Let us return to the grocers to see how multiplying more than two numbers should work. Not a list this time but a practical example.
A grocer sells boxes of six apples .
A customer orders five of these boxes to be delivered in a box as a gift.
How many apples are delivered in the gift box?
Five boxes of six apples, so 5 x 6 apples, which is 30 apples.
A restaurant seeing this gift box orders four of them, how many apples will the restaurant receive?
Four gift boxes, each containing five boxes of six apples.
Four gift boxes, each containing 5 x 6 = 30 apples.
Four gift boxes of 30 apples, so 4 x 30 = 120 apples.
Using () to show calculations in a box we have worked out 4 x (5 x 6) = 4 x 30 = 120.
We could have done this differently.
Each gift box will have five boxes in it
So there will be four gift boxes each containing five boxes, so 4 x 5 = 20 boxes.
Each of these twenty boxes will contain six apples, so 20 x 6 = 120 apples.
We have worked out (4 x 5) x 6 =120.
We can only multiply two numbers at a time however we have found out that which pair to start with does not matter.
4 x (5 x 6) = (4 x 5) x 6 = 4 x 5 x 6 =120
As it is with boxes of apples so it is with cards and holders.
Consider this hand
Using a new holder to hold these four piles we get
and as gives the same result as
gives the same result as which results in one hundred and twenty black counters.
We could have done this differently is four holders each containing five holders or a holder of twenty
So gives the same result as which also results in one hundred and twenty counters.
is recorded as 4 x 5 x 6 =120 no () needed since we can multiply any pair first.
When we have four gift boxes containing five boxes we have a total of twenty boxes .
These can be re-arranged as five gift boxes containing four boxes.
In this case there is still a total of twenty boxes .
We have already seen that and give the same results.
Thinking about the boxes above it is also true that four holders holding five holders will be the same as five holders holding four holders .
All of the following give the same result.
Recorded as 5 x 4 x 6 = 4 x 5 x 6 = 4 x 6 x 5 = 6 x 4 x 5 = 6 x 5 x 4 = 5 x 6 x 4 = 120
When multiplying numbers the numbers may be multiplied in any order.