Long division into the decimals

The equivalent-fraction method needs a denominator that divides a power of ten. For something like 10/3 that is impossible, so we need a general method. Long division — division using place value — is it.

Recall how it works on whole numbers with 1324 ÷ 4. One thousand cannot be split into 4 parts, so regroup it as 10 hundreds, giving 13 hundreds. 13 hundreds ÷ 4 gives 3 hundreds each with 1 hundred left.

Regroup that hundred as 10 tens, giving 12 tens; 12 tens ÷ 4 gives 3 tens each. Then 4 ones ÷ 4 gives 1 one each. So 1324 ÷ 4 = 3 hundreds + 3 tens + 1 one = 331, exactly.

Now try 1325 ÷ 4. Everything goes the same way until the ones: 5 ones ÷ 4 gives 1 one each, with 1 ONE left over. How do you split one unit among four?

Regroup again — that is the whole trick. One one becomes 10 TENTHS. 10 tenths ÷ 4 gives 2 tenths each, with 2 tenths left. When we regroup ones into tenths, we place a decimal point in the quotient.

Still 2 tenths left, so regroup once more: 2 tenths become 20 hundredths. 20 hundredths ÷ 4 gives 5 hundredths each, with nothing left. Done.

So 1325 ÷ 4 = 3 hundreds + 3 tens + 1 one + 2 tenths + 5 hundredths = 331.25. Check it the other way: (1325 × 25)/(4 × 25) = 33125/100 = 331.25. Same answer. Try 237 ÷ 8 the same way and you will find 29.625.