Functional Skills: Proportion

Proportional Quantities

Example: Antonia has a recipe for a fruit drink shown below. She wants to make a big batch of this drink. Assuming the ingredients remain in proportion, calculate how much of the drink she makes if she uses $1100$ ml of lemonade.

The ingredients remaining “in proportion” means that even after increasing their amounts, their ratio stays the same.

We need to determine by what scale factor the amount of lemonade has increased:

$textcolor{purple}{1100} div 200 = textcolor{limegreen}{5.5}$

So, we can now calculate the amount of orange juice and cranberry juice used:

orange juice:       $150 times textcolor{limegreen}{5.5} = 825$ ml

cranberry juice:     $80 times textcolor{limegreen}{5.5} = 440$ ml

Therefore, the total amount the big batch of drink is:

$1100 + 825 + 440 = textcolor{black}{2365}$ ml

Note: This is an example of direct proportion.

Direct Proportion

Two quantities are directly proportional if as one increases, the other one increases at the same rate, e.g. as one is doubled, the other is doubled.

Example: Toni uses $textcolor{red}{150text{ g}}$ of chocolate to make $textcolor{blue}{6}$ cookies. How much chocolate would Toni need to make $textcolor{purple}{20}$ cookies?

Step 1: Divide the amount of chocolate by $6$ to find the amount needed for $1$ cookie.

$textcolor{blue}{150} div textcolor{red}{6} = 25$ g of chocolate

Step 2: Multiply the amount of chocolate needed for $1$ cookie by the $textcolor{purple}{20}$ cookies needed.

$textcolor{purple}{20} times 25 = 500$ g of chocolate

Inverse Proportion

Two quantities are inversely proportional if as one increases, the other one decreases at the same rate, e.g. as one is doubled, the other is halved.

Example: It takes $textcolor{red}{8}$ workers $textcolor{blue}{25}$ months to build $10$ houses. Assuming they all work at the same rate, how long would it take $textcolor{limegreen}{20}$ workers to build the same number of houses?

Step 1: Multiply the number of workers by the number of months, to find the time it would take $1$ worker to build $10$ houses:

$textcolor{red}{8} times textcolor{blue}{25} = 200$ months

Step 2: Divide the time it takes $1$ worker to build $10$ houses by $textcolor{limegreen}{20}$ workers, to get the answer:

$200 div textcolor{limegreen}{20} = textcolor{black}{10}$ months