Mixed-Number Addition and What Is Left

One and one fifth is one whole and one fifth. A whole is five fifths, so it is six fifths altogether. That is the conversion: multiply the whole by the denominator and add the numerator. Going back is a division: seven quarters is seven divided by four, one remainder three, one and three quarters. Whole numbers are fractions as well — three is fifteen fifths, or twelve quarters, whichever pieces you need. For adding and subtracting, improper fractions are usually easier, because everything becomes counting pieces of one size. For the final answer, give the form the question asks for.

To add mixed numbers, add the wholes and add the fractions separately. Two and three quarters plus one and a half: the wholes make three; the fractions, three quarters plus a half, need matching pieces — three quarters plus two quarters is five quarters. Five quarters is more than a whole, so carry: one and a quarter, plus the three, is four and a quarter. Or turn both into quarters from the start — eleven quarters plus six quarters, seventeen quarters, four and a quarter — the same answer by a different road. Estimate with the wholes: nearly three plus one and a half is about four and a bit.

Four pizzas, and three people eat four fifths, one and a fifth, and three fifths. How much is left for the fourth? Write everything in fifths. Four pizzas are twenty fifths. One and a fifth is six fifths. Eaten: four plus six plus three, thirteen fifths. Left: twenty fifths minus thirteen, seven fifths, which is one and two fifths of a pizza. The slice picture says the same thing: cut each pizza into five, twenty slices, thirteen eaten, seven left. Use whichever you find clearer, and use the other one to check.

The selective paper does not stop at what is left; it asks how much more the fourth person ate than the first. That is a subtraction of the two shares: seven fifths minus four fifths, three fifths of a pizza more. Compared with Fred's six fifths the difference is one fifth; with Sally's three fifths, four fifths. The trap is that it is a two-step question — you have to find Quoc's share first, and only then compare it. Keep the shares as improper fractions until the subtraction is done.

One problem, three pictures. Fractions: twenty fifths minus thirteen fifths. Slices: twenty slices minus thirteen slices. A strip of twenty cells: shade four, six and three, and count the seven unshaded. They are the same calculation in different clothes, and having more than one is how you check yourself under time pressure. The strip is especially good at catching the commonest slip — forgetting to convert the mixed number — because you cannot shade one and a fifth cells. Quiz and exercises now.