Share 3 bars among 4 people
Four people share 3 bars equally. There are fewer bars than people, so nobody can have a whole bar. Each bar has to be cut.
Cut every bar into 4 equal pieces, one piece for each person. Each piece is a quarter of a bar, , because 1 bar shared among 4 people is each. Then each person takes one piece from each of the 3 bars.
Each of the 3 bars is cut into 4 quarters. The colored quarter in each bar is the first person's piece.
One share is
Put one person's pieces together: a quarter from each of 3 bars is 3 quarters, of a bar. So 3 shared among 4 is each: .
The line in a fraction is a division sign. means 3 ÷ 4: the numerator is the amount being shared, and the denominator is the number of equal shares.
Check it: the 4 people each have 3 quarters, which is 4 × 3 = 12 quarters. 12 quarters fill 12 ÷ 4 = 3 whole bars, which is what there was to share.
One person's 3 quarters, put together, fill of a bar.
each of the 4 people gets one piece of every bar: 3 pieces of 1/4, so 3 ÷ 4 = 3/4
Share 3 bars among 4 people
Hand out the bars one at a time: every person gets one quarter of each bar. After all 3, each holds 3 quarters, so .
Any division can be written as a fraction
The same is true of every division. , , and .
When the amount shared is bigger than the number of shares, each share is more than one whole. Share 7 bars among 4 people. Each person can have 1 whole bar, which uses 4 bars. The 3 bars left over are shared as before, a quarter from each, so each person also gets . Each share is .
Cutting all 7 bars into quarters gives the same answer. Each person takes a quarter from each of the 7 bars, which is 7 quarters, . And , because 4 of the 7 quarters make a whole bar.
One person's share of 7 bars: 7 quarters, which fill one whole bar and of another. .
The remainder is shared too
In whole numbers, 7 ÷ 4 = 1 remainder 3. As a fraction, the 3 left over is not left over: it is shared out as well, each. So the remainder becomes the numerator, and 1 remainder 3 becomes .
The remainder is not a decimal either. 1 remainder 3 is not 1.3, because the 3 is shared among 4, not cut into tenths.
Which number goes on top
The usual mistake is to turn the fraction upside down. 3 bars shared among 4 people is , not . The amount being shared goes on top. A quick check: with fewer bars than people, each share must be less than one bar, and is less than 1 while is more.
is before 1 and is past it. Sharing 3 bars among 4 people gives each less than one bar, so the share is .
Worked example: Pizzas Shared Equally at Two Picnic Tables, Compared
Question At a school picnic, all the pizzas are the same size. At the first table, 3 pizzas are shared equally among 4 children. At the second table, 2 pizzas are shared equally among 3 children. (a) What fraction of a pizza does each child at the first table get? (b) Does each child at the first table or each child at the second table get more pizza, and how much more?
1.First table: cut each of the 3 pizzas into 4 equal parts, one part for each of the 4 children. Each part is 14 of a pizza.
Table 1: three pizzas, each cut into 4 equal parts, one part for each of the 4 children. 2.(a) Each child takes one part from every pizza, so each child gets 3 quarters: 3 ÷ 4 = 34 of a pizza.
(a) One child takes one part of every pizza: 3 quarters, 3 ÷ 4 = 34 of a pizza. 3.Second table: cut each of the 2 pizzas into 3 equal parts. Each child takes one part from each pizza, so each child gets 2 thirds: 2 ÷ 3 = 23 of a pizza.
Table 2: two pizzas, each cut into 3 equal parts. One child takes 2 thirds, 2 ÷ 3 = 23. 4.Draw the two shares on bars one pizza long and cut both bars into twelfths: 34 = 912 and 23 = 812.
Each share on a bar one pizza long, cut into twelfths: 34 = 912 and 23 = 812. 5.(b) The first share is one twelfth longer, so each child at the first table gets 912 − 812 = 112 of a pizza more. Check: 4 × 34 = 3 pizzas and 3 × 23 = 2 pizzas.
(b) A child at the first table gets 912 − 812 = 112 of a pizza more.
Answer: (a) 34 of a pizza; (b) each child at the first table, by 112 of a pizza
Common mistakes
- Writing the share at the first table as 43, children over pizzas. The amount being shared, 3 pizzas, is divided by the number of children, 4, so it goes on top: 3 ÷ 4 = 34. With fewer pizzas than children, each share must be less than one pizza.
- Deciding that each child at the second table gets more because there are fewer children there. There are also fewer pizzas; only the two shares, written over the same denominator, show which is larger.