Finding the whole when only a fraction of it is known

When you reach for it

A class is painting a mural on the school wall, one section at a time. Every question about it has three numbers in it: the whole job, a part of the job, and the fraction of the whole that the part is. They are tied together by one multiplication:

When you know the whole and the fraction, you multiply, as in finding a fraction of a number: \displaystyle \frac{3}{4} of a 16 square meter mural is 12 square meters.

This lesson runs the other way. You can see the part, and you are told what fraction of the whole it is, but the whole is hidden. Beatriz has painted 12 square meters, and that is \displaystyle \frac{3}{4} of the mural. How big is the mural? Reach for this move whenever an amount comes with the words “which is ... of the whole”, “that is ... of what we need”, or “so far we have done ...”.

The procedure

A tape diagram

Draw the whole mural as a long bar, a tape. The fraction \displaystyle \frac{3}{4} says the mural is cut into 4 equal pieces and the part fills 3 of them, so cut the tape into 4 equal pieces and shade 3. The 12 square meters go over the shaded pieces.

Now read the tape. Three equal pieces hold 12 square meters, so one piece holds 12 \div 3 = 4. The pieces are equal, which is what lets you share the 12 evenly. The whole is 4 of those pieces, so it is 4 \times 4 = 16 square meters.

That is the whole idea: the numerator tells you how many pieces the part fills, so dividing by it gives one piece; the denominator tells you how many pieces make the whole, so multiplying by it gives the whole.

Dividing by the fraction

The multiplication above has the whole as its missing factor:

A missing factor is found by division, because division undoes multiplication. So

Dividing by \displaystyle \frac{3}{4} is multiplying by its reciprocal \displaystyle \frac{4}{3}, as dividing by multiplying by the reciprocal shows, and why flipping works explains. The tape shows the same two moves: multiplying by \displaystyle \frac{4}{3} divides by 3 and multiplies by 4, which is “find one piece, then build the whole”. The tape is the picture and the division is the shortcut; they always agree.

In general, if a part p is \displaystyle \frac{a}{b} of the whole, then

When the part is a mixed number, write it as an improper fraction first, as in dividing mixed numbers.

Checking by multiplying back

The answer can always be tested, because the forward direction is easy. Take the fraction of your whole and see whether it gives the part back:

It does, so 16 square meters is right. A quick size check helps too. When the fraction is less than 1, the part is smaller than the whole, so the whole must come out bigger than the part. An answer smaller than the part is wrong before you multiply anything.

Worked examples

Boundaries

The fraction must belong to the number you were given. If the fraction describes one part and the number describes another, as in Example 5, find the fraction that goes with the number first.

The fraction must be of the whole you want. In Example 7 the sky is \displaystyle \frac{2}{3} of a panel, not of the mural, so one division gives the panel and a second gives the mural.

The pieces must be equal. A fraction means equal shares, which is why one piece can be found by sharing the part evenly. When paint and area are mixed in one problem, the move also needs the paint to go on evenly, so that a fraction of the area uses the same fraction of the tins.

The fraction cannot be 0. “No part of the mural is 5 square meters” makes no sense, and there is no dividing by 0.

A fraction bigger than 1 still divides, but the word “part” no longer fits. If 15 tins is \displaystyle \frac{5}{4} of what is needed, then \displaystyle 15 \div \frac{5}{4} = 12 tins are needed, fewer than you have. The amount is more than the whole, so the whole comes out smaller.

Where this goes next

Finding the whole is the “how much in one whole” meaning of division, the one in sharing into equal parts. The next lesson, word problems that divide fractions, mixes it with the “how many fit” meaning and asks you to tell them apart. A tape cut into equal pieces is also the main tool of tape diagrams for ratios. Later, the same backwards move with hundredths in place of other fractions becomes finding the whole from a part and a percent.

Exercises

Tape diagrams

Draw a tape diagram for each, then find the whole.

1. Beatriz paints 7 square meters, which is \displaystyle \frac{1}{6} of the mural. How big is the mural?
2. The class has used 18 tins of paint, which is \displaystyle \frac{3}{5} of the paint the mural needs. How many tins does the mural need?
3. Ola’s border is 20 square meters, which is \displaystyle \frac{4}{9} of the mural. How big is the mural?
4. Rahim has painted 14 square meters, which is \displaystyle \frac{7}{10} of the mural. How big is the mural?

Dividing by the fraction

Find the whole by dividing, then check by multiplying back.

5. 24 square meters is \displaystyle \frac{6}{7} of a mural. Find the area of the mural.
6. Nia uses \displaystyle \frac{2}{3} of a tin of yellow, which is \displaystyle \frac{1}{9} of all the paint. How much paint is there?
7. Elena uses \displaystyle \frac{5}{6} of a tin of green, which is \displaystyle \frac{5}{8} of all the green the mural needs. How much green does it need?
8. Soren’s section is \displaystyle 3\frac{3}{4} square meters, which is \displaystyle \frac{3}{8} of the mural. Find the area of the mural.
9. The class has used \displaystyle 6\frac{2}{3} tins, which is \displaystyle \frac{5}{6} of the paint the mural needs. How many tins does it need?
10. Beatriz’s sunflower is \displaystyle 4\frac{1}{2} square meters, which is \displaystyle \frac{9}{20} of the mural. Find the area of the mural.

Which fraction goes with the number

11. \displaystyle \frac{2}{7} of the mural is painted, and 25 square meters are still to paint. How big is the mural?
12. After the first day, \displaystyle \frac{5}{12} of the mural is painted and 21 square meters are left. How big is the mural, and how much was painted on the first day?
13. Elena has used \displaystyle \frac{3}{5} of her tins and has 8 tins left. How many did she start with?

Two steps

14. Soren paints \displaystyle \frac{1}{3} of the mural and Nia paints \displaystyle \frac{1}{4} of it. Together they paint 28 square meters. How big is the mural?
15. A tree takes up 9 square meters, which is \displaystyle \frac{3}{5} of the left panel. The left panel is \displaystyle \frac{1}{4} of the mural. How big is the mural?
16. Beatriz paints \displaystyle \frac{1}{2} of the mural, Ola paints \displaystyle \frac{1}{3}, and Rahim paints the remaining 7 square meters. How big is the mural?
17. Rahim has painted \displaystyle \frac{3}{8} of the mural with 6 tins. Paint goes on evenly. How many more tins does he need to finish?
18. The class bought 21 tins of paint, which turns out to be \displaystyle \frac{7}{6} of the paint the mural needs. How many tins does the mural need, and how many will be left over?

Reasoning

19. Beatriz says the 10 square meters finished so far are \displaystyle \frac{2}{5} of the mural. Ola says the 6 square meter border is \displaystyle \frac{1}{4} of the same mural. Can both be right? Explain.
20. Explain, using a tape diagram, why \displaystyle 12 \div \frac{3}{4} can be worked out as “divide by 3, then multiply by 4”.
21. Nia says that 12 square meters is \displaystyle \frac{3}{5} of the mural, so the mural is 8 square meters. Without working out the answer, explain how you know she is wrong.
22. Explain why multiplying the fraction by your answer checks a “find the whole” problem.
23. Write a mural problem whose answer is \displaystyle 12 \div \frac{2}{3}, then solve it.