Decimals: Rules, Formulas, Tricks and Decimal Conversion

Decimals represent parts of a whole using powers of 10. This topic covers place values, comparison, addition, subtraction, multiplication, division, decimal-to-fraction conversion, recurring decimals and rounding. The rules and solved examples help calculate decimal questions accurately and avoid errors in signs, zeros and decimal-point placement.

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What Are Decimals?

A decimal is a number written with a decimal point to show a whole part and a fractional part based on powers of 10. In 24.375, 24 is the whole-number part, while 375 represents 3 tenths, 7 hundredths and 5 thousandths.

The digits to the right of the decimal point have place values 10⁻¹, 10⁻², 10⁻³ and so on. Thus, 24.375 = 24 + 3/10 + 7/100 + 5/1000. Zeros at the end of a decimal do not change its value, so 4.50 = 4.5.

Decimals Formula & Tricks

Important Formulas

Decimal to fraction
0.abc = abc/1000

For a terminating decimal with three digits after the decimal point, remove the decimal point and place 1000 in the denominator. Reduce the fraction if possible.

Fraction to decimal
a/b = a ÷ b

Divide the numerator by the denominator. The decimal terminates when the reduced denominator has only 2 and/or 5 as prime factors.

Percentage to decimal
p% = p/100

Divide a percentage by 100 to write it as a decimal. For example, 37% = 0.37.

Decimal multiplication by a power of 10
x × 10ⁿ = move the decimal point n places right

Multiplication by 10, 100 and 1000 moves the decimal point 1, 2 and 3 places to the right respectively.

Decimal division by a power of 10
x ÷ 10ⁿ = move the decimal point n places left

Division by 10, 100 and 1000 moves the decimal point 1, 2 and 3 places to the left respectively.

Quick Tricks

Align decimal points in addition and subtraction

Write numbers one below another with their decimal points in the same column. Add zeros to the right when needed, then calculate normally.

Example: 7.5 + 2.36 = 7.50 + 2.36 = 9.86.
Count decimal places in multiplication

Multiply the numbers as whole numbers, then place the decimal point so that the product has the total number of decimal places present in both factors.

Example: 0.24 × 0.3: 24 × 3 = 72, and there are three decimal places in the factors, so the answer is 0.072.
Use fractions for recurring decimals

For a recurring decimal, multiply by a suitable power of 10 and subtract the original equation to eliminate the repeating part.

Example: If x = 0.̅3, then 10x − x = 3, so 9x = 3 and x = 1/3.
Compare decimals by equalising places

Append zeros to decimals with fewer digits after the point. Compare the digits from left to right.

Example: 0.7 = 0.70, and 0.68 < 0.70; therefore, 0.68 < 0.7.

Decimals Concepts

Place Value and Expanded Form

Each position after the decimal point represents a negative power of 10: tenths, hundredths, thousandths and so on.

In 5.406, 4 is in the tenths place, 0 is in the hundredths place and 6 is in the thousandths place. Therefore, 5.406 = 5 + 4/10 + 0/100 + 6/1000. A zero between non-zero decimal digits has value and cannot be removed.

Example: 3.072 = 3 + 0/10 + 7/100 + 2/1000.

Addition and Subtraction of Decimals

To add or subtract decimals, align their decimal points and then operate column by column.

The decimal points remain in the same column in the answer. Trailing zeros may be added to make the number of decimal places equal. For subtraction, borrow from the next place exactly as in whole-number subtraction.

Example: 12.04 − 3.786 = 12.040 − 3.786 = 8.254.

Multiplication and Division of Decimals

For multiplication, multiply without decimal points and restore the total decimal places; for division, make the divisor a whole number by shifting the decimal point equally in both numbers.

In 2.5 × 0.04, calculate 25 × 4 = 100 and restore three decimal places to get 0.100 = 0.1. For 6.48 ÷ 0.12, shift both decimal points two places right: 648 ÷ 12 = 54.

Example: 0.36 ÷ 0.09 = 36 ÷ 9 = 4.

