LogicGrades 5–6
Some children go to the chess club, some to football, some to both, and some to neither. In these tasks the groups overlap, and the trick is to count everyone exactly once. Two circles drawn on paper turn a confusing question into four numbers.
chess football ( 7 ( 3 ) 9 )
Adding the two groups counts the children in both of them twice, once in each. Take them away once, and everyone is counted once.
Nobody has to be in a group. The whole class is those in at least one group and those in none.
If nobody says how many are in both groups, that number can only be bounded: the more the groups share, the fewer children are in at least one of them.
30 pupils: 14 sing in the choir, 11 go to the art club, and 4 do both. Fill the picture from the middle out.
| The part | How many | How it is found |
|---|---|---|
| In both | 4 | given |
| Choir only | 14 − 4 = 10 | the choir without those in both |
| Art only | 11 − 4 = 7 | the art club without those in both |
| In neither | 30 − 10 − 7 − 4 = 9 | everyone else |
Draw two circles that overlap. Write the middle first, those in both, then each circle without the middle, and last those outside. When the middle is not given, try the values it could take and see which fit. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 5–6
In a group of 24 children, 15 have a sister, 11 have a brother, and 6 have both a sister and a brother. How many children have neither?
sister brother
( 9 ( 6 ) 5 )
outside: 4Answer 4 children.
Example 2 · Grades 5–6
In a group of 25 children, 18 like apples and 16 like pears. What is the smallest number of children who could like both?
Answer 9 children.
Example 3 · Grades 5–6
Each of 20 pupils likes at least one of three subjects: maths, reading and art. 12 like maths, 10 like reading and 9 like art. Nobody likes all three. How many pupils like exactly two subjects?
Answer 11 pupils.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
The child adds the two groups, 15 and 11, and takes the sum for the number of children: those in both groups are counted twice.
What to say
“Is anybody counted in both circles?”
The child finds a part of the picture and gives it as the answer, though the question asked about another: “in both” instead of “in neither”, or “choir only” instead of the whole choir.
What to say
“Point to the part of the picture the question asks about.”
A part of the picture is left out: one group is taken away from everyone and the other forgotten, or those in all three groups are not added back.
What to say
“Is every part of the picture filled in?”
Lay two hoops or two loops of string on the floor so that they overlap, and sort toys into them: “red” and “soft”. A soft red bear goes in the middle.
When the child answers “six”, ask “six who?”. Six in both clubs and six in neither are different answers.
Ask the family two questions, such as who likes tea and who likes cocoa, and draw the circles together. The middle is where the counting usually goes wrong.