Number & Algebra

Percentage Change and Rates

Year 8

✓By the end of this lesson students will be able to calculate percentage increase and decrease.
✓By the end of this lesson students will be able to solve problems involving percentage change in real-world contexts.
✓By the end of this lesson students will be able to define and identify different types of rates.
✓By the end of this lesson students will be able to calculate and compare unit rates.
✓By the end of this lesson students will be able to apply rates to solve practical problems.

Key concepts

Percentage Change

Percentage change describes how much a quantity has changed relative to its original value. It can be an increase or a decrease. It is always calculated using the original value as the base.

Percentage Change = (Change / Original Value) × 100%

Percentage Increase

When a quantity goes up, we calculate the percentage increase. The 'increase' is the difference between the new value and the original value.

Percentage Increase = (Increase / Original Value) × 100%

Percentage Decrease

When a quantity goes down, we calculate the percentage decrease. The 'decrease' is the difference between the original value and the new value.

Percentage Decrease = (Decrease / Original Value) × 100%

Rates

A rate is a comparison of two quantities with different units. It tells us how much of one quantity there is for a certain amount of another quantity. Common examples include speed (km/h), cost per item ($/item), or flow rate (L/min).

Unit Rates

A unit rate is a rate where the second quantity is 1 unit. This makes it easy to compare different rates. To find a unit rate, divide the first quantity by the second quantity so that the denominator becomes 1.

Unit Rate = Quantity 1 / Quantity 2 (where Quantity 2 becomes 1 unit)

Key facts to remember

1Percentage change is always calculated relative to the original value.
2Percentage Increase = (Increase / Original Value) × 100%.
3Percentage Decrease = (Decrease / Original Value) × 100%.
4A rate compares two quantities with different units (e.g., km/h, $/kg, L/min).
5A unit rate has a denominator of 1 (e.g., 60 km/1 hour, $2.50/1 kg).
6To find a unit rate, divide the first quantity by the second quantity.
7Always include the correct units when stating rates and unit rates.

Worked examples

Example 1

A shirt originally cost $40. During a sale, its price increased to$ 50. Calculate the percentage increase in price.

IIdentify the original value: Original Value = $40

IIIdentify the new value: New Value = $50

IIICalculate the increase in price: Increase = New Value - Original Value =

50 -

40 = $10

IVApply the percentage increase formula: Percentage Increase = (Increase / Original Value) × 100%

VSubstitute the values: Percentage Increase = (

10 /

40) × 100%

VICalculate the result: Percentage Increase = 0.25 × 100% = 25%

Answer

25%

Always use the original value as the denominator when calculating percentage change.

Example 2

A car was bought for $25 000 and sold a year later for$ 20 000. Calculate the percentage decrease in its value.

IIdentify the original value: Original Value = $25 000

IIIdentify the new value: New Value = $20 000

IIICalculate the decrease in value: Decrease = Original Value - New Value =

25 000 -

20 000 = $5 000

IVApply the percentage decrease formula: Percentage Decrease = (Decrease / Original Value) × 100%

VSubstitute the values: Percentage Decrease = (

5 000 /

25 000) × 100%

VICalculate the result: Percentage Decrease = 0.2 × 100% = 20%

Answer

20%

The original value is the starting price or amount before the change occurred.

Example 3

Shop A sells 3 kg of apples for $7.50. Shop B sells 2 kg of apples for$ 5.20. Which shop offers the better value per kilogram?

ICalculate the unit rate for Shop A:

IICost per kg (Shop A) = Total Cost / Quantity =

7.50 / 3 kg =

2.50/kg

IIICalculate the unit rate for Shop B:

IVCost per kg (Shop B) = Total Cost / Quantity =

5.20 / 2 kg =

2.60/kg

VCompare the unit rates:

2.50/kg is less than

2.60/kg.

Answer

Shop A offers the better value at $2.50 per kilogram.

When comparing 'better value' for price, you are looking for the lower unit price.

Common mistakes

✗Using the new value instead of the original value as the denominator when calculating percentage change.
✗Confusing percentage increase with percentage decrease, leading to incorrect calculations for the 'change' amount.
✗Forgetting to multiply by 100 to express the change as a percentage (leaving it as a decimal).
✗Not including the correct units when stating a rate or unit rate (e.g., just writing '2.50' instead of '$2.50/kg').
✗Incorrectly simplifying rates or unit rates, such as dividing in the wrong order.

Exam tips

★Always identify the 'original value' clearly before starting any percentage change calculation.
★Show all steps in your calculations, especially for percentage change problems, as this can earn partial marks.
★Pay close attention to the units given in rate problems and ensure your final answer includes the correct units.
★When comparing different rates, convert them to unit rates first to make the comparison straightforward.
★Read the question carefully to determine if you need to find a percentage increase, a percentage decrease, or a unit rate.

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