Rate and Proportion Problems

Rate and proportion problems are a staple of QR. They include speed/distance/time, cost per unit, density, flow rates, and any situation where one quantity depends on another. This lesson covers the core methods and the specific rate problems that appear most often.

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What Is a Rate?

A rate is a ratio that compares two different units. Common rates in QR:

Rate	Formula	Units
Speed	Distance ÷ Time	km/h, mph, m/s
Price per unit	Total Cost ÷ Quantity	£/kg, £/item, p/litre
Density	Mass ÷ Volume	g/cm³, kg/m³
Flow rate	Volume ÷ Time	litres/min, ml/s
Fuel consumption	Distance ÷ Fuel Used	km/litre, miles/gallon
Wage rate	Pay ÷ Hours	£/hour

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Speed, Distance, and Time

The Formula Triangle

To Find	Formula
Distance	Speed × Time
Speed	Distance ÷ Time
Time	Distance ÷ Speed

Worked Example: Simple Journey

Question: A car travels at 60 mph for 2 hours 30 minutes. How far does it travel?

• Time = 2.5 hours (not 2.3!)
• Distance = 60 × 2.5 = 150 miles

Worked Example: Finding Time

Question: A train covers 240 km at an average speed of 96 km/h. How long does the journey take?

• Time = 240 ÷ 96 = 2.5 hours = 2 hours 30 minutes

Worked Example: Average Speed for Multi-Leg Journey

Question: A cyclist rides 30 km in 1 hour, then 20 km in 1.5 hours. What is the average speed for the whole journey?

• Total distance = 30 + 20 = 50 km
• Total time = 1 + 1.5 = 2.5 hours
• Average speed = 50 ÷ 2.5 = 20 km/h

Common Trap: Average speed is not the average of the two speeds. The first leg's speed was 30 km/h and the second was 13.3 km/h, but the average speed is 20 km/h, not (30 + 13.3)/2 = 21.7 km/h. Always use total distance ÷ total time.

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Cost Per Unit (Unit Price)

Finding the Best Value

Data:

Pack	Quantity	Price
Small	6	£2.40
Medium	10	£3.50
Large	15	£4.80

Question: Which pack offers the lowest price per item?

• Small: £2.40 ÷ 6 = £0.40 per item
• Medium: £3.50 ÷ 10 = £0.35 per item
• Large: £4.80 ÷ 15 = £0.32 per item

Answer: Large pack (£0.32 per item)

Total Cost from Unit Price

Question: Oranges cost £1.80 per kg. How much do 3.5 kg of oranges cost?

• 3.5 × £1.80 = 3 × 1.80 + 0.5 × 1.80 = 5.40 + 0.90 = £6.30

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Density Problems

Formula: Density = Mass ÷ Volume

To Find	Formula
Density	Mass ÷ Volume
Mass	Density × Volume
Volume	Mass ÷ Density

Worked Example

Question: A block of metal has a mass of 540 g and a volume of 200 cm³. What is its density?

• Density = 540 ÷ 200 = 2.7 g/cm³

Follow-up: What would be the mass of a 350 cm³ block of the same metal?

• Mass = 2.7 × 350 = 945 g

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Flow Rate Problems

Formula: Flow Rate = Volume ÷ Time

Worked Example

Question: A tap fills a 60-litre tank in 4 minutes. What is the flow rate?

• Flow rate = 60 ÷ 4 = 15 litres per minute

Follow-up: How long would it take to fill a 225-litre tank at the same rate?