A decimal fraction is simply a fraction whose denominator is a power of ten written compactly with a dot. In the CDS and OTA paper, decimals appear inside percentage, average, ratio and simplification questions, so fluency here saves you precious minutes. This page rebuilds the topic from first principles and trains you on the exact patterns the examiner repeats year after year.
What a Decimal Fraction Really Is
A fraction with denominator 10, 100, 1000 and so on is called a decimal fraction. Instead of writing 7⁄10 or 37⁄100, we use a decimal point: 0.7 and 0.37. The dot separates the whole-number part on the left from the fractional part on the right. Every digit after the point sits in a place that is ten times smaller than the digit on its left, exactly mirroring how units, tens and hundreds grow tenfold to the left.
So 0.37 means 3 tenths plus 7 hundredths, i.e. 3⁄10 + 7⁄100 = 37⁄100. Reading a decimal aloud as a sum of place values is the single best habit for avoiding silly errors, because it forces you to respect each digit’s true weight.
Place values to the right of the point are: tenths (10−1), hundredths (10−2), thousandths (10−3), and so on. The number of digits after the point is called the number of decimal places.
A whole number such as 25 is itself a decimal with an unwritten point: 25 = 25.0. Likewise a decimal smaller than one, such as 0.4, has a zero whole-number part. Writing that leading zero before the point (0.4 rather than .4) is the convention examiners expect and helps you not lose the point while copying.
Adding zeros to the extreme right of the decimal part never changes the value: 0.5 = 0.50 = 0.500. These are called like decimals when made to have equal decimal places, which is exactly how we line them up for addition. Decimals with unequal numbers of places, such as 0.5 and 0.625, are unlike decimals until we pad the shorter one with trailing zeros.
Converting a Fraction to a Decimal
To turn a vulgar fraction into a decimal, simply divide the numerator by the denominator. Keep adding zeros after the decimal point of the dividend and continue the long division until the remainder becomes zero or a block starts repeating. If the denominator can be written using only the prime factors 2 and 5, the decimal terminates; otherwise it recurs.
The reason is neat: a terminating decimal is just a fraction whose denominator is a power of ten, and ten factorises as 2 × 5. So a denominator made only of 2s and 5s can always be scaled into a power of ten. For instance, 3⁄8 = 3 × 125⁄1000 = 375⁄1000 = 0.375.
A fraction in lowest terms gives a terminating decimal only if its denominator has no prime factor other than 2 or 5. Example: 3⁄8 (8 = 23) terminates, but 1⁄6 (6 = 2 × 3) does not.
To convert a decimal back to a fraction, write the digits without the point over 1 followed by as many zeros as there were decimal places, then reduce. So 0.625 = 625⁄1000 = 5⁄8. Always finish by reducing to lowest terms, because CDS options are usually given in simplest form.
Memorise the common ones: 1⁄8=0.125, 1⁄4=0.25, 3⁄8=0.375, 1⁄20=0.05. They reappear in percentage and profit-loss sums.
Recurring (Repeating) Decimals
When a digit or block of digits repeats forever, we call it a recurring decimal (also a repeating or circulating decimal) and mark it with a bar or dots over the repeating part. For example, 1⁄3 = 0.333… written as 0.3. The repeating block is called the period of the decimal. Such numbers are still rational, because every recurring decimal can be written as a fraction.
Pure recurring decimal
Here every digit after the point repeats. To convert it to a fraction, put the repeating block over an equal number of nines, then reduce. The logic is that multiplying by the right power of ten and subtracting eliminates the endless tail, leaving a clean fraction.
0.ab = ab⁄99. So 0.27 = 27⁄99 = 3⁄11.
Mixed recurring decimal
Here some digits do not repeat before the repeating block starts, e.g. 0.166. The rule: numerator equals (whole figure after the point) minus (non-repeating figure); denominator equals as many nines as repeating digits followed by as many zeros as non-repeating digits.
0.166 = (166 − 16)⁄900 = 150⁄900 = 1⁄6.
Adding and Subtracting Decimals
Addition and subtraction are easy once the numbers are written as like decimals. Pad the shorter numbers with trailing zeros so all have equal decimal places, then line up the decimal points one under another and operate as with whole numbers. Carry and borrow exactly as you would in ordinary integer addition; the decimal point in the answer drops straight down from the column above.
Find 12.5 + 3.07 + 0.625.
Subtraction follows the same alignment rule. To compute 8 − 2.46, treat 8 as 8.00, then subtract to get 5.54. Padding the whole number with a point and zeros removes any confusion about borrowing.
Never align numbers by their right-hand edge. Always align the decimal points. 12.5 + 3.07 is 15.57, not 12.82.
Multiplying Decimals
Ignore the decimal points, multiply the numbers as whole numbers, then place the point so that the product has as many decimal places as the total of decimal places in the two factors.
Evaluate 0.6 × 0.05.
Multiplying by 10, 100, 1000 just shifts the point right by 1, 2, 3 places. Dividing shifts it left. So 4.7 × 100 = 470 and 4.7 ÷ 100 = 0.047.
Dividing Decimals
To divide one decimal by another, shift the point in both numbers equally until the divisor becomes a whole number, then divide normally. This works because a division is really a fraction, and multiplying its top and bottom by the same power of ten leaves the value untouched while clearing the decimal from the divisor.
Evaluate 0.0048 ÷ 0.24.
