International Mathematics (0607) | Rates

Your IGCSE International Mathematics (0607) Study Notes: Chapter on Rates

Hello everyone! Welcome to the exciting world of Rates. This chapter is essential because rates connect the pure mathematics of numbers directly to the real world—from calculating how much you earn to planning a long journey. Don't worry if measuring speed or comparing currencies seems complex; we'll break it down into simple, manageable pieces. By the end of this, you’ll be a pro at handling measurements that involve "per"!

1. Understanding the Concept of Rate

What is a Rate?

A rate is a way of comparing two quantities that are measured in different units.
Unlike a ratio, which compares two quantities with the same unit (like 3 apples to 5 apples), a rate must involve different types of measurements, such as distance and time, or money and weight.

The key word in rates is "per," which mathematically means "divided by" (\(\div\)).

• Example: Speed is measured in kilometres per hour (km/h) or metres per second (m/s).
• Example: Cost of fuel is often dollars per litre (\$/L).

Rate Notation (The Units)

When writing a rate, the units are usually written as a fraction or using the "/" or "per" notation.

Important Notation from the Syllabus:

• Metres per second: \(\text{m/s}\)
• Grams per cubic centimetre (density): \(\text{g/cm}^3\)
• Kilometres per hour: \(\text{km/h}\)

Quick Review: A rate always links two measurements with different units using division.

2. Common Measures of Rate (The "Per" Club)

Rates show up everywhere in practical calculations. The syllabus requires you to be comfortable calculating and using several specific measures of rate.

A. Hourly Rates of Pay (Money per Time)

This is how we calculate earnings.

Formula:

$$ \text{Total Pay} = \text{Hourly Rate} \times \text{Hours Worked} $$

Example: If you earn \(\$15\) per hour, and you work for 6 hours, your total pay is \(15 \times 6 = \$90\).

B. Exchange Rates (Currency per Currency)

Exchange rates allow you to convert money from one currency to another when travelling or shopping internationally.

Did you know? An exchange rate tells you the value of 1 unit of one currency in terms of another. For example, if the rate is 1 USD = 0.85 EUR, it means 1 US Dollar is worth 0.85 Euros.

Rule of Thumb:

• To convert into the currency shown as "1 unit" in the rate, you usually multiply.
• To convert out of the currency shown as "1 unit", you usually divide.

Example: The exchange rate is 1 GBP = 1.30 USD. How many USD do you get for 500 GBP?
Calculation: \(500 \times 1.30 = 650 \text{ USD}\).

C. Flow Rates (Volume per Time)

Flow rates measure how much liquid (volume) passes a point over a certain time.

Formula:

$$ \text{Volume Flow Rate} = \frac{\text{Volume}}{\text{Time}} $$

Example: A tap fills a swimming pool at a rate of 50 litres per minute (\(50 \text{ L/min}\)). How long will it take to fill a 1000 litre tank?
Calculation: \(\text{Time} = \frac{\text{Volume}}{\text{Rate}} = \frac{1000 \text{ L}}{50 \text{ L/min}} = 20 \text{ minutes}\).

D. Fuel Consumption (Distance per Volume)

This measures how efficiently a vehicle uses fuel, often given in kilometres per litre (\(\text{km/L}\)) or sometimes litres per 100 kilometres (\(\text{L/100km}\)).

Example: If a car achieves \(15 \text{ km/L}\), how much fuel is needed for a 300 km journey?
Calculation: \(\text{Volume needed} = \frac{\text{Total Distance}}{\text{Consumption Rate}} = \frac{300 \text{ km}}{15 \text{ km/L}} = 20 \text{ litres}\).

Key Takeaway for Section 2: All practical rates follow the same fundamental structure: \(\text{Rate} = \frac{\text{Quantity 1}}{\text{Quantity 2}}\).

3. The Most Important Rate: Average Speed

Speed is the rate of change of distance over time. This is a core topic, and you must know the relationship between speed, distance, and time.

The Speed, Distance, Time Triangle (The Memory Aid!)

The easiest way to remember the three formulas is using the triangle method. Cover the variable you want to find, and the remaining variables show you the formula.

