14.03 Find part or whole measures with ratios

We can now apply our skills to finding part or whole measures of quantities using ratios. This is useful for when we want to find an unknown value that is in a ratio with a known ratio.

 

Exploration

Consider a bag containing $56$56 red and blue marbles where the ratio of red marbles to blue marbles is $3:5$3:5.
How many red marbles are there? How many blue marbles are there?

To find the number of red and blue marbles we would normally use equivalent ratios by increasing or decreasing the ratio by some multiple. However, in this case, we only know the total number of marbles so this method won't work. So what do we do?

To solve our problem we can make the total number of marbles a component in our ratio. To do this we need to know how many ratio parts correspond to the total number of marbles. Since there are $3$3 parts red marbles and $5$5 parts blue marbles, the total number of marbles is simply $3+5=8$3+5=8 parts.

Taking this information, our ratio of red to blue to the total number of marbles is written $3:5:8$3:5:8.

Now we can use equivalent ratios to solve our problem.

Since we get $56$56 from $8$8 by increasing by a multiple of $7$7, we can find the number of red and blue marbles by increasing their ratio parts by the same multiple.

 

Red

to

Blue

to

Total

Ratio parts

$3$3

:

$5$5

:

$8$8

$\times7$×7

$\times7$×7

$\times7$×7

No. marbles

$21$21

:

$35$35

:

$56$56

 

Using the equivalent ratio we find that there are $21$21 red marbles and $35$35 blue marbles.

 

Part to whole ratios

In the exploration we used the ratio $3:5:8$3:5:8 to represent the ratio of red to blue to the total number of marbles. This is an example of a part to whole ratio. A part to whole ratio is a ratio that shows the ratio of one component compared to the whole and is useful when considering what fraction of the total represents that component.

For example, we can write the part to whole ratio of blue marbles to the total as $5:8$5:8 which is equivalent to saying that $\frac{5}{8}$58​ of the marbles are blue. This can then be used to find the number of blue marbles from the total with the calculation

Number of blue marbles$=$=$56\times\frac{5}{8}$56×58​$=$=$35$35

This is equivalent to finding the number of marbles in one part, that is we divide the total, $56$56, by $8$8, and then multiplying by $5$5 to find the number of marbles in $5$5 parts. This is known as the unitary method.

Number of blue marbles$=$=$\frac{56}{8}\times5$568​×5$=$=$35$35

Notice that the only difference is which number we divide by $8$8,

We can use the ratio $3:8$3:8 to perform a similar calculation to find the number of red marbles.

 

 

Can we use these types of ratios to find the total quantities when we only know the quantity of the component?

Yes, we can do this by multiplying the quantity of the component by the reciprocal of the fraction equivalent to the ratio.

For example, if we know that there are $21$21 red marbles and the ratio of red to the total number of marbles is $3:8$3:8 we can find the total number of marbles with the calculation

Total number of marbles $=$=$21\times\frac{8}{3}$21×83​$=$=$56$56

which returns the expected answer.

We can also solve this using the unitary method.

If $21$21 red marbles is $3$3 parts, we can divide $21$21 by $3$3 to find the size of one part, and then multiply this amount by $8$8 to find the total number of marbles,

Total number of marbles $=$=$\frac{21}{3}\times8$213​×8$=$=$56$56

 

Let's have a look at how we can use our part to whole ratios to split total lengths into parts.

 

Worked example 

example 1

A plank of wood is cut so that the length of the two pieces are in the ratio $3:11$3:11.
If the plank of wood was $280$280 cm long, what is the length of the shorter piece?

Think: Since we are given the total length and want to find the length of the short piece, we want to find the length of one part first, and then we can find the length of $3$3 parts, that is, the length of the shorter piece.

Do: We can find how many ratio parts are in the total component by adding the ratio components of the two pieces together. This gives us:

Ratio parts for the total $=$= $3+11$3+11 $=$= $14$14

So we can find the length of one part by dividing the total length by $14$14.

We can then multiply this by $3$3, to find the total length of the shorter piece.

Length of the short piece $=$= $\frac{280}{14}\times3$28014​×3 $=$= $20\times3$20×3 $=$= $60$60

So the length of the short piece is $60$60 cm.

Reflect: We found the total component for the ratio so that we could write the appropriate part to whole ratio. Then we used the unitary method to find the length of one part, and used this to find the length of three parts.

It should be noted that we could have instead constructed a parts to whole ratio and used equivalent ratios. Using this method we could write "short piece to long piece to total" as $3:11:14$3:11:14 and multiply each component by $20$20 to get $60:220:280$60:220:280. This also tells us that the shorter piece has a length of $60$60 cm.

 

Increase a quantity in a given ratio (Extended)

 

To increase a quantity in a given ratio, say $a:b,$a:b,we can multiply the quantity by the fraction $\frac{a}{b}$ab​, where $a>b$a>b. 

 

Worked example

Example 2

Increase the quantity $\$600$$600 in the ratio $5:4$5:4.

Think: We must multiply the quantity by a fraction larger than one to increase it. So we should multiply it by $\frac{5}{4}.$54​.

Do: 

Increased quantity	$=$=	$\$600\times\frac{5}{4}$$600×54​
	$=$=	$\$750$$750

 

Decrease a quantity in a given ratio (Extended)

 

To increase a quantity in a given ratio, say $c:d,$c:d,we can multiply the quantity by the fraction $\frac{c}{d}$cd​, where $cc<d. 

 

Example 3

Decrease the quantity $1200\text{ mL}$1200 mLin the ratio $5:6$5:6.

Think: We must multiply the quantity by a fraction smaller than one to decrease it. So we should multiply it by $\frac{5}{6}.$56​.

Do: 

Decreased quantity	$=$=	$1200\times\frac{5}{6}$1200×56​
	$=$=	$1000\text{ mL}$1000 mL

 

Practice questions

Question 1

Question 2

Question 3