Last lesson we saw that we could express a ratio as a parttopart relationship or a parttowhole relationship. For example, if we wanted to describe the ratio of green dots to blue dots in the image above, we would write it as ratio $2:3$2:3. And alternatively, if we wanted to express the ratio of green dots to the total amount of dots we could use the ratio $2:5$2:5.
We can also express ratios as fraction in these two ways. The ratio of green to blue dots is $\frac{2}{3}$23, which we can interpret as there are twothirds as many green dots as blue dots. We can also express the green dots as a ratio out of the total amount of dots with the ratio $\frac{2}{5}$25, that is twofifths of the total number of dots are green. When expressing ratios take care if you want a direct comparison between parts or the proportion those parts make up of a whole.
A cordial mix requires the ratio of $30$30 ml of cordial to $100$100 ml of water.
(a) Express the ratio of cordial to water as a fraction in simplest form.
Think: As we want the ratio of cordial to water, the amount of cordial will be in the numerator and water will be in the denominator.
Do:
Ratio cordial to water  $=$=  $\frac{30}{100}$30100  
$=$=  $\frac{30\div10}{100\div10}$30÷10100÷10 
Divide top and bottom by a common factor of $10$10 to simplify the fraction. 

$=$=  $\frac{3}{10}$310 
This means there is $\frac{3}{10}$310 the amount of cordial compared to water.
(b) Write the ratio of cordial in a mixed drink as a fraction in simplest form.
Think: This time we want the ratio of cordial to the total liquid in a mixed drink. We have $30$30 ml of cordial and $100$100 ml of water, so the total will be $130$130 ml.
Do:
Ratio cordial to total  $=$=  $\frac{30}{130}$30130  
$=$=  $\frac{30\div10}{130\div10}$30÷10130÷10 
Divide top and bottom by a common factor of $10$10 to simplify the fraction. 

$=$=  $\frac{3}{13}$313 
Thus, in a drink made to the given ratio cordial will be $\frac{3}{13}$313 of the total drink.
The table shows the amount of several ingredients in a pack of $150$150gram biscuits.
Number of grams in one pack of biscuits  

fat  $14$14 grams 
sugar  $16$16 grams 
milk  $15$15 grams 
wheat  $18$18 grams 
State the ratio of sugar to fat as a fraction in simplest form.
State the ratio of wheat to milk as a fraction in simplest form.
Xavier and Quiana scored goals in their netball game in the ratio $8:3$8:3.
What fraction of the total number of goals was scored by Xavier?
$\frac{\editable{}}{\editable{}}$
Express $\frac{7}{35}$735 as a ratio of integers in the form $a:b$a:b
Just as we can make equivalent fractions and can simplify fractions we can also do this with ratios written in the form $a:b$a:b. To make an equivalent fraction, recall we can multiply or divide the numerator and denominator by the same value. Having seen above how we can express a ratio as a fraction, this would be the same as multiplying or dividing both sides of a ratio by the same number. Multiplication and division by the same number preserves the proportion of the values in the ratio.
Consider the following example: a cake recipe that uses $1$1 cup of milk and $4$4 cups of flour, that is, the ratio of milk to flour is $1:4$1:4. What if we want to make two cakes? We would need to double the amount of milk and flour we use. This means we will need $2$2 cups of milk and $8$8 cups of flour. Now the ratio of milk to flour is $2:8$2:8.
But do we get two different ratios from the same recipe? No, the two ratios actually represent the same proportion of milk to flour. We say that $1:4$1:4 and $2:8$2:8 are equivalent ratios.
A ratio is a simplified ratio if there is no equivalent ratio with smaller integer values. This is the same as saying that the two integers in the ratio have a highest common factor of $1$1. Just as we simplify fractions by dividing by the numerator and denominator by a common factor we can divide both sides of a ratio expression by a common factor.
A simplified ratio uses only integers. A ratio that includes fractions or decimals is not yet fully simplified and can be increased or decreased by an appropriate multiple to simplify it.
A recipe for a salad dressing includes $20$20 ml of vinegar and $60$60 ml of olive oil.
(a) Fill in the table shown below to make equivalent ratios for a larger amount of dressing.
Vinegar  to  Olive Oil 

$20$20  :  $60$60 
$30$30  :  $90$90 
$40$40  :  $\editable{}$ 
$\editable{}$  :  $300$300 
Think: What do you need to multiply one side of the ratio by to get the known value? Multiply both sides by this value to get an equivalent ratio.
Do:
Vinegar  to  Olive oil  Vinegar  to  Olive oil  
$20$20  :  $60$60  $20$20  :  $60$60  
$\times2$×2  $\times2$×2  $\times5$×5  $\times5$×5  
$40$40  :  $\editable{}$  $\editable{}$  :  $300$300 
Hence, the completed table is:
Vinegar  to  Olive Oil 

$20$20  :  $60$60 
$30$30  :  $90$90 
$40$40  :  $120$120 
$100$100  :  $300$300 
(b) What is the simplified ratio of vinegar to olive oil in the dressing?
Think: What is the highest common factor of $20$20 and $60$60? Divide both sides of the ratio by this number.
Do:Vinegar  to  Olive oil 
$20$20  :  $60$60 
$\div20$÷20  $\div20$÷20  
$1$1  :  $3$3 
The simplified ratio of vinegar to olive oil is $1:3$1:3. A simplified ratio is great for simple recipes, this one tells us we need three times as much olive oil than vinegar. So if rather than measuring in millilitres we put $1$1 tablespoon on vinegar we know we need to put $3$3 tablespoons of olive oil.
Complete the table of equivalent ratios and use it to answer the following questions.
Dogs  to  Cats 

$9$9  :  $5$5 
$18$18  :  $10$10 
$27$27  :  $\editable{}$ 
$45$45  :  $\editable{}$ 
$\editable{}$  :  $50$50 
If there are $270$270 dogs, how many cats are there expected to be?
$150$150
$30$30
$270$270
$266$266
$150$150
$30$30
$270$270
$266$266
Which of the following is the fully simplified ratio for $270:150$270:150?
$135:75$135:75
dogs$:$:cats
$2:1$2:1
$9:5$9:5
$135:75$135:75
dogs$:$:cats
$2:1$2:1
$9:5$9:5
Write $50$50cents to $\$2.10$$2.10 as a fully simplified ratio by first converting to the same units, and then simplifying.
Simplify the ratio $5.4:0.75$5.4:0.75
Simplify this ratio:
$\frac{6}{5}$65:$\frac{7}{10}$710
understand the relationship between fractions and ratio
express a ratio in simplest form