Proportion

Example 1: Direct Proportion

Sarah is baking some brownies. Here are the ingredients needed to make 6 brownies:

Sarah wants to make 10 brownies. Work out how what quantity of each ingredient Sarah will need.

[3 marks]

We can use what we learnt about direct proportion to work this out:

Overall method: divide what you know to find the quantity for one unit, and then multiply by the amount you need to find.

Let’s start with butter, for one brownie, Sarah would need:

180\text{ g}\div 6=30\text{ g}

So, for 10 brownies:

30\text{ g}\times 10=300\text{ g}

Now the same method for chocolate:

210\text{ g}\div 6=35\text{ g}

35\text{ g}\times 10=350\text{ g}

Flour:

90\text{ g}\div 6=15\text{ g}

15\text{ g}\times 10=150\text{ g}

Eggs:

3\div 6 = 0.5

0.5\times 10 = 5 eggs

Sugar:

270\text{ g}\div 6 = 45\text{ g}

45\text{ g}\times 10=450\text{ g}

 

So, overall:

300\text{ g} butter

350\text{ g} chocolate

150\text{ g} flour

5 eggs

450\text{ g} sugar

Example 2: Inverse Proportion

Noah and Mike are trying to build a fence, but they have worked out it would take them 8\text{ hours} if they built it on their own, and they want to get it built within 2\text{ hours}.

a) Work out how many people will have to help Noah and Mike to build their fence in 2\text{ hours}.

b) What assumption are you making when answering this question?

[3 marks]

a) We can use the method we learnt about for inverse proportion:

Overall method: multiply what you know to find the quantity for one unit, and then divide by the amount you need to find.

 

So, if Noah or Mike were building the fence on their own, it would take:

8\text{ hours}\times 2 = 16\text{ hours}

 

The next stage is to divide this by the number of people helping now, but we do not have this information. Instead, we know we want it to take 2 hours or less. So,

2\text{ hours} = 16\text{ hours}\div\text{number of people}

Which can be rearranged,

\text{Number of people} =16\text{ hours}\div 2\text{ hours}

\text{Number of people}= 8

 

So, Noah and Mike will need 6 people to help them (to make the total number of people 8).

 

b) We are assuming that all of the people build the fence at the same rate.

Question 3: Direct Proportion with Algebra

y is directly proportional to the square of x. 

When y=27, x=3.

Work out the value of y when x=11

[3 marks]

For this question, we will use the constant of proportionality, k, that we saw earlier.

y is directly proportional to the square of x can be written as,

y\propto x^2

Or,

y=kx^2

We need to first work out what the value of k is in this scenario, using what we know:

When y=12, x=3

So,

27=k(3^2)

27=9k

k=3

Now, we want to work out the value of y when x=11,

y=3(11^2)

y=363

Example 4: Inverse Proportion with Algebra

y is inversely proportional to the square root of x.

When y=1, x=4.

Work out the value of y when x=9

[3 marks]

For this question, we will use the constant of proportionality, k, that we saw earlier.

y is inversely proportional to the square of x can be written as,

y\propto \dfrac{1}{\sqrt{x}}

Or,

y=\dfrac{k}{\sqrt{x}}

We need to first work out what the value of k is in this scenario, using what we know:

When y=1, x=4

So,

1=\dfrac{k}{\sqrt{4}}

2=k

Now, we want to work out the value of y when x=9,

y=\dfrac{2}{\sqrt{9}}

y=\dfrac{2}{3}