Fractions of quantities II

In A Part of Something, we were introduced to the idea of finding a fraction of a quantity when the fraction was just a unit fraction.  We discovered that $\frac{1}{4}$14​ of something was the same as dividing by $4$4, or that $\frac{1}{9}$19​ of something was the same as dividing by $9$9.  In fact $\frac{1}{n}$1n​ of something is the same as dividing by $n$n.  It doesn't matter what $n$n is!

We are going to extend this idea a little bit more now by finding fractions of quantities when the fractions are not a unit fraction. so this means there is a number other than 1 on the top (numerator).

Fortunately because we can find the unit fraction quantity, this next step is a simple one. 

Remember our definition of a fraction,$\frac{\text{numerator }}{\text{denominator }}=\frac{\text{number of parts we want }}{\text{number of parts the whole is divided into }}$numerator denominator ​=number of parts we want number of parts the whole is divided into ​. 

This means that $\frac{3}{5}$35​ is $3$3 of the fifth size pieces, and $\frac{7}{9}$79​ is $7$7 of the ninth size pieces.

Lets see how to use this in a question

Example

Find:$\frac{3}{4}$34​ of $24$24

Think:  What is $\frac{1}{4}$14​ of $24$24?  

Do:  $\frac{1}{4}$14​ of $24$24 is $6$6 (remember we divide $24$24 by $4$4).

What we really want is $3$3 of the quarters, so $3\times6=18$3×6=18

$\frac{3}{4}$34​ of $24=18$24=18

 

Mixed Fractions

Lets learn how to deal with these by looking at a question in two different ways.

Find:  $2\frac{4}{5}$245​ of $20$20 litres.

 

Method 1 - splitting the mixed number apart

The first way is to split the mixed fraction into its whole and fraction parts.

$2\frac{4}{5}$245​ is the same as $2$2 wholes and then another $\frac{4}{5}$45​.

We find $2$2 groups of the $20$20 and $\frac{4}{5}$45​ of the $20$20 and add them together at the end.

$2\times20=40$2×20=40  this takes care of the 'wholes' component

$\frac{4}{5}\times20$45​×20, we find $\frac{1}{5}$15​ which is $4$4, ($20\div5=4$20÷​5=4) and then multiply by $4$4 to get $4\times4=16$4×4=16, so $\frac{4}{5}\times20=16$45​×20=16

Our answer is $40+16$40+16 which is $56$56.

DON'T FORGET THE UNITS! 

That's right, our question had units so the answer is $56$56 litres.

 

METHOD 2 - converting the mixed number to a single fraction

The second method is to convert the mixed number to a single fraction.

$2\frac{4}{5}$245​ is the same as $\frac{14}{5}$145​.

We find $\frac{1}{5}$15​ of $20$20 which is $4$4. ($20\div5=4$20÷​5=4)

and then we need $14$14 of these, so $14\times4=56$14×4=56.

So the answer is $56$56 litres.

Worked Examples

QUESTION 1

QUESTION 2