Why 0.2 Times 0.3 Is Not 0.6
Learn how the decimal point shifts when multiplying and dividing by 10, 100 and 1000, how to place the point in any decimal product, how to divide by a decimal, and how to turn fractions into decimals.
Why is 0.2 times 0.3 not 0.6?
Because you are taking two-tenths of three-tenths, and a part of a part is smaller than either. Written as fractions the answer is obvious:
The digits are 6, but the place value is hundredths, not tenths. This page covers everything in the CBSE Class 7 Mathematics chapter on decimal operations: multiplying and dividing by 10, 100 and 1000, multiplying decimals together, dividing by a decimal, and converting fractions to decimals.
The digits are 6, but the place value is hundredths, not tenths. This page covers everything in the CBSE Class 7 Mathematics chapter on decimal operations: multiplying and dividing by 10, 100 and 1000, multiplying decimals together, dividing by a decimal, and converting fractions to decimals.
What happens to the decimal point when you multiply by 10, 100 or 1000?
It moves to the right — one place for 10, two for 100, three for 1000 — because every digit shifts up to a bigger place value.
When the digits run out, fill with zeros. That is where the 4560 comes from: 4.56 has only two decimal places, so the third shift needs a zero to land on.
For a whole-number multiplier, multiply as usual and keep the same number of decimal places:
Check it by thinking in paise. Buying 7 pencils at ₹3.40 each costs 7 lots of 340 paise, which is 2380 paise, or ₹23.80.
The misconception to clear up early: multiplying by 10 does not "add a zero". That works for whole numbers only. Adding a zero to 4.56 gives 4.560, which is the same number — the point has to move instead.
When the digits run out, fill with zeros. That is where the 4560 comes from: 4.56 has only two decimal places, so the third shift needs a zero to land on.
For a whole-number multiplier, multiply as usual and keep the same number of decimal places:
Check it by thinking in paise. Buying 7 pencils at ₹3.40 each costs 7 lots of 340 paise, which is 2380 paise, or ₹23.80.
The misconception to clear up early: multiplying by 10 does not "add a zero". That works for whole numbers only. Adding a zero to 4.56 gives 4.560, which is the same number — the point has to move instead.
How do you multiply one decimal by another?
Multiply as if there were no decimal points, then count the decimal places in both numbers and give the answer that many.
Worked example: . Ignore the points and multiply . The factors have 1 and 2 decimal places, so the answer needs :
The rule comes straight from fractions, which is the surest way to see it:
The denominators multiplied to 1000, and three zeros mean three decimal places.
Estimate before you calculate and the point can never end up badly wrong. For , round to , so the exact answer 15.19 is believable while 1.519 or 151.9 are not.
The boundary case is when the product ends in zeros. gives digits-wise, placed as , which is written 0.2 — so count the places first and tidy the trailing zero afterwards, never the other way round.
Worked example: . Ignore the points and multiply . The factors have 1 and 2 decimal places, so the answer needs :
The rule comes straight from fractions, which is the surest way to see it:
The denominators multiplied to 1000, and three zeros mean three decimal places.
Estimate before you calculate and the point can never end up badly wrong. For , round to , so the exact answer 15.19 is believable while 1.519 or 151.9 are not.
The boundary case is when the product ends in zeros. gives digits-wise, placed as , which is written 0.2 — so count the places first and tidy the trailing zero afterwards, never the other way round.
How do you divide a decimal, including by another decimal?
Dividing by 10, 100 or 1000 moves the point to the left — one, two or three places:
A zero goes in before the digits when they run out, so the place values stay correct.
By a whole number, divide as usual and keep the point above where it was:
By a decimal, first make the divisor a whole number by multiplying both numbers by the same power of 10. For , multiply both by 100:
That step is legal because multiplying both parts of a division by 100 leaves the answer unchanged, in the same way and are the same fraction.
And notice the answer, 30, is far bigger than 4.5. Dividing by a number less than 1 always gives a larger result, because you are asking how many small pieces fit — a useful check that catches a misplaced point immediately.
A zero goes in before the digits when they run out, so the place values stay correct.
By a whole number, divide as usual and keep the point above where it was:
By a decimal, first make the divisor a whole number by multiplying both numbers by the same power of 10. For , multiply both by 100:
That step is legal because multiplying both parts of a division by 100 leaves the answer unchanged, in the same way and are the same fraction.
And notice the answer, 30, is far bigger than 4.5. Dividing by a number less than 1 always gives a larger result, because you are asking how many small pieces fit — a useful check that catches a misplaced point immediately.
