Fractions, Decimals and Percentages Are the Same Thing
Learn decimal place value, the four operations on decimals, and how to move freely between fractions, decimals, ratios and percentages in exam questions.
What does each digit after the decimal point mean?
A decimal extends place value past the ones column. Moving right from the point, the places are tenths, hundredths and thousandths — each one-tenth of the place before it. So in 4.75 the digits after the point are genuine fractions written in columns: seven tenths and five hundredths.
This page covers everything in the ICSE Class 6 Mathematics chapter on decimals and percentage: place value, converting to and from fractions, the four operations, and moving between fractions, decimals, ratios and percentages.
This page covers everything in the ICSE Class 6 Mathematics chapter on decimals and percentage: place value, converting to and from fractions, the four operations, and moving between fractions, decimals, ratios and percentages.
How do you convert between decimals and fractions?
In a decimal number every digit still has a place value. Take 45.276: the 2 is in the tenths place, the 7 in the hundredths and the 6 in the thousandths, so
To turn a decimal into a fraction, write the digits after the point over 1 followed by that many zeros, then reduce:
To turn a fraction into a decimal, divide the numerator by the denominator: .
Like decimals have the same number of decimal places; unlike decimals do not. To compare unlike decimals, add zeros on the right until they match — this changes nothing, because . So comparing 0.5 and 0.47 becomes comparing 0.50 and 0.47, and .
That is the classic trap: students see 47 as bigger than 5 and choose wrongly. Compare place by place from the left, not by counting digits.
To turn a decimal into a fraction, write the digits after the point over 1 followed by that many zeros, then reduce:
To turn a fraction into a decimal, divide the numerator by the denominator: .
Like decimals have the same number of decimal places; unlike decimals do not. To compare unlike decimals, add zeros on the right until they match — this changes nothing, because . So comparing 0.5 and 0.47 becomes comparing 0.50 and 0.47, and .
That is the classic trap: students see 47 as bigger than 5 and choose wrongly. Compare place by place from the left, not by counting digits.
How do the four operations work on decimals?
To add or subtract, line the decimal points up vertically and fill gaps with zeros:
To multiply, ignore the points, multiply as whole numbers, then place the point so the answer has as many decimal places as both numbers together. For : , and with two decimal places in total the answer is 0.36.
Multiplying or dividing by 10, 100 or 1000 just shifts the point. Multiplying moves it right, dividing moves it left, by as many places as there are zeros:
In real life this is everyday money and measurement work. Three items at Rs 24.50 come to , and 2.5 kg shared among 5 people gives kg each.
The error to avoid is misplacing the point when multiplying — count the decimal places in both factors and add them.
To multiply, ignore the points, multiply as whole numbers, then place the point so the answer has as many decimal places as both numbers together. For : , and with two decimal places in total the answer is 0.36.
Multiplying or dividing by 10, 100 or 1000 just shifts the point. Multiplying moves it right, dividing moves it left, by as many places as there are zeros:
In real life this is everyday money and measurement work. Three items at Rs 24.50 come to , and 2.5 kg shared among 5 people gives kg each.
The error to avoid is misplacing the point when multiplying — count the decimal places in both factors and add them.
Formula
How do you convert a percentage into a fraction, decimal and ratio?
Per cent means per hundred, so a percentage is a fraction with denominator 100:
Take 25%. As a fraction it is ; as a decimal, ; as a ratio, .
Going the other way, multiply by 100:
And a ratio converts through its fraction: becomes , which is 40%.
So the rule is one line in each direction: to a percentage, multiply by 100; from a percentage, divide by 100.
A few pairs are worth knowing by sight because they recur constantly: , , , and .
Take 25%. As a fraction it is ; as a decimal, ; as a ratio, .
Going the other way, multiply by 100:
And a ratio converts through its fraction: becomes , which is 40%.
So the rule is one line in each direction: to a percentage, multiply by 100; from a percentage, divide by 100.
A few pairs are worth knowing by sight because they recur constantly: , , , and .
How do you find a percentage of a quantity?
To find a percentage of a quantity, convert the percentage to a fraction and multiply.
Find 15% of Rs 800:
To express one quantity as a percentage of another, divide the first by the second and multiply by 100. A student scoring 45 marks out of 60 has
For an increase or decrease, take the percentage of the original amount and then add or subtract it. If a price of Rs 500 rises by 10%, the rise is , so the new price is Rs 550.
The boundary case that catches students: a percentage is always of some specific quantity, so 10% of Rs 500 and 10% of Rs 900 are different amounts of money. Identify the base quantity before calculating.
Find 15% of Rs 800:
To express one quantity as a percentage of another, divide the first by the second and multiply by 100. A student scoring 45 marks out of 60 has
For an increase or decrease, take the percentage of the original amount and then add or subtract it. If a price of Rs 500 rises by 10%, the rise is , so the new price is Rs 550.
The boundary case that catches students: a percentage is always of some specific quantity, so 10% of Rs 500 and 10% of Rs 900 are different amounts of money. Identify the base quantity before calculating.
Exam tip
The mistake most students make with percentage increase
When a quantity rises by 20% and then falls by 20%, students answer that it returns to the original value. It does not.
Start with Rs 1,000. A 20% rise adds , giving Rs 1,200. The 20% fall is now taken on Rs 1,200, not on Rs 1,000, so it removes , leaving Rs 960.
The reason is that the base changed between the two steps. Before every percentage calculation, write down which quantity you are taking the percentage of — that single habit prevents most errors in this topic.
Start with Rs 1,000. A 20% rise adds , giving Rs 1,200. The 20% fall is now taken on Rs 1,200, not on Rs 1,000, so it removes , leaving Rs 960.
The reason is that the base changed between the two steps. Before every percentage calculation, write down which quantity you are taking the percentage of — that single habit prevents most errors in this topic.
Did you know
Why does moving the decimal point multiply by ten?
Each place in our number system is worth ten times the place to its right. Shifting every digit one place left therefore multiplies the whole number by ten, and shifting right divides it by ten.
We draw this as "moving the point", but strictly it is the digits that move into new columns — the point simply marks where the whole numbers end and the fractions begin.
We draw this as "moving the point", but strictly it is the digits that move into new columns — the point simply marks where the whole numbers end and the fractions begin.
Key takeaways
Decimals and percentage in 30 seconds
- Decimal places continue place value as tenths, hundredths and thousandths; add zeros on the right to compare unlike decimals fairly.
- Convert a decimal to a fraction by writing the digits over a power of ten and reducing; convert a fraction to a decimal by dividing.
- Line up the points to add or subtract; when multiplying, the answer has as many decimal places as both factors combined.
- Multiplying or dividing by 10, 100 or 1000 shifts the point right or left by the number of zeros.
- Per cent means per hundred: multiply by 100 to reach a percentage, divide by 100 to leave one, and always identify the quantity the percentage is taken of.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Convert a decimal to a fraction by writing the digits over a power of ten and reducing; convert a fraction to a decimal by dividing.
- Line up the points to add or subtract; when multiplying, the answer has as many decimal places as both factors combined.
- Multiplying or dividing by 10, 100 or 1000 shifts the point right or left by the number of zeros.
- Per cent means per hundred: multiply by 100 to reach a percentage, divide by 100 to leave one, and always identify the quantity the percentage is taken of.
You will remember all of this far better after answering five questions on it than after reading it twice.