Number Rounding error ?

Question

Hellooooooooooo

I was doing problem in Manhattan GMAT math series (inequalities) and they way they rounded the number threw me off at first, i think i understand it now but would like your opinion

It's about a biker crossing a distance of 225 miles rounded to the nearest mile, in 5 hours rounded to the nearest hours, what was his lowest possible and highest possible speed

I interpreted the 225 miles rounding to mean that he must have cycled 224.5 (minimum to round up to 225) and 225.4

In the solution they say its between 224.5 and 225.5

So the difference is in the upper limit

After thinking about it the mention that it is strictly between those numbers, and those numbers aren't included ? so Speed < 225.5 is that what they mean ? Cause 225.5 rounds up to 226, but a fraction less than 225.5 is 225.499... which rounds down to 225

Is that the correct logic or did I miss something ?

whiplash2411 · Answer

I think what they meant is using the upper limit of one side of the inequality and the lower limit of the other side of the inequality. In essence, they're taking the bound of values on each side of 225 that would be rounded to 225, i.e.

224.5 <= 225 < 225.5

225.499 would round to 225.5 and that would also round to 226. The largest possible fraction to round to 225 would be 225.44444444 .... 49.

I am not really sure what logic they're using. But I might just guess that estimating this kind of a recurring decimal is hard, and they just decided to go with the 225.5.

Can someone else answer this?

kuharido · Answer

Hey thanks whiplash, i think your explanation is probably the case in terms of how they thought about it

My remaining doubt would be regarding how you round 225.4999 , if a question, not related lets say, just asked to round 225.4999 to the nearest units digits, i was under the impression that it would round to 225.0 because the 4 drops no ? even if it is followed by 9, or does the rounding "cascade" so to speak ?

what do you think ?

whiplash2411 · Answer

You are right in assuming that 225.4999 would be rounded to 225.

I was talking about 225.49, for instance. I am not really sure how the rule would work in this case since 225.49 can be rounded to 225.5 which would eventually be rounded to 226. I am not really sure as to what constitutes the break off point for rounding something DOWN instead of UP

Bunuel · Answer

Rounding rules

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

So if  x  was rounded to the nearest mile and the result is 225, then it must be true that  224.5\leq{x}<225.5 .

225.49 rounded to the nearest mile would be 225 since the dropped 4 (the first dropped digit) is less than 5.

Hope it's clear.

Number Rounding error ?

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