8.02 Multiplication as scaling
Are you ready?
Do you remember how multiplication can be thought of as a comparison?
Use the multiplication number sentence $36=4\times9$36=4×9 to choose the correct statement.
Nine is thirty six times as many as four
AThirty six is four times as many as nine
BFour is nine times as many as thirty six
C
Learn
When we multiply any starting number by $1$1, the value does not change. $3\times1=3$3×1=3, $7\times1=7$7×1=7, and so on.
If we multiply by any other whole number, the result is larger than the starting number. We sometimes think of this as taking multiple groups of the starting number. For example, $4\times2$4×2 is two groups of $4$4, which has a total value of $8$8.
But notice that all whole numbers (other than $1$1) are larger than $1$1. What happens if we multiply by something smaller than $1$1 instead?
Let's look at multiplying $10\times\frac{1}{2}$10×12. If we continue to think about multiplication as groups of, then this is like taking half of a group of $10$10.
Half of a group is less than a whole group (in this case, half of $10$10 is $5$5), and so multiplying by this fraction has resulted in a value that is smaller than the starting number.
In general, if we multiply by a value between $0$0 and $1$1, the result is smaller than what we started with. If we multiply by a value larger than $1$1 (even if it is not a whole value), the result is larger than what we started with.
Apply
Question
Which of the following is true about the value of $\frac{8}{6}\times\frac{5}{4}$86×54?
$\frac{8}{6}\times\frac{5}{4}$86×54 is larger than $\frac{8}{6}$86.
A$\frac{8}{6}\times\frac{5}{4}$86×54 is smaller than $\frac{8}{6}$86.
B$\frac{8}{6}\times\frac{5}{4}$86×54 and $\frac{8}{6}$86 have the same value.
C
When we multiply by a number:
- If the number is greater than $1$1, the result is larger than the starting value
- If the number is less than $1$1, the result is smaller than the starting value
- If the number is equal to $1$1, the result is the same as the starting value