1000 can be easily written as decimals.

  • $\frac{3}{10} = 0.3$
  • $\frac{47}{100} = 0.47$
  • $\frac{8}{1000} = 0.008$

• To convert a fraction to a decimal:

  • Rewrite it as an equivalent fraction with denominator 10, 100, or 1000 if possible, or
  • Divide the numerator by the denominator.
  • Example:
    • $\frac{3}{4} = 3 \div 4 = 0.75$

• To convert a decimal to a fraction:

  1. Write the digits after the decimal point over the correct place value.
  2. Simplify the fraction if needed.
  • Example:
    • $0.6 = \frac{6}{10} = \frac{3}{5}$
    • $0.25 = \frac{25}{100} = \frac{1}{4}$

• Some decimals terminate.

  • A terminating decimal ends after a certain number of digits.
  • Example: $0.4, 0.75, 1.125$

• Some fractions give recurring (repeating) decimals — the digits repeat forever.

  • $\frac{1}{3} = 0.333\ldots$
  • $\frac{1}{6} = 0.1666\ldots$
  • $\frac{1}{7} = 0.142857142857\ldots$
  • For PSLE, you will either be given a rounded answer or asked to round to a specific decimal place.
  • You do not need to use dot notation (e.g. $0.\dot{3}$) at PSLE level.

Rounding decimals

• Rounding means changing a number to the nearest required place value.

• Common place values for rounding:

  • nearest whole number
  • nearest tenth
  • nearest hundredth

• Steps for rounding:

  1. Find the digit at the place you are rounding to.
  2. Look at the digit immediately to its right.
  3. If that digit is 5 or more, round up.
  4. If that digit is 4 or less, keep the digit the same.
  5. Change all digits to the right into zeroes if rounding a whole number, or remove them if writing a decimal answer.

• Examples:

  • $4.36$ rounded to the nearest tenth = 4.4
  • $7.24$ rounded to the nearest whole number = 7
  • $9.995$ rounded to the nearest hundredth = 10.00 or 10.0 depending on required form

• Be careful:

  • The digit you look at is the one immediately to the right of the place you are rounding to.

Decimal word problems

• Decimal word problems often involve:

  • money
  • mass
  • length
  • volume
  • distance
  • measurement comparisons

• Important strategies:

  • Read the question carefully.
  • Underline key information and units.
  • Decide whether to add, subtract, multiply, or divide.
  • Convert all values to the same unit before calculating if needed.
  • Check whether the answer should be rounded.

• Common clues:

  • each, every, per → often multiplication or division
  • total, altogether → often addition
  • difference, how much more, how much less → subtraction
  • shared equally → division

• In money problems:

  • Write answers correctly to 2 decimal places when using dollars and cents.
  • Example: $3.50, not $3.5 in final money form

Important Definitions

• Decimal: a number that represents a whole number and/or part of a whole using place value and a decimal point.
• Decimal point: the dot that separates the whole number part from the fractional part of a decimal.
• Place value: the value of a digit depending on its position in a number.
• Tenths: parts out of 10; the first digit to the right of the decimal point.
• Hundredths: parts out of 100; the second digit to the right of the decimal point.
• Thousandths: parts out of 1000; the third digit to the right of the decimal point.
• Multiply: to find the total when equal groups of the same amount are combined.
• Divide: to split into equal parts or find how many times one number is contained in another.
• Fraction: a number that represents part of a whole, written as $\frac{\text{numerator}}{\text{denominator}}$.
• Equivalent fractions: different fractions that have the same value.
• Terminating decimal: a decimal that ends after a fixed number of digits.
• Round off / Rounding: changing a number to the nearest value of a stated place.
• Nearest tenth: closest value with 1 digit after the decimal point.
• Nearest hundredth: closest value with 2 digits after the decimal point.
• Quotient: the answer when one number is divided by another.
• Product: the answer when numbers are multiplied.
• Reasonable answer: an answer that makes sense based on estimation and the numbers given.

