Mathematics (0580) | Rates

Hello IGCSE Mathematicians! Ready to Master Rates?

Welcome to the "Rates" chapter, a crucial part of the Number section! Don't worry if this sounds complicated; rates are simply how we compare two quantities that have different units. Think of it as "how much of A happens for every B."

Mastering rates isn't just for exams—it’s practical math you use every day, whether you're calculating your holiday spending money (exchange rates), planning a road trip (average speed), or figuring out your earnings (hourly pay).

Let's break down this topic into simple, powerful steps!

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Section 1: Understanding What a Rate Is

1.1 Definition and Structure

A Rate measures how one quantity changes in relation to another quantity. Crucially, the units involved are not the same (unlike ratio, which compares quantities with the same units).

• Example: If you travel 100 kilometres in 2 hours, your rate (speed) is 50 kilometres per hour.

The word "per" is the key indicator of a rate, and mathematically, it usually means division.

The general formula for finding a rate is:

Rate \( = \frac{\text{Quantity 1}}{\text{Quantity 2}} \)

Think of the units: If Quantity 1 is Distance (km) and Quantity 2 is Time (h), the resulting rate unit is \(\frac{\text{km}}{\text{h}}\) (kilometres per hour).

Key Takeaway: A rate is a measure of change, defined by division, and its units compare two different types of measurement.

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Section 2: Common Measures of Rate (C1.11, E1.11)

The syllabus requires you to be comfortable calculating and using several specific real-world rates:

2.1 Hourly Rates of Pay

This is the rate at which someone earns money per unit of time (usually per hour).

• Unit: Dollars per hour (\$/h) or local currency per hour.
• Calculation: Total Pay \( = \) Hourly Rate \(\times\) Hours Worked

Example: If you earn $12.50 per hour and work for 8 hours:

Total Pay \( = \$12.50 \times 8 = \$100\)

2.2 Exchange Rates Between Currencies

An exchange rate is the value of one currency in terms of another. This is used when travelling or shopping internationally.

• Unit: Currency A per Currency B (e.g., 1 USD = 0.85 EUR).

Step-by-Step Conversion Trick:

1. Look at the rate: e.g., 1 GBP = 1.30 USD.
2. To convert GBP to USD: Multiply by the rate (1.30). (Since 1.30 > 1, the number of dollars is higher than the number of pounds).
3. To convert USD to GBP (the reverse): Divide by the rate (1.30). (Since we want fewer pounds for the same value, we divide).

Common Mistake to Avoid: Always check which direction you are converting. If you end up with a huge number, you probably multiplied when you should have divided!

2.3 Flow Rates

Flow rate measures the volume of a liquid (or sometimes gas) that passes a point over a period of time.

• Unit: Volume per time (e.g., litres/minute, cm³/second, m³/h).
• Calculation: Total Volume \( = \) Flow Rate \(\times\) Time

Example: A tap fills a container at a rate of 5 litres per minute. How long does it take to fill a 30-litre container?

Time \( = \frac{\text{Total Volume}}{\text{Flow Rate}} = \frac{30 \text{ litres}}{5 \text{ litres/min}} = 6 \text{ minutes}\)

2.4 Fuel Consumption

This rate tells you how efficient a vehicle is. It relates the distance travelled to the fuel used.

• Units:
  • Distance per volume (e.g., kilometres per litre, km/L). Higher number is better.
  • Volume per distance (e.g., litres per 100 kilometres, L/100 km). Lower number is better.

Did you know? Different countries prefer different units! In the UK/US, miles per gallon (mpg) is common, but in IGCSE and much of Europe, km/L or L/100km are used.

Key Takeaway: All these practical rates follow the same fundamental structure: Rate = Quantity 1 / Quantity 2. Ensure your units match the context.

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Section 3: Average Speed, Distance, and Time

This is the most frequent type of rate calculation in IGCSE Mathematics. Speed is simply the rate of distance travelled over time taken.

3.1 The Fundamental Formula

You must know the relationship between speed, distance, and time:

Average Speed \( = \frac{\text{Total Distance Travelled}}{\text{Total Time Taken}} \)

We can rearrange this formula to find any of the three variables. An easy way to remember this is the "DST Triangle" (or "Distance, Speed, Time" Triangle):

The DST Triangle Memory Aid

Imagine a triangle with D (Distance) at the top, and S (Speed) and T (Time) at the bottom:

If you cover the variable you want to find, the remaining letters show you the formula:

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

3.2 The Importance of Consistent Units

When calculating speed, the units of distance and time must be compatible to give you the correct speed unit.

If distance is in kilometres (km) and time is in hours (h), the speed is in km/h.

If distance is in metres (m) and time is in seconds (s), the speed is in m/s.

You often need to convert time, especially when dealing with minutes and hours.

Step-by-Step Example: Time Conversion

Problem: A cyclist travels 45 km in 3 hours 45 minutes. What is their average speed?

Step 1: Convert time into a single unit (hours or minutes).
The distance is in km, so let's aim for km/h. We must convert 45 minutes into hours.

• Since there are 60 minutes in an hour:
• \( 45 \text{ minutes} = \frac{45}{60} \text{ hours} = 0.75 \text{ hours} \) (or \(\frac{3}{4}\) hours).
• Total Time \( T = 3 \text{ hours} + 0.75 \text{ hours} = 3.75 \text{ hours} \)

Step 2: Apply the speed formula.

• Distance \( D = 45 \text{ km} \)
• Speed \( = \frac{D}{T} = \frac{45}{3.75} \)

Step 3: Calculate the answer and include units.

• Speed \( = 12 \text{ km/h} \)

Key Takeaway: Before applying the Speed, Distance, Time formulas, always ensure your time and distance units are consistent!

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Section 4: Understanding Rate Notation and Units

In mathematics and science, rates are often written using special notation. You must be familiar with this for the exam.

4.1 Using the Slash Notation (per)

The slash (/) represents the word "per" or "divided by" and is the standard way to write rate units.

• Metres per second: m/s
• Kilograms per square metre: kg/m²
• Grams per cubic centimetre: g/cm³ (This particular notation relates mass and volume, which is the rate known as density.)

4.2 Unit Conversion within Rate Problems

Sometimes you need to convert a rate from one unit to another (e.g., km/h to m/s).

Converting km/h to m/s (Hours to Seconds)

To convert distance (km) and time (h) units simultaneously, remember the conversion factors:

• 1 km = 1000 m
• 1 hour = 60 minutes = \( 60 \times 60 = 3600 \) seconds

Step-by-Step Conversion: Convert 72 km/h to m/s.

\[ 72 \text{ km/h} = \frac{72 \text{ km}}{1 \text{ h}} \]

1. Convert km to m:

\[ \frac{72 \times 1000 \text{ m}}{1 \text{ h}} = \frac{72000 \text{ m}}{1 \text{ h}} \]

2. Convert h to s:

\[ \frac{72000 \text{ m}}{3600 \text{ s}} \]

3. Simplify:

\[ 72000 \div 3600 = 20 \text{ m/s} \]

Key Takeaway: Treat the units separately. Convert the numerator unit, convert the denominator unit, and then divide.

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Section 5: Summary and Study Tips

5.1 Common Mistakes to Avoid

Struggling students often lose marks on rates problems due to simple errors related to units and time.

1. Mixing Time Units: Never mix hours and minutes in the same equation! You must convert all time to hours (decimals are safest) or all time to minutes.
   Example: 2 hours 30 minutes is 2.5 hours (NOT 2.3 hours).
2. Incorrect Exchange Rate Application: If the currency is getting "stronger" (the converted value is higher), you should multiply. If it's getting "weaker," you should divide. Use common sense to check your result!
3. Forgetting Units: In any rate calculation, the final answer MUST include the correct units (e.g., km/h, g/cm³, \$/h).

5.2 Final Key Takeaways

• Rates compare two quantities with different units.
• The key calculation for rates is always division: Quantity A per Quantity B.
• For speed calculations, use the formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
• Always ensure units are compatible before calculating. If a distance is in metres and time is in hours, you must convert one or both.

You've got this! Practice converting time carefully, and you will ace the rate questions!