Compound Measures: Speed, Density and Unit Pricing

Welcome to Compound Measures!

Have you ever wondered how fast a sprinter runs, why a massive steel ship floats while a small pebble sinks, or which box of cereal at the supermarket is truly the best bargain? The answer to all three questions comes down to compound measures!

A compound measure is simply a measure that combines two or more different standard units (such as distance, time, mass, or money) to tell us about a rate of change. (Note: For converting between basic single units like minutes to hours or metres to kilometres, take a quick look at the Unit Conversions chapter.)

Don't worry if this sounds tricky at first. In these notes, we will break down the three main compound measures you need to master: speed, density, and unit pricing.

1. Understanding Compound Measures and Their Units

In everyday life, single measures describe just one quantity, such as a length of \(5\text{ m}\) or a time of \(10\text{ s}\). A compound measure involves a relationship between two different quantities.

The secret weapon with compound measures is looking closely at the unit: the unit always tells you the formula!

Whenever you see the word per or a slash symbol (/), it means "divided by":

• Miles per hour (\(\text{mph}\) or \(\text{miles/hour}\)) means \(\text{distance} \div \text{time}\).
• Grams per cubic centimetre (\(\text{g/cm}^3\)) means \(\text{mass} \div \text{volume}\).
• Pence per gram (\(\text{p/g}\)) means \(\text{cost} \div \text{mass}\).

Key Takeaway

If you forget a formula in a question, read the unit carefully. The unit gives away the exact division you need to do!

2. Speed: Distance and Time

Speed is a compound measure that tells us how much distance is covered in a certain amount of time.

The Speed Formulas

We use three related equations depending on what we need to calculate:

\(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)

\(\text{Distance} = \text{Speed} \times \text{Time}\)

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

Memory Aid: The Formula Triangle

Imagine a triangle split into three parts, with \(D\) (Distance) on the top, and \(S\) (Speed) and \(T\) (Time) side-by-side on the bottom:

• Cover \(S\) with your finger: you see \(D\) over \(T\), so \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\).
• Cover \(D\): you see \(S\) next to \(T\), so \(\text{Distance} = \text{Speed} \times \text{Time}\).
• Cover \(T\): you see \(D\) over \(S\), so \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).

Common Units of Speed

• Metres per second (\(\text{m/s}\))
• Kilometres per hour (\(\text{km/h}\))
• Miles per hour (\(\text{mph}\))

Step-by-Step Example 1: Finding Speed

A cyclist rides a distance of \(45\text{ km}\) in \(3\text{ hours}\). What is her average speed?

Step 1: Identify the values given: \(\text{Distance} = 45\text{ km}\), \(\text{Time} = 3\text{ hours}\).
Step 2: Choose the formula: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\).
Step 3: Calculate: \(\text{Speed} = \frac{45}{3} = 15\text{ km/h}\).
The cyclist's speed is \(15\text{ km/h}\).

Watch Out: Time as a Decimal!

This is the most common mistake students make: time in minutes is not a decimal of an hour!

Because there are \(60\text{ minutes}\) in an hour (not \(100\)), you must divide the minutes by \(60\) to turn them into hours:

• \(30\text{ minutes} = \frac{30}{60} = 0.5\text{ hours}\) (NOT \(0.3\text{ hours}\))
• \(15\text{ minutes} = \frac{15}{60} = 0.25\text{ hours}\)
• \(45\text{ minutes} = \frac{45}{60} = 0.75\text{ hours}\)
• \(1\text{ hour and } 12\text{ minutes} = 1 + \frac{12}{60} = 1 + 0.2 = 1.2\text{ hours}\)

Step-by-Step Example 2: Finding Distance with Time Conversion

A car travels at an average speed of \(60\text{ mph}\) for \(2\text{ hours and } 30\text{ minutes}\). How far does the car travel?

Step 1: Convert the time into hours: \(30\text{ minutes} = 0.5\text{ hours}\), so \(\text{Time} = 2.5\text{ hours}\).
Step 2: Choose the formula: \(\text{Distance} = \text{Speed} \times \text{Time}\).
Step 3: Calculate: \(\text{Distance} = 60 \times 2.5 = 150\text{ miles}\).
The car travels \(150\text{ miles}\).

Key Takeaway

Speed measures distance covered per unit of time. Always check that your time units match your speed units before multiplying or dividing!

3. Density: Mass and Volume

Density is a measure of how tightly packed the matter inside an object is. It tells us how much mass is contained within a specific volume.

Analogy: Think of a box filled with feathers compared to the same-sized box packed tightly with lead weights. Both boxes have the same volume, but the box of lead has much more mass, so it has a much higher density.

The Density Formulas

Just like speed, we can write three related equations:

\(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\)

\(\text{Mass} = \text{Density} \times \text{Volume}\)

\(\text{Volume} = \frac{\text{Mass}}{\text{Density}}\)

Memory Aid: The DMV Triangle

Picture a triangle with \(M\) (Mass) on top, and \(D\) (Density) and \(V\) (Volume) on the bottom:
• Cover \(D\) to get \(\frac{M}{V}\).
• Cover \(M\) to get \(D \times V\).
• Cover \(V\) to get \(\frac{M}{D}\).

