Ratios and Proportion

 

What is Direct Proportion?

A direct proportion means that as one quantity increases, the other quantity increases at the same rate.

Similarly, if one quantity decreases, the other decreases at the same rate.

The ratio between the two quantities stays constant.

Example: Direct Proportion in a Recipe

If 2 boxes of cereal contain 800g of cornflakes, then:

• 4 boxes will contain 1600g
• 6 boxes will contain 2400g

 

How to Solve Direct Proportion Problems

Identify the two related quantities.

• E.g. hours worked and pay received.

Find the scaling factor.

• If the amount is doubled, multiply by 2.
• If the amount is tripled, multiply by 3.
• If not given, calculate it as: $$\text{Factor} = \frac{\text{New Value}}{\text{Old Value}}$$​

Multiply the other quantity by this factor.

Give the final answer in context.

• Include correct units and round appropriately.

 

Example: Direct Proportion - Bonuses

An employee’s bonus is directly proportional to the company’s profit.

• The company pays a $\pounds 250$ bonus per $\pounds 3000$ profit.

 

(i) Find the bonus if the company makes $\pounds 18,000$ profit.

Step 1: Find the scaling factor:

$$\frac{18,000}{3,000} = 6$$

Step 2: Multiply the bonus by this factor:

$$250 \times 6 = 1500$$

Final Answer: The employee receives $\pounds 1500$.

 

(ii) Find the lowest possible bonus if no bonus is paid below $\pounds 600$ profit.

Step 1: Find the factor:

$$\frac{600}{3,000} = \frac{1}{5}$$

 

Step 2: Find the bonus:

$$250 \times \frac{1}{5} = 50$$

 

Final Answer: The lowest bonus is $\pounds 50$.

 

What is the Unitary Method?

The unitary method involves finding the value of one unit and then scaling up.

 

Example: Finding the Weight of 7 Boxes If 8 boxes weigh 60kg, find the weight of 7 boxes.

Find the weight of 1 box:

$$60 \div 8 = 7.5 \text{ kg}$$

 

Multiply by 7:

$$7.5 \times 7 = 52.5 \text{ kg}$$

 

Final Answer: 7 boxes weigh 52.5kg.

 

What is Inverse Proportion?

An inverse proportion means that as one quantity increases, the other quantity decreases at the same rate.

If one triples, the other is divided by 3. If one halves, the other doubles.

 

Example: Inverse Proportion in a Factory

• 2 robots take 15 hours to build a car.
• If 6 robots are used, they will take 5 hours (since 6 is 3 times more, we divide 15 by 3).

 

How to Solve Inverse Proportion Problems

Identify the two related quantities.

• E.g. number of pumps and time to fill a swimming pool.

Find the scaling factor:

$$\text{Factor} = \frac{\text{New Value}}{\text{Old Value}}$$

 

Divide the other quantity by this factor to find the required amount.

Give the final answer in context.

 

Example: Inverse Proportion - Filling a Pool

A swimming pool is filled with 3 pumps in 12 hours.

(i) Find the time needed with 9 pumps.

(ii) Find the extra time needed if only 2 pumps are available.

 

(i) Find the time needed with 9 pumps.

Step 1: Find the factor:

$$\frac{9}{3} = 3$$

Step 2: Divide the time by this factor:

$$12 \div 3 = 4 \text{ hours}$$

Final Answer: With 9 pumps, it will take 4 hours.

 

(ii) Find the extra time needed if only 2 pumps are available.

Step 1: Find the factor:

$$\frac{2}{3} = 0.666...$$

Step 2: Divide the time by this factor:

$$12 \div \left(\frac{2}{3}\right) = 12 \times \frac{3}{2} = 18 \text{ hours}$$

Final Answer: 18 hours (which is 6 extra hours).