Rates, the Unitary Method, and Making Comparisons Revision Notes for HSC SSCE Mathematics Standard

Rates, the Unitary Method, and Making Comparisons

What are rates?

A rate is a way of comparing two quantities that have different units of measurement. For example, when we compare the distance travelled with the time taken, we're using a rate. The key feature of a rate is that the two units being compared are different, and both must be specified when writing the rate.

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When writing rates, the order matters. A rate is always expressed as the first amount per one unit of the second amount. Think of "per" as meaning "for each one" - this helps you understand the order.

For instance:

$2.99/kg means $2.99 per one kilogram
80 km/h means 80 kilometres per one hour

Rates help us understand how things change over time and are used constantly in everyday life. Here are some common examples:

Growth rate: The average growth rate of a child from birth to 15 years of age
Running rate: Your running pace measured in metres per second
Typing rate: Your typing speed measured in words per minute
Wage rate: The amount of money you earn per hour of work

Converting rates

Sometimes we need to convert a rate from one set of units to another. This is a systematic process that involves four steps:

Step 1: Write the rate as a fraction, with the first quantity as the numerator and 1 as the denominator

Step 2: Convert the first amount (numerator) to the required unit

Step 3: Convert the second amount (denominator) to the required unit

Step 4: Simplify the fraction to get your final answer

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Worked Example: Converting Speed

Let's convert $55\,200$ m/h to m/min.

Step 1: Write as a fraction

$\frac{55\,200\text{ m}}{1\text{ h}}$

Step 2: The numerator is already in metres, so no conversion needed

Step 3: Convert 1 hour to minutes by multiplying by 60

$\frac{55\,200\text{ m}}{1 \times 60\text{ min}} = \frac{55\,200\text{ m}}{60\text{ min}}$

Step 4: Simplify the fraction

$\frac{55\,200}{60} = 920\text{ m/min}$

Therefore, $55\,200$ m/h = 920 m/min.

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Worked Example: Converting Price per Mass

Now let's convert $6.50/kg to c/g.

Step 1: Write as a fraction

$\frac{\$6.50}{1\text{ kg}}$

Step 2: Convert $6.50 to cents by multiplying by 100

$\frac{6.50 \times 100\text{ c}}{1\text{ kg}} = \frac{650\text{ c}}{1\text{ kg}}$

Step 3: Convert 1 kg to grams by multiplying by 1000

$\frac{650\text{ c}}{1 \times 1000\text{ g}} = \frac{650\text{ c}}{1000\text{ g}}$

Step 4: Simplify the fraction

$\frac{650}{1000} = 0.65\text{ c/g}$

Therefore, $6.50/kg = 0.65 c/g.

The unitary method

The unitary method is a powerful technique for solving rate problems. It involves finding the value of one unit first (by division), then multiplying this result to find the answer to the problem. This method is particularly useful when making comparisons between different options.

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The word "unitary" comes from "unit" - meaning one. This method always starts by finding the value of one unit before scaling up or down. This makes complex comparisons much simpler!

How to use the unitary method

Step 1: Find one unit of an amount by dividing by the total number of units

Step 2: Multiply the result from step 1 by the number of units you need

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Worked Example: Distance and Fuel Consumption

A car travels 360 km on 30 L of petrol. How far does it travel on 7 L?

Step 1: Find the distance travelled on 1 L of petrol

$30\text{ L} = 360\text{ km}$

$1\text{ L} = \frac{360}{30}\text{ km} = 12\text{ km}$

Step 2: Multiply by 7 to find the distance on 7 L

$7\text{ L} = \frac{360}{30} \times 7\text{ km} = 84\text{ km}$

Therefore, the car travels 84 km on 7 L of petrol.

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Worked Example: Typing Rate

Bella can touch type at 70 words per minute. How many words can she type in 20 minutes?

Solution:

The typing rate tells us that Bella types 70 words in one minute.

To find the total for 20 minutes:

$\text{Number of words} = 70 \times 20 = 1400$

Therefore, Bella can type 1400 words in 20 minutes.

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Worked Example: Comparing Prices

A brand of 400 mL soft drink cans are sold in three ways:

Singly for $2.40
In a six-pack for $11.95
In a carton of 24 for $39.95

Part (a): Compare the cost of one can in each option, to the nearest cent.

Single can: $2.40

Six-pack:

$\text{Cost per can} = \frac{\$11.95}{6} \approx \$1.99$

Carton of 24:

$\text{Cost per can} = \frac{\$39.95}{24} \approx \$1.66$

A can costs $2.40 when bought singly, $1.99 each in a six-pack, and $1.66 each in a carton.

Part (b): What is the cost of 14 cans at the cheapest option?

The cheapest option is the carton at $1.66 per can.

$\text{Cost of 14 cans} = \$1.66 \times 14 = \$23.24$

Therefore, 14 cans at the cheapest price option will cost $23.24.

Using rates to make comparisons

Rates are essential tools for solving practical problems involving wages, best buys, and various costs. Let's explore how to apply rates in these real-world situations.

Finding the best buy

When comparing different purchasing options, the best buy is the option with the lowest unit cost. Use the unitary method to calculate the cost per item for each option.

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Always calculate the unit cