Make the divisor whole without changing the quotient
Before this page: be confident with multiplying by powers of ten, then dividing with short division or long division.
Standard written division needs an integer divisor. A decimal divisor can be made into an integer by multiplying it by 10, 100 or 1,000—but the dividend must be multiplied by exactly the same number.
Scaling both numbers equally changes their sizes but not their ratio, so it does not change the quotient.
a ÷ b=(a × k) ÷ (b × k)
Use the same positive scale factor k on both numbers.
Scale both numbers together
The divisor 0.46 has two decimal places. One multiplication by 10 is not enough; multiply both numbers by 10 again, then stop because the divisor is an integer.
55.2 ÷ 0.46
55.2 ÷ 0.46 = 5,520 ÷ 46 = 120
Why equal scaling preserves the answer
The quotient counts how many divisor-sized groups make the dividend. Scaling both numbers makes every group larger and makes the total larger by the same factor, so the number of groups does not change.
Here, 120 groups of 0.46 make 55.2. After both numbers are multiplied by 100, the same 120 groups of 46 make 5,520.
Choose the smallest sufficient power of ten
Count the divisor's decimal places after ignoring trailing zeroes. Use the first power of ten that makes that divisor an integer.
One decimal place
7.2 ÷ 0.3 = 72 ÷ 3 = 24
Two decimal places
1.8 ÷ 0.04 = 180 ÷ 4 = 45
The dividend may stay decimal
8.75 ÷ 2.5 = 87.5 ÷ 25 = 3.5
Ignore a trailing zero
6.4 ÷ 0.20 = 64 ÷ 2 = 32
0.20 has the same value as 0.2, so one multiplication by 10 is sufficient.Only the divisor needs to become an integer. In 8.75 ÷ 2.5, 87.5 is a valid dividend for written division, so multiplying both numbers by another 10 would add unnecessary work.
Scale a chosen division
Enter a positive dividend and a positive, non-integer decimal divisor. The equivalent calculation updates using the smallest sufficient power of ten; an ellipsis marks a decimal that continues beyond the displayed digits.
Multiply both by 100
12.6 ÷ 0.35 = 36
Multiplying both numbers by 100 gives 1,260 divided by 35, so the quotient is 36.
Why the quotient can exceed the dividend
Division can ask how many equal pieces of a given size fit into an amount. If each piece is smaller than 1 unit, more than one piece fits into each whole unit.
55.2 metres cut into 0.46 metre lengths
Each length is less than 1 metre, so more than one length fits into each metre. Exactly 120 lengths can be cut.
This size effect is explored further on multiplying and dividing by numbers below one.
Check the original calculation
Check with multiplication
120 × 0.46 = 55.2, so 55.2 ÷ 0.46 = 120.
Check the size
Because 0.46 is below 1, the quotient should exceed 55.2. The answer 120 has the expected size.
Common mistakes
- Scaling only the divisorMultiplying one number changes the quotient. Apply the same factor to both numbers.
- Stopping while the divisor is still decimalIn 552 ÷ 4.6, another equal multiplication by 10 is still required.
- Forcing both numbers to be integersStop once the divisor is an integer. A decimal dividend can be divided using the methods already learned.
- Using different scale factorsThe two numbers must move through the same number of place-value columns.
Summary
- Choose the smallest power of ten that makes the decimal divisor an integer.
- Multiply the dividend and divisor by exactly the same power of ten.
- Stop once the divisor is an integer; the dividend may still be decimal.
- Use short or long division on the equivalent calculation.
- Check with multiplication and predict the answer's size from the original divisor.