(Imagine the triangle with D at the top, and S and T sharing the bottom row, separated by a line that means division, and S and T separated by a multiplication line).

• Distance (\(D\)): \(D = S \times T\)
• Speed (\(S\)): \(S = \frac{D}{T}\)
• Time (\(T\)): \(T = \frac{D}{S}\)

Calculating Average Speed

When solving problems involving journeys, remember that Average Speed is calculated using the total distance covered and the total time taken.

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

It is NOT simply the average of different speeds!

Step-by-Step Example (Syllabus Example)

Question: A cyclist travels \(45 \text{ km}\) in \(3\) hours and \(45\) minutes. What is their average speed in \(\text{km/h}\)?

Step 1: Convert Time to consistent units.
The distance is in km, and the required speed is in km/hour, so we must convert the minutes into a decimal fraction of an hour.

• 45 minutes = \(\frac{45}{60}\) hours = 0.75 hours.
• Total Time (\(T\)) = 3 hours + 0.75 hours = 3.75 hours.

Step 2: Apply the Average Speed formula.
Total Distance (\(D\)) = \(45 \text{ km}\).

$$ \text{Average Speed} = \frac{45 \text{ km}}{3.75 \text{ h}} $$
Calculation: \(45 \div 3.75 = 12\)

Answer: The average speed is \(12 \text{ km/h}\).

Common Mistake Alert!

Do NOT calculate time like this: \(3 \text{ hours } 45 \text{ minutes} = 3.45\) hours. This is incorrect! Time is based on 60 minutes, not 100. Always convert minutes to hours by dividing by 60.

4. Working with Different Units (Conversions)

Rate problems often require you to convert units before or during the calculation. This is crucial for matching the required notation (e.g., finding speed in m/s when given km and hours).

The Chain Conversion Method

To convert rates, we multiply by fractions that are equal to 1, ensuring the units we want to cancel appear on the opposite side of the fraction bar.

• \(\text{1 km} = 1000 \text{ m}\)
• \(\text{1 hour} = 60 \text{ minutes} = 3600 \text{ seconds}\)

Step-by-Step Conversion Example: Converting Speed

Question: Convert \(72 \text{ km/h}\) into \(\text{m/s}\).

Step 1: Convert Distance (km to m).

$$ 72 \frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} $$

(Notice km is now cancelled out, leaving us with \(72 \times 1000 = 72000 \text{ m/h}\)).

Step 2: Convert Time (h to s).

$$ 72000 \frac{\text{m}}{\text{h}} \times \frac{1 \text{ h}}{3600 \text{ s}} $$

(Notice h is now cancelled out, leaving us with \(\frac{72000}{3600}\) in units of m/s).

Final Calculation: \(72000 \div 3600 = 20\)
Answer: \(20 \text{ m/s}\).

Tip for Speed: To go from \(\text{km/h}\) to \(\text{m/s}\), you always divide by 3.6. To go from \(\text{m/s}\) to \(\text{km/h}\), you multiply by 3.6.

E. Density and Concentration (Mass/Volume)

Although not explicitly called "density" in the core syllabus (C1.11), the required notation \(\text{g/cm}^3\) deals with mass per volume, which is density.

Formula:

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

Example: A block of metal has a mass of \(150 \text{ g}\) and a volume of \(50 \text{ cm}^3\).
Density: \(\frac{150 \text{ g}}{50 \text{ cm}^3} = 3 \text{ g/cm}^3\).

5. Final Review and Problem-Solving Strategy

When tackling any rate problem in the exam, follow these steps to ensure you get the units correct:

1. Identify the Rate: What rate are you asked to find or use? (Speed, pay, flow, density?)
2. Check Units: What units are you given, and what units are required in the final answer?
3. Convert Time/Measurements: If necessary, perform all unit conversions (especially time—minutes to hours!) before calculating the final rate.
4. Apply Formula: Use the appropriate formula (\(S = D/T\), \(\text{Rate} = Q1/Q2\)).

Remember, rates are just ratios where the two items being compared have different labels. Keep track of your units, and you will master this topic! Good luck!