Division with Remainder

Question

When I divide 48/12 by hand, it's 0 Remainder 12. But when I divide it in a calculator, it's .25. Why is this? How does 0R12 turn into .25? Thanks.

ANSWER (system won't let me answer my own question because this is my first post)

I get it! if you have 12/48, it's not really divisible. Instead, you treat it as a fraction: 12/48 simplified is 1/4, which is .25 in a calculator! I can't believe that took me the whole day to figure out.

Thanks guys for keeping me thinking. I was helping my nephew (obviously been out of school for a while).

Answer 1

When you divide 48/12 by hand, it should be 4 Remainder 0. Also, 48/12 in a calculator should come out to 4, not .25. Perhaps you accidentally did 12/48 in your calculator instead?

Anyways, either way, your result comes out to 4 with 0 remainder. Hope that helps.

Answer 2

Either answer is correct, depending on what number system and arithmetic system you are working in.

If you are working with just integers, using standard grade-school integer arithmetic, then it is completely correct to say "12 divided by 48 results in 0 remainder 12." There are no fractional integers, so numbers like $0.25$ don't exist in that arithmetic system.

If you are working in rational numbers or real numbers, then $12/48 = 1/4 = 0.25$. Most calculators are designed to work with decimal numbers with fractional parts, which is a kind of rational number, so their answer to the division will be $0.25$.

You can use an ordinary calculator to help you do integer arithmetic, but you have to realize what the results mean to you. The integer quotient is only the part to the left of the decimal point; anything to the right of the decimal point is an indication of a remainder. But you have to multiply that fraction by your original divisor to get the true integer remainder. For example, on the calculator,

$$ 72/48 = 1.5.$$

So the integer quotient is $1$. To find the remainder, we take just the "fractional" part of the calculator's result, namely $0.5$ in this case, and multiply by the original divisor:

$$ 0.5 \times 48 = 24.$$

Therefore 72 divided by 48 results in 1 remainder 24 in integer arithmetic.