Fractions of quantities I

Method 1 - Using words

You have probably already had experience with questions like, 

What is a half of $16$16?  This is because halves come up all the time, especially if you have ever had to share with a a brother or a sister!  To find a half of something we know that we divide it by $2$2.  

We can also write  a  half of 16 using symbolic fraction notation.  It would look like this $\frac{1}{2}\times16$12​×16

So if we have to do questions like $\frac{1}{2}$12​ of $32$32 or $\frac{1}{4}$14​ of $16$16 we might find it easier to say a half of $32$32, or a quarter of $16$16.  The answers are sometimes more obvious!

 

Method 2 - Reversing the Order

What about questions like $\frac{1}{3}$13​ of $4$4 or $\frac{1}{4}$14​ of $9$9.  Saying these in words doesn't help much here.  A third of $4$4, or a quarter of $9$9 - so I still don't know the answer!

So let's try something else, because multiplication is commutative we can change the order around. You know how $2\times4=4\times2$2×4=4×2, well this works when you are multiplying anything - even fractions.

Lets see how this might help

$4\times\frac{1}{3}$4×13​ we would say as $4$4 thirds..... just like how $2\times4$2×4 we say $2$2 fours.  

Well $4$4 thirds,  is the answer!  We can write $4$4 thirds as $\frac{4}{3}$43​. 

 

The whole idea

Now lets connect some dots.

$\frac{1}{3}$13​ of $4$4 = $\frac{1}{3}\times4$13​×4 = $4\times\frac{1}{3}$4×13​ = $4$4 thirds = $\frac{4}{3}$43​ = $4\div3$4÷​3

One third, ended up being the same as dividing by $3$3.

Lets look at another one,

$\frac{1}{5}$15​of $20$20 = $\frac{1}{5}\times20$15​×20=$20\times\frac{1}{5}$20×15​ = $20$20 fifths = $\frac{20}{5}$205​ = $20\div5$20÷​5 , so$\frac{1}{5}$15​is the same as dividing by $5$5.

This comes right back to our definition of a fraction, the denominator is how many parts we divide up into.

So $\frac{1}{10}$110​ of $80$80 is $80\div10=8$80÷​10=8

$\frac{1}{4}$14​ of $20$20 is $20\div4=5$20÷​4=5

and 

$\frac{1}{7}$17​ of $21$21 is $21\div7=3$21÷​7=3

Practice Questions

QUESTION 1

QUESTION 2

QUESTION 3