When you divide a decimal by a whole number directly, place the decimal point in the quotient straight above its position in the dividend, then divide as usual. For 6.4 ÷ 8, the point sits after the first quotient digit, giving 0.8.
When numerator and denominator are multiplied by the same power of ten, the quotient is unchanged. That is why shifting the point in both is allowed.
Simplification and BODMAS
CDS simplification sums mix decimals with brackets and the word ‘of’. Always follow BODMAS: Brackets, Of, Division, Multiplication, Addition, Subtraction.
‘Of’ behaves like multiplication but is performed before ordinary division in the VBODMAS reading. Resolve innermost brackets first.
Simplify 0.5 of 0.8 + 1.2 ÷ 0.4 − 0.3.
Comparing Decimal Fractions
To compare decimals, first compare the whole-number parts. If they are equal, compare digits after the point place by place, from left to right. To rank a list quickly, convert all to like decimals.
Arrange 0.7, 0.07, 0.77, 0.707 in ascending order.
Do not assume a longer decimal is larger. 0.7 is greater than 0.0707, even though the latter has more digits. Length of the tail tells you nothing about size; only the place values do.
When fractions and decimals are mixed in the same list, convert everything to decimals first. Comparing 5⁄8 with 0.6 is instant once you note 5⁄8 = 0.625, which is larger than 0.6.
Speed Tricks Examiners Reward
The CDS clock is tight, so build a few reflexes for decimal-heavy questions.
- Estimate first. Round each decimal to one place and check your final answer lies near the estimate.
- Cancel before you compute. In a product like 0.36 × 0.5⁄0.6, cancel 0.36 and 0.6 to 0.6 before multiplying.
- Use known fraction values. Replace 0.125 by 1⁄8 and 0.0625 by 1⁄16 to avoid long division.
- Powers of recurring nines. Any pure recurring block over equal nines reduces instantly, e.g. 0.142857 = 1⁄7.
If options are far apart, a quick estimate often picks the answer without exact calculation — a real time-saver under exam pressure.
Where Decimals Show Up in CDS
Decimal skill rarely appears as a standalone question; it powers other arithmetic topics.
- Percentage: 37.5% is just 0.375, so a 37.5% discount multiplies the price by 0.375.
- Average: dividing a sum by the count almost always yields a decimal answer.
- Ratio and proportion: converting a ratio like 3 : 8 to 0.375 helps you compare unlike ratios fast.
- Mensuration: π taken as 3.14 turns area and circumference sums into decimal arithmetic.
Even data-interpretation tables in the GS-adjacent reasoning often quote figures to two decimals, so being unfazed by the dot lets you read and compare values without slowing down. The habit of mentally rewriting a decimal as a familiar fraction, and back again, is what separates a fast solver from a slow one.
Confidence with decimals indirectly lifts your score across half the arithmetic section, not just the decimal sums themselves. Treat decimal mastery as an investment that pays dividends in every numerical question.
Previous-Year Style Practice
Q. The value of (0.96)3 − (0.1)3⁄(0.96)2 + 0.096 + (0.1)2 is:
Answer: Using the identity (a3 − b3) = (a − b)(a2 + ab + b2), the numerator and denominator share the factor (a2 + ab + b2) where a = 0.96 and b = 0.1 (note ab = 0.096). So the expression reduces to (a − b) = 0.96 − 0.1 = 0.86.
Notice how recognising the algebraic identity removed all heavy decimal computation. CDS examiners love this exact disguise, dressing a simple identity in decimals to test whether you panic or pattern-match. Whenever you see cubes or squares of decimals stacked in a fraction, scan for a hidden identity before reaching for long multiplication.
Q. What is the value of 0.6 + 0.3 ?
Answer: 0.6 = 6⁄9 = 2⁄3 and 0.3 = 3⁄9 = 1⁄3. Their sum is 2⁄3 + 1⁄3 = 1. A handy check: any two complementary single-digit recurring decimals over 9 add to exactly 1.
Quick Revision
- Decimal fraction = fraction with denominator a power of 10; digits after the point are tenths, hundredths, etc.
- Fraction terminates only if its reduced denominator has factors 2 and 5 only.
- Pure recurring: block over equal nines. Mixed recurring: (whole − non-repeating) over nines-then-zeros.
- Add or subtract using like decimals aligned at the point.
- Multiply: add decimal places. Divide: shift the point equally in both numbers.
- Apply BODMAS, estimate first, and reuse memorised fraction values for speed.
Frequently asked questions
How do I know if a fraction will give a terminating decimal?
Reduce the fraction to lowest terms and factorise the denominator. If the only prime factors are 2 and 5, the decimal terminates; any other prime factor (like 3 or 7) makes it recur.
What is the fastest way to convert 0.45 with 45 repeating into a fraction?
It is a pure recurring decimal, so place the repeating block over an equal number of nines: 0.45 (45 repeating) equals 45/99, which reduces to 5/11.
Why must I align decimal points when adding rather than the last digits?
Because addition must combine digits of the same place value. Tenths add to tenths and hundredths to hundredths, which only happens when the decimal points sit in one vertical line.
How many decimal places does a product have?
The product has exactly as many decimal places as the sum of the decimal places in the two factors. For example 0.2 times 0.03 gives 1 plus 2 equals 3 places, so 0.006.
Are decimal questions asked directly in CDS Maths?
Sometimes directly as simplification or comparison, but more often decimals are embedded in percentage, average, ratio and mensuration sums, so the skill pays off across the whole paper.
Related CDS / OTA Maths topics
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