Common Units of Density

• Grams per cubic centimetre (\(\text{g/cm}^3\))
• Kilograms per cubic metre (\(\text{kg/m}^3\))

Step-by-Step Example 3: Finding Density

A block of wood has a mass of \(240\text{ g}\) and a volume of \(300\text{ cm}^3\). Calculate its density.

Step 1: Identify the given values: \(\text{Mass} = 240\text{ g}\), \(\text{Volume} = 300\text{ cm}^3\).
Step 2: Choose the formula: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\).
Step 3: Calculate: \(\text{Density} = \frac{240}{300} = \frac{4}{5} = 0.8\text{ g/cm}^3\).
The density of the wood is \(0.8\text{ g/cm}^3\).

Did You Know? Sinking and Floating

Pure water has a density of roughly \(1\text{ g/cm}^3\). Objects with a density less than \(1\text{ g/cm}^3\) will float in water, while objects with a density greater than \(1\text{ g/cm}^3\) will sink! Because our wooden block has a density of \(0.8\text{ g/cm}^3\), it will float.

Step-by-Step Example 4: Finding Mass

A solid silver coin has a volume of \(4\text{ cm}^3\). The density of silver is \(10.5\text{ g/cm}^3\). What is the mass of the coin?

Step 1: Identify values: \(\text{Volume} = 4\text{ cm}^3\), \(\text{Density} = 10.5\text{ g/cm}^3\).
Step 2: Choose the formula: \(\text{Mass} = \text{Density} \times \text{Volume}\).
Step 3: Calculate: \(\text{Mass} = 10.5 \times 4 = 42\text{ g}\).
The coin has a mass of \(42\text{ g}\).

Key Takeaway

Density tells us how heavy an object is for its size. Always ensure mass and volume use consistent units (e.g. grams with \(\text{cm}^3\), or kilograms with \(\text{m}^3\)).

4. Unit Pricing: Best Buys and Value for Money

Unit pricing is a compound measure connecting money and quantity. It tells us the price per single unit (such as price per item, price per gram, or price per litre).

Supermarkets often sell products in different sizes. To find out which size gives you the best value for money, you calculate and compare the cost of a single shared unit.

The Unit Price Formula

\(\text{Unit Price} = \frac{\text{Total Cost}}{\text{Quantity}}\)

When comparing prices, the item with the lowest cost per unit gives you the best value!

Method 1: Cost per Single Item or Unit

Let's look at an everyday shopping problem:

A supermarket sells eggs in two packs:
• Pack A: \(6\text{ eggs}\) for \(£1.50\)
• Pack B: \(10\text{ eggs}\) for \(£2.30\)
Which pack offers better value for money?

Step 1: Find the cost of \(1\text{ egg}\) in Pack A:
\(\text{Cost per egg} = \frac{£1.50}{6} = £0.25\) (or \(25\text{p}\)).

Step 2: Find the cost of \(1\text{ egg}\) in Pack B:
\(\text{Cost per egg} = \frac{£2.30}{10} = £0.23\) (or \(23\text{p}\)).

Step 3: Compare the two prices:
\(23\text{p}\) is less than \(25\text{p}\), so Pack B gives better value for money.

Method 2: Comparing a Common Amount (e.g. per 100g)

Sometimes dividing down to a single gram gives a tiny decimal that is hard to work with. In that case, finding the cost per \(100\text{ g}\) or per \(1\text{ kg}\) is much easier!

Two jars of coffee are on offer:
• Small Jar: \(200\text{ g}\) for \(£3.60\)
• Large Jar: \(500\text{ g}\) for \(£8.50\)
Which jar gives better value?

Step 1: Calculate the cost per \(100\text{ g}\) for the Small Jar:
To get from \(200\text{ g}\) to \(100\text{ g}\), divide by \(2\):
\(\frac{£3.60}{2} = £1.80\text{ per } 100\text{ g}\).

Step 2: Calculate the cost per \(100\text{ g}\) for the Large Jar:
To get from \(500\text{ g}\) to \(100\text{ g}\), divide by \(5\):
\(\frac{£8.50}{5} = £1.70\text{ per } 100\text{ g}\).

Step 3: Compare:
\(£1.70\) is less than \(£1.80\), so the Large Jar is the better buy.

Method 3: Quantity per Unit Cost (e.g. grams per £1)

Instead of finding the price for one unit of goods, you can also find out how much product you get for \(£1\):

\(\text{Amount per } £1 = \frac{\text{Quantity}}{\text{Total Cost}}\)

When using this method, the item that gives you the highest quantity per pound is the best value!

Key Takeaway

To compare offers fairly, always adjust both options so they describe the exact same quantity. Look for either the lowest price for the same amount, or the greatest amount for the same price.

5. Summary and Quick Checklist

Whenever you tackle a compound measure question, follow these simple steps:

1. Identify the measure: Is it speed, density, or unit pricing?
2. Check your units: Are all lengths in the same unit? Has time been correctly converted into decimal hours if needed?
3. Pick the right formula: Use your formula triangles (\(D = S \times T\) or \(M = D \times V\)).
4. Check your final answer: Does the answer make sense in the real world? Always write the correct compound unit at the end (e.g. \(\text{km/h}\), \(\text{g/cm}^3\), or \(\text{p/item}\))!