Worked Examples

Example 1: Multiplying decimals

Find $3.6 \times 0.4$.

Step 1: Ignore the decimal points first.
$36 \times 4 = 144$

Step 2: Count total decimal places.

• $3.6$ has 1 decimal place
• $0.4$ has 1 decimal place
• Total = 2 decimal places

Step 3: Put the decimal point into the product.
$144 \rightarrow 1.44$

Answer: $3.6 \times 0.4 = 1.44$

Check:
$3.6 \times 0.4$ is less than $3.6$, so $1.44$ is reasonable.

Example 2: Converting between fractions and decimals

Convert $\frac{7}{20}$ to a decimal.

Method 1: Make the denominator 100

Answer: $\frac{7}{20} = 0.35$

Now convert 0.48 to a fraction.

Step 1: Write it over 100.

Step 2: Simplify.

Answer: $0.48 = \frac{12}{25}$

Example 3: Decimal word problem

A bottle contains 1.5 L of juice. Mei drinks 0.35 L and her brother drinks 0.4 L. How much juice is left?

Step 1: Find the total amount drunk.

Step 2: Subtract from the original amount.

Answer: $0.75\text{ L}$ of juice is left.

Check:

• They drank less than 1 L in total.
• The amount left should be less than 1.5 L.
• $0.75\text{ L}$ is reasonable.

Common Mistakes to Avoid

• Putting the decimal point in the wrong place after multiplication.
• Forgetting to count the total number of decimal places in both factors.
• Moving the decimal point in the wrong direction when multiplying or dividing by 10, 100, or 1000.
• Forgetting to multiply both numbers when dividing a decimal by a decimal.
• Writing a fraction from a decimal but not simplifying it.
  • Example: leaving $0.5$ as $\frac{5}{10}$ instead of $\frac{1}{2}$
• Rounding using the wrong digit.
  • Always check the digit immediately to the right.
• Ignoring zeroes that are important for place value.
  • Example: $0.07$ is 7 hundredths, not 7 tenths
• Mixing up units in word problems.
  • Example: adding metres and centimetres without converting
• Forgetting to line up decimal points when adding or subtracting decimals.
• Writing money answers incorrectly.
  • Example: write $2.30, not $2.3 if the question is about money

Exam Tips

• Always write decimals in place value form carefully.
• For addition and subtraction, line up the decimal points.
• For multiplication:
  • multiply first as whole numbers
  • then count decimal places
• For division by a decimal:
  • change the divisor into a whole number first
  • multiply both dividend and divisor by the same power of 10
• Use estimation to check your answer.
  • Example: $4.9 \times 2.1$ is about $5 \times 2 = 10$
• In word problems:
  • write the operation statement
  • include the correct unit
  • check whether the final answer needs rounding
• For money:
  • write to 2 decimal places
  • include the dollar sign if needed
• Mark-earning habits:
  • show working clearly
  • do not skip steps in multistep problems
  • box or underline your final answer
• Helpful phrases to write in solutions:
  • “Convert to the same unit first.”
  • “Round to the nearest tenth/hundredth.”
  • “Therefore, …”
  • “So, the amount left is …”
  • “Hence, the total cost/mass/length is …”

Quick Summary

• A decimal uses a decimal point to separate whole numbers from parts of a whole.
• The first 3 decimal places are tenths, hundredths, thousandths.
• Multiplying by 10, 100, 1000 moves the decimal point right.
• Dividing by 10, 100, 1000 moves the decimal point left.
• To multiply decimals, multiply as whole numbers first, then place the decimal point using the total number of decimal places.
• To divide by a decimal, first change the divisor into a whole number by multiplying both numbers by the same power of 10.
• Fractions and decimals can represent the same value.
• Decimals can be written as fractions using place value, then simplified.
• To round decimals, look at the digit immediately to the right of the required place value.
• In word problems, use the correct operation and convert all values to the same unit.
• Line up decimal points carefully in addition and subtraction.
• Check whether your answer is reasonable and write the correct unit.