How do you convert a fraction to a decimal and use it in word problems?
Divide the numerator by the denominator, adding a point and zeros as needed.
For , divide 3 by 8: it gives 0.375, since . Some fractions convert more directly — and — because the denominator is already a power of 10.
Word problems then become ordinary decimal arithmetic, provided the units stay attached.
Money. 2.5 kg of dal at ₹48.50 per kg costs
Speed. A car covering 45 km in 1.5 hours travels at
Mass. A packet of 0.25 kg repeated 12 times is kg.
The trap in these questions is mixed units. Adding 750 g to 1.2 kg needs one unit throughout: convert 750 g to 0.75 kg, giving 1.95 kg. Adding 1.2 and 750 without converting gives an answer that means nothing at all.
For , divide 3 by 8: it gives 0.375, since . Some fractions convert more directly — and — because the denominator is already a power of 10.
Word problems then become ordinary decimal arithmetic, provided the units stay attached.
Money. 2.5 kg of dal at ₹48.50 per kg costs
Speed. A car covering 45 km in 1.5 hours travels at
Mass. A packet of 0.25 kg repeated 12 times is kg.
The trap in these questions is mixed units. Adding 750 g to 1.2 kg needs one unit throughout: convert 750 g to 0.75 kg, giving 1.95 kg. Adding 1.2 and 750 without converting gives an answer that means nothing at all.
Exam tip
Exam tip: estimating first so the point lands in the right place
Almost every lost mark in this chapter is a decimal point in the wrong place, and a five-second estimate catches it.
Before calculating, round both numbers to something easy and write the rough answer in the margin. For , note "about ", so 28.56 is right and 2.856 is not.
For multiplication, count the decimal places in both factors and add them — that total is exactly how many the answer needs, before you tidy any trailing zeros.
For division by a decimal, show the step where you multiply both numbers by 10, 100 or 1000. It earns method marks and makes the long division far easier to write.
And convert all quantities to the same unit before doing anything else. Metres with centimetres, or kilograms with grams, in the same calculation is a guaranteed wrong answer.
Before calculating, round both numbers to something easy and write the rough answer in the margin. For , note "about ", so 28.56 is right and 2.856 is not.
For multiplication, count the decimal places in both factors and add them — that total is exactly how many the answer needs, before you tidy any trailing zeros.
For division by a decimal, show the step where you multiply both numbers by 10, 100 or 1000. It earns method marks and makes the long division far easier to write.
And convert all quantities to the same unit before doing anything else. Metres with centimetres, or kilograms with grams, in the same calculation is a guaranteed wrong answer.
Did you know
Why does multiplying two decimals give more decimal places?
Because the hidden denominators multiply together.
A number with one decimal place is really something over 10, and two places is something over 100. Multiply them and the denominator becomes — three zeros, so three decimal places.
That is the whole rule in one line: the zeros in the denominators add up, so the decimal places add up too. Which also explains the opening puzzle, where tenths times tenths landed in the hundredths.
A number with one decimal place is really something over 10, and two places is something over 100. Multiply them and the denominator becomes — three zeros, so three decimal places.
That is the whole rule in one line: the zeros in the denominators add up, so the decimal places add up too. Which also explains the opening puzzle, where tenths times tenths landed in the hundredths.
Key takeaways
Multiplying and dividing decimals: quick revision
- Multiplying by 10, 100 or 1000 moves the point right by 1, 2 or 3 places; dividing moves it left the same amount, padding with zeros as needed.
- For a decimal times a decimal, multiply without the points and give the answer as many decimal places as both factors have together: .
- Multiplying by 10 is not "adding a zero" — that only works for whole numbers.
- To divide by a decimal, multiply both numbers by the same power of 10 to make the divisor whole: .
- Dividing by a number below 1 gives a bigger answer, so use that to check the point's position.
- Convert a fraction by dividing, as , and always bring mixed units to one unit before calculating.
You will remember all of this far better after answering five questions on it than after reading it twice.
- For a decimal times a decimal, multiply without the points and give the answer as many decimal places as both factors have together: .
- Multiplying by 10 is not "adding a zero" — that only works for whole numbers.
- To divide by a decimal, multiply both numbers by the same power of 10 to make the divisor whole: .
- Dividing by a number below 1 gives a bigger answer, so use that to check the point's position.
- Convert a fraction by dividing, as , and always bring mixed units to one unit before calculating.
You will remember all of this far better after answering five questions on it than after reading it twice.