Fraction word problems

Read what the story asks you to find. For two known amounts of the same unit, finding their combined total uses addition. Finding what remains after an amount is removed uses subtraction. Choose from the relationship, not one story word alone.

To choose the nearest of 0, 1/2 and 1, use matching pieces. Double the fraction's denominator and numerator. This new denominator is even, so half is half its piece count. Zero is zero pieces, and 1 is the full piece count. Compare the distances from the new numerator to those three counts. The smallest distance is nearest. Equal distances mean a tie.

Adding a positive amount moves a total above the starting amount. A sum of two positive fractions exceeds each fraction separately. This gives a useful lower bound before you compute exactly.

To decide whether addition passes 1, compare the added amount with the gap from the starting fraction to 1. A larger added amount passes 1; an equal amount reaches 1; a smaller amount stays below 1. Nearness to 1 alone does not settle this.

For a mixed distance, keep complete miles separate from fraction parts. Match the fraction denominators and add those parts. If they make another whole mile, include that whole in the total too. Keep miles as the unit throughout.

Check an addition error by examining its numerator and denominator separately. The denominator must come from matching piece sizes, not from adding the original denominators. A proposed sum below either positive addend also breaks the lower bound.

For a remainder, subtract the amount removed from the starting amount. Both fractions must refer to the same whole or unit. Rewrite them with matching denominators first. A fraction of a whole bar is not automatically the same as that fraction of the remaining piece.

For three amounts, find a positive common multiple of all three denominators. Rewrite every amount with it, then add all three numerators. Use the same reference whole and separate, nonoverlapping portions when a sharing story counts parts of one object.

Two quarters make 1/2. Compare each of two addends with 1/4. If both are at least 1/4 and one is larger, their sum exceeds 1/2. Replacing a quarter with a larger amount makes the total larger. If both addends are below 1/2, their sum is below 1. This last rule is for two addends, not three. If the first addend is below 1/2, you can also compare the second with its gap to 1/2.

For successive removals, subtract each amount from what remains. Convert all the amounts to one shared piece size before starting. Check that the removed amounts do not exceed the amount available.