Math Detours · Grade 5 · Sharpen · 6

Why you cannot add thirds to quarters

Not until the pieces are the same size. Cut both into twelfths and the sum is easy.

Name ____________________Date ____________

Tonight's idea

Two apples and three apples make five apples. Two apples and three pears do not make five of anything. You can only count things together when they are the same kind.

The bottom number of a fraction tells you the kind of piece. In \(\tfrac13 + \tfrac14\) the pieces are different sizes, so you cannot just count them.

The fix is to cut both into smaller pieces that match. Thirds and quarters both cut neatly into twelfths, because \(3 \times 4 = 12\).

13=412 14=312 712in allboth cut into twelfths, laid end to end
A third is 4 twelfths. A quarter is 3 twelfths. Now the pieces match and you can count them.

Now it is 4 twelfths plus 3 twelfths. That is 7 twelfths: \(\tfrac13 + \tfrac14 = \tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12}\).

Rename both fractions so they have the same bottom number. Then add the tops. The bottom stays as it is.

Taking away works the same way. To find a bottom number that suits both, you can always multiply the two bottoms together.

Watch one: \(\tfrac34 - \tfrac16\)

Quarters and sixths do not match. Try \(4 \times 6 = 24\), so cut both into twenty-fourths.

\(\tfrac34 = \tfrac{18}{24}\), multiplying top and bottom by 6. \(\tfrac16 = \tfrac{4}{24}\), multiplying top and bottom by 4.

\[\tfrac{18}{24} - \tfrac{4}{24} = \tfrac{14}{24} = \tfrac{7}{12}\]

The last step divides top and bottom by 2. Twelfths would have worked from the start, since 4 and 6 both go into 12.

Pick two fractions and press "cut smaller": mathdetours.com/demos/fractionbars.html.

Your turn

  1. (a) \(\tfrac27 + \tfrac37\); (b) \(\tfrac12 + \tfrac14\); (c) \(\tfrac12 + \tfrac13\).
  2. (a) \(\tfrac23 + \tfrac15\); (b) \(\tfrac34 - \tfrac23\); (c) \(\tfrac56 - \tfrac14\).
  3. Ana drinks \(\tfrac14\) of a bottle of juice. Ben drinks \(\tfrac25\) of the same bottle. What fraction has been drunk? What fraction is left?
  4. Leo says \(\tfrac12 + \tfrac13 = \tfrac25\). He added the tops and added the bottoms. Do not work out the right answer. Explain how you can tell that \(\tfrac25\) must be wrong.

Stretch: the smallest sum

Put the digits 1, 2, 3 and 4 in the boxes, using each once: \(\tfrac{\square}{\square} + \tfrac{\square}{\square}\). Make the sum as small as you can.

Answers

  1. (a) \(\tfrac57\). The pieces already match. (b) \(\tfrac24 + \tfrac14 = \tfrac34\). (c) \(\tfrac36 + \tfrac26 = \tfrac56\).
  2. (a) \(\tfrac{10}{15} + \tfrac{3}{15} = \tfrac{13}{15}\). (b) \(\tfrac{9}{12} - \tfrac{8}{12} = \tfrac{1}{12}\). (c) \(\tfrac{10}{12} - \tfrac{3}{12} = \tfrac{7}{12}\).
  3. \(\tfrac{5}{20} + \tfrac{8}{20} = \tfrac{13}{20}\) drunk. \(\tfrac{20}{20} - \tfrac{13}{20} = \tfrac{7}{20}\) left.
  4. \(\tfrac25\) is less than a half, because half of 5 is \(2\tfrac12\). But Leo started with a half and added more. The answer has to be bigger than a half.

Stretch. \(\tfrac13 + \tfrac24 = \tfrac{4}{12} + \tfrac{6}{12} = \tfrac{10}{12} = \tfrac56\). Small tops and big bottoms make small fractions, so 1 and 2 go on top. The other choice, \(\tfrac14 + \tfrac23 = \tfrac{11}{12}\), is bigger.

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