Direct proportion

Topic summary

Direct proportion describes a relationship where two quantities increase or decrease at the same rate. If one quantity is multiplied or divided by a factor, the other quantity is multiplied or divided by the same factor. This concept is widely used in real-life scenarios such as finding the best value for money, scaling recipes, and converting currencies.

1. Understanding Direct Proportion:

In direct proportion, the relationship between two variables can be expressed as:

\[ y = kx \]

Here, \(k\) is the constant of proportionality. If you know one variable and the constant, you can calculate the other.

2. Best Buys:

To determine the best buy, compare the unit prices of items. The unit price is calculated by dividing the total cost by the quantity.

\[ \text{Unit price} = \frac{\text{Total cost}}{\text{Quantity}} \]

Example: You are comparing two brands of rice:

• Brand A: 2 kg for £3.60.
• Brand B: 3 kg for £5.10.

Calculate the unit prices:

• Brand A: \(\frac{3.60}{2} = £1.80\) per kg.
• Brand B: \(\frac{5.10}{3} = £1.70\) per kg.

Brand B offers better value as its unit price is lower.

3. Scaling Recipes:

To scale a recipe, multiply or divide the ingredients by the same factor as the change in the number of servings.

Example: A recipe for 4 servings requires 200 g of flour. You need enough for 6 servings. Use direct proportion:

\[ \text{New quantity} = \frac{\text{Required servings}}{\text{Original servings}} \times \text{Original quantity} \]

Substitute the values:

\[ \text{New quantity} = \frac{6}{4} \times 200 = 1.5 \times 200 = 300 \, \text{g} \]

You will need 300 g of flour for 6 servings.

4. Exchange Rates:

Direct proportion is used to convert between currencies. Multiply or divide by the exchange rate to find the equivalent amount.

Example: The exchange rate is £1 = €1.15. How many euros can you get for £50?

\[ \text{Euros} = \text{Pounds} \times \text{Exchange rate} \]

Substitute the values:

\[ \text{Euros} = 50 \times 1.15 = 57.50 \]

You can get €57.50 for £50.

Reverse conversion: To convert €80 back to pounds, divide by the exchange rate:

\[ \text{Pounds} = \frac{\text{Euros}}{\text{Exchange rate}} = \frac{80}{1.15} = 69.57 \, \text{(rounded to 2 decimal places)} \]

5. Summary:

• Direct proportion involves two quantities that increase or decrease at the same rate.
• For best buys, calculate unit prices to determine which item offers better value.
• For recipes, scale ingredient quantities by the ratio of servings required to the original servings.
• For exchange rates, multiply or divide by the exchange rate to convert between currencies.

Direct proportion is a versatile concept used to solve real-world problems efficiently.