Hi Kenneth.

It is unusual for decimals and fractions to be written in this way, so we don't see this very often.

I agree that the expression 0.24 ½ is 0.245. But this is an odd expression. The closest real-world situation might be something like "multiply the $30 cost of the shirt by 5 ½ % to find the sales tax" which would correspond to 30 × 0.055, because 5 ½ % = 0.05 ½ (per cent meaning "divided by 100").

But the space in this case binds more tightly than the percent sign, so really

5 ½ % = (5 ½) % = 5 % + ½ % = 0.05 + 0.5% = 0.05 + 0.005 = 0.055 anyway.

So this interpretation does work, but isn't really necessary.

The point is that 5 1000^th's = ½ 100^th.

So if you write it with the fraction, it is the 100^th place that holds the ½, not the 1000^th place, otherwise you have changed it from 0.245 to 0.2405.

Thus in 0.24 ½, you have a 0 in the one's place, 2 in the tenths place, and 4½ in the hundredths place. This is equivalent to 0 in the ones, 2 in the tenths, 4 in the hundredths and 5 in the thousandths, but not equivalent to 0 in the ones, 2 in the tenths, 4 in the hundredths and ½ in the thousandths.

Remember that when writing numbers, the goal is to communicate effectively and unambiguously. If a student gave me an answer on a test and wrote 0.24 ½ instead of 0.245, I would probably take off half a mark for the lack of clarity.

Hope this helps,
Stephen La Rocque.