Multiplication can Increase or Decrease a Number
Question: Does multiplication always increase a
Yes it does; take the number 8, for example:
2 Ã— 8 = 16
3 Ã— 8 = 24
4 Ã— 8 = 32
In each it is getting larger, so, yes, multiplication
clearly increases a number.
No it increases a number only under certain
Multiplying any positive number by a whole number
greater than 1 will always increase its value see
the example opposite; but consider
; here the number 8 is reduced.
So, multiplying can have a reducing effect when multiplying a
positive number by a fraction which is less than one. But this
can still be confusing. While we accept the above, the concept of
' a number times 8' continues to be perceived as an increase. How
then can we attach a meaning to so that this will be
perceived as decreasing?
When multiplying by a whole positive number, e.g. 6 times 5,
we understand this as being 5 added over and over again, how ever
many times six times in this example. But this
interpretation of times does not quite work with fractions. If we
ask how many times , the answer is "not quite once".
Again we need to put the term multiplying into a context with
which we can identify, and which will then make the situation
We want to buy 30 roses which are sold in bunches of 5, so we
ask for "6 of the 5-rose bunches". In this way, the
word times also often means of. If we try using the word of when
times appears to have an unclear meaning, we get of 8 rather than times 8. Indeed we know
what of 8 means
So, by using of instead of times we are able to understand the
concept of multiplying by a fraction and how this can have a
reducing effect when the fraction is smaller than 1.
This also helps us to understand how we multiply by a
fraction, and why the method works:
the 4 which results from (or of 8) can be reached by
dividing 8 by 2;
similarly, the 5 which results from (or of 15), (or a third of
fifteen) can be reached by dividing 15 by 3.
Generalising this result gives:
When your bank balance is +4 pounds you have Â£4.
When your bank balance is 4 pounds you owe Â£4.
Owing is the opposite of having , so we find that we can
associate the concept of 'minus' with '(the) opposite (of)'. This
also works in reverse.
Thus, ( -4) Ã— 8 means " owing Â£4, eight times
over" or "owing Â£32" which is - Â£32 .
Now - 32 is smaller than 8, so we have illustrated another
case where multiplying has a reducing effect, i.e. when
multiplying by a negative number.
Note that, using the method shown above, it follows that -1 Ã—
8 = - 8 , and vice versa.
Are the following statements:
- sometimes false
- always true and always false
- sometimes true
- Multiplication of a positive number by a number greater
than 1 always increases the number.
- Multiplication of a positive number by a positive number
between 0 and 1 always increases the number.
- Multiplication of a negative number by a positive number
always increases the first number.
- Always true
- Always false, as multiplication of a positive number by a
number between 0 and 1 will always reduce the number.
- Sometimes false and sometimes true; e.g. for the number
8, 2 Ã— (-8) = -16, so the number is decreased,
whereas the number increases in the example below: