Rates

1. Overview

Rates are fundamental in IGCSE Mathematics, describing how one quantity changes relative to another. This often involves different units, such as speed (distance per time) or price per item. You'll learn to solve real-world problems involving travel, wages, currency, consumption, and more. The key is understanding the relationship between quantities and their units, and applying the correct formulas.

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Key Definitions

• Rate: A comparison of two quantities with different units (e.g., $km/h$, $$ /kg$).
• Average Speed: The total distance traveled divided by the total time taken for a journey.
• Unit Rate: A rate expressed as a quantity of one (e.g., price per 1 litre).
• Exchange Rate: The ratio at which one currency can be exchanged for another.
• Constant Rate: A rate that does not change over time.

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Core Content

3.1 Common Measures of Rate

Common rates include speed, hourly wages, and price per unit.

Worked example 1 — Calculating Hourly Wage

A worker earns $$450$ for working $36$ hours. Calculate the hourly rate.

Step-by-Step Working:

1. Identify the formula: $\text{Rate} = \frac{\text{Total Amount}}{\text{Total Units}}$
2. Substitute the values: $\text{Hourly Rate} = \frac{450}{36}$
3. Calculate: $450 \div 36 = 12.5$
4. State with units: $\boxed{$12.50 \text{ per hour}}$

Worked example 2 — Calculating Price Per Kilogram

A 5kg bag of rice costs $$8.50$. What is the price per kilogram?

Step-by-Step Working:

1. Identify the formula: $\text{Price per kg} = \frac{\text{Total Cost}}{\text{Total Weight}}$
2. Substitute the values: $\text{Price per kg} = \frac{8.50}{5}$
3. Calculate: $8.50 \div 5 = 1.70$
4. State with units: $\boxed{$1.70 \text{ per kg}}$

3.2 Applying Other Measures of Rate

This involves fuel consumption ($L/100km$), flow rates ($L/min$), or density ($g/cm^3$).

Worked example 3 — Fuel Consumption

A car uses $48$ litres of petrol to travel $600\text{ km}$. Calculate the rate of fuel consumption in litres per $100\text{ km}$.

Step-by-Step Working:

1. Find the fuel used per $1\text{ km}$: $48 \div 600 = 0.08\text{ L/km}$
2. Multiply by $100$ to find the rate per $100\text{ km}$: $0.08 \times 100 = 8$
3. Final Answer: $\boxed{8\text{ L/100km}}$

Worked example 4 — Calculating Flow Rate

A tap fills a 9-litre bucket in 2 minutes. Calculate the flow rate in litres per minute.

Step-by-Step Working:

1. Identify the formula: $\text{Flow Rate} = \frac{\text{Volume}}{\text{Time}}$
2. Substitute the values: $\text{Flow Rate} = \frac{9}{2}$
3. Calculate: $9 \div 2 = 4.5$
4. State with units: $\boxed{4.5 \text{ litres per minute}}$

3.3 Average Speed

Average speed is not the average of the speeds. It is always the total distance over the total time.

Worked example 5 — Basic Average Speed

A cyclist travels $30\text{ km}$ in $1\text{ hour } 30\text{ minutes}$. Calculate the average speed in $km/h$.

Step-by-Step Working:

1. Convert time to hours: $30\text{ minutes} = 0.5\text{ hours}$, so total time = $1.5\text{ hours}$.
2. Use the formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
3. Substitute: $S = \frac{30}{1.5}$
4. Calculate: $30 \div 1.5 = 20$
5. Final Answer: $\boxed{20\text{ km/h}}$

Worked example 6 — Average Speed with Mixed Units

A train travels 120 km in 2 hours and 30 minutes. Calculate the average speed in km/h.

Step-by-Step Working:

1. Convert minutes to hours: $30 \text{ minutes} = \frac{30}{60} = 0.5 \text{ hours}$
2. Calculate total time in hours: $2 \text{ hours} + 0.5 \text{ hours} = 2.5 \text{ hours}$
3. Apply the average speed formula: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$
4. Substitute the values: $\text{Average Speed} = \frac{120}{2.5}$
5. Calculate: $120 \div 2.5 = 48$
6. State the final answer with units: $\boxed{48 \text{ km/h}}$

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Extended Content (Extended Only)

4.1 Advanced Average Speed

Extended questions often involve a journey with two or more parts with different speeds. The key is to calculate the total distance and the total time, then divide.

Worked example 7 — Multi-part Journey

A car travels for $2\text{ hours}$ at $60\text{ km/h}$ and then for $3\text{ hours}$ at $80\text{ km/h}$. Calculate the average speed for the whole journey.

Step-by-Step Working:

1. Calculate Distance 1: $D_1 = S_1 \times T_1 = 60 \times 2 = 120\text{ km}$
2. Calculate Distance 2: $D_2 = S_2 \times T_2 = 80 \times 3 = 240\text{ km}$
3. Calculate Total Distance: $120 + 240 = 360\text{ km}$
4. Calculate Total Time: $2 + 3 = 5\text{ hours}$
5. Use Average Speed formula: $\frac{360}{5} = 72$
6. Final Answer: $\boxed{72\text{ km/h}}$

Worked example 8 — Journey with Different Speeds and Distances

A train travels 150 km at a speed of 50 km/h, then travels 180 km at a speed of 90 km/h. Calculate the average speed for the entire journey.

Step-by-Step Working:

1. Calculate the time for the first part of the journey: $T_1 = \frac{D_1}{S_1} = \frac{150}{50} = 3 \text{ hours}$
2. Calculate the time for the second part of the journey: $T_2 = \frac{D_2}{S_2} = \frac{180}{90} = 2 \text{ hours}$
3. Calculate the total distance: $D_{\text{total}} = D_1 + D_2 = 150 + 180 = 330 \text{ km}$
4. Calculate the total time: $T_{\text{total}} = T_1 + T_2 = 3 + 2 = 5 \text{ hours}$
5. Calculate the average speed: $\text{Average Speed} = \frac{D_{\text{total}}}{T_{\text{total}}} = \frac{330}{5} = 66$
6. State the final answer with units: $\boxed{66 \text{ km/h}}$

4.2 Converting Compound Units

Converting units like $km/h$ to $m/s$ requires two steps. It's vital to show your working clearly, especially in non-calculator papers.

Worked example 9 — Convert $90\text{ km/h}$ to $m/s$

1. Convert $km$ to $m$: $90 \text{ km} \times 1000 \frac{\text{m}}{\text{km}} = 90,000\text{ m}$
2. Convert hours to seconds: $1\text{ hour} \times 60 \frac{\text{min}}{\text{hour}} \times 60 \frac{\text{sec}}{\text{min}} = 3600\text{ seconds}$
3. Calculate the new rate: $\frac{90,000\text{ m}}{3600\text{ s}}$
4. Simplify: $900 \div 36 = 25$
5. Final Answer: $\boxed{25\text{ m/s}}$

Worked example 10 — Converting Fuel Consumption Units

A car has a fuel consumption of 7 litres per 100 km. Convert this to millilitres per meter (mL/m).

Step-by-Step Working:

1. Convert litres to millilitres: $7 \text{ litres} \times 1000 \frac{\text{mL}}{\text{litre}} = 7000 \text{ mL}$
2. Convert kilometers to meters: $100 \text{ km} \times 1000 \frac{\text{m}}{\text{km}} = 100,000 \text{ m}$
3. Calculate the new rate: $\frac{7000 \text{ mL}}{100,000 \text{ m}}$
4. Simplify: $7000 \div 100,000 = 0.07$
5. Final Answer: $\boxed{0.07 \text{ mL/m}}$

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Key Equations

• Average Speed: $\text{Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

  • $S$ in $km/h$ or $m/s$; $D$ in $km$ or $m$; $T$ in $h$ or $s$.
  • Not on formula sheet - MEMORISE

• Density: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

  • $\rho$ in $g/cm^3$ or $kg/m^3$.
  • Not on formula sheet - MEMORISE

• Unit Price: $\text{Price} = \frac{\text{Total Cost}}{\text{Quantity}}$

  • Not on formula sheet - MEMORISE

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Common Mistakes to Avoid

• ❌ Wrong: Calculating the average speed of two parts of a journey by adding the two speeds and dividing by 2. For example, travelling 50 km/h then 70 km/h, and saying the average speed is 60 km/h.

  • ✓ Right: Always divide Total Distance by Total Time.

• ❌ Wrong: Rounding exchange rates or intermediate values to 2 decimal places in the middle of a calculation, leading to inaccurate final answers, especially in multi-step currency conversions.

  • ✓ Right: Keep the full number on your calculator until the very final step to avoid "rounding errors."

• ❌ Wrong: Assuming $2.4\text{ hours}$ is $2\text{ hours and } 40\text{ minutes}$.

  • ✓ Right: $2.4\text{ hours} = 2\text{ hours and } (0.4 \times 60) = 2\text{ hours and } 24\text{ minutes}$.

• ❌ Wrong: Converting $mm$ to $m$ by dividing by $100$.

  • ✓ Right: Divide by $1000$ ($1000\text{ mm in a metre}$).

• ❌ Wrong: Forgetting to convert units before calculating. For example, using km for distance and seconds for time when the question requires m/s.

  • ✓ Right: Convert all quantities to the same units before applying any formulas. Highlight the required units in the question.

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Exam Tips

• Units: Always check the units requested in the question. If the question gives distance in $m$ and time in $s$, but asks for $km/h$, convert at the start or the end. Use the "fraction chain" method to convert compound units reliably.
• Calculator vs Non-Calculator: In non-calculator papers, rates are usually chosen to cancel out (e.g., multiples of $12$ or $60$). In calculator papers, do not round until the end.
• Command Words:
  • "Calculate": Show every step of your arithmetic.
  • "Show that": You are given the answer; you must show the full method to reach it.
• Contexts: Expect questions on "Best Buy" (comparing price per $100g$), currency conversion for holidays, and travel timetables.
• Time Conversions: Remember the magic number 3600 (the number of seconds in one hour). To convert $km/h$ to $m/s$ quickly, you are effectively dividing by $3.6$. Conversely, to convert from m/s to km/h, multiply by 3.6.