Decimal, Fraction and Percentage Conversion

A terminating decimal can be converted into a fraction by using 10, 100, 1000 or the relevant power of 10 as the denominator.

For 0.625, 0.625 = 625/1000 = 5/8. To convert a decimal to a percentage, multiply by 100 and add the percent sign. Thus, 0.625 = 62.5%. To convert a fraction to a decimal, divide the numerator by the denominator.

Example: 3/8 = 0.375 = 37.5%.

Recurring Decimals

A recurring decimal has one or more digits that repeat indefinitely, such as 0.̅6 or 0.1̅2.

For a single repeating digit, if x = 0.̅6, then 10x = 6.̅6. Subtracting x gives 9x = 6, so x = 2/3. For 0.1̅2, let x = 0.1222..., then 10x − x = 1.1, giving 9x = 1.1 and x = 11/90.

Example: 0.̅9 = 1 because 10x − x = 9 gives 9x = 9 and x = 1.

Comparison and Rounding of Decimals

Compare decimals from the leftmost digit after making their decimal lengths equal; round a decimal by checking the digit immediately after the required place.

To round to two decimal places, retain two digits after the point. If the third digit is 5 or more, increase the second retained digit by 1; otherwise, leave it unchanged. For negative numbers, the same absolute-value rounding rule applies, with the sign retained.

Example: 8.376 rounded to two decimal places is 8.38, while 8.374 rounded to two decimal places is 8.37.

Decimals Video Lessons

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Decimal Fraction Percent Conversion

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Practice Decimals Questions

Practise published questions related to this topic.

1The value of: (3 + √8) + ( 1/(3 - √8) ) - (6 + 4√2) is:→ 2The product of 5xy2z and 7xz2 is:→ 3Simplify: (√75 - √108 + √6075)²→ 4If (0.04)² ÷ (0.008) × (0.2)⁶ = (0.2)^x , then find x.→ 5Simplify: ( (16⁻¹ × 8²)/(4⁻³ × 2⁵) )^(1/3)→ 6If 5^(x+1) = 1 , what is the value of x?→ 7Rationalize the denominator: 1/√(7+2)→ 8Find the value of m which satisfies (14/5)¹¹ × (5/14)¹⁷ × (14/5)¹⁴ = (5/14)^(3m+7)→ 9A clerk calculates √45 + √20 .Simplify to surd form.→ 10A clerk calculates √98 .Simplify the surd form.→

Decimals Quick Quiz

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Quick Revision Notes

Decimals Revision Points

Use these rules to recall decimal operations, conversions and special cases.

  • Digits after the decimal point have place values 1/10, 1/100, 1/1000 and so on.
  • Trailing zeros do not change value: 6.20 = 6.2.
  • Align decimal points before addition or subtraction.
  • In multiplication, the product has as many decimal places as the total in the factors.
  • To divide by a decimal, shift the decimal point in both dividend and divisor until the divisor is whole.
  • A terminating decimal becomes a fraction whose reduced denominator contains only factors 2 and 5.
  • Convert a decimal to a percentage by multiplying by 100.
  • For rounding, inspect the first discarded digit: 5 or more means increase the last retained digit by 1.

Decimals FAQs

How do you convert 0.375 into a fraction?

Write 0.375 as 375/1000 and reduce by 125: 375/1000 = 3/8.

What is 0.45 as a percentage?

Multiply by 100: 0.45 × 100 = 45%. Thus, 0.45 = 45%.

How do you calculate 2.4 × 0.35?

Ignore decimal points first: 24 × 35 = 840. There are three decimal places in the factors, so the result is 0.840 = 0.84.

How do you divide 4.68 by 0.12?

Shift both decimal points two places right: 4.68 ÷ 0.12 = 468 ÷ 12 = 39.

Which fraction gives a terminating decimal?

After reduction, the denominator must contain no prime factors other than 2 and 5. For example, 7/40 terminates because 40 = 2³ × 5, while 1/3 is recurring.

What is the difference between 0.5 and 0.50?

There is no difference in value. Appending a zero to the right of a decimal does not change the number, so 0.5 = 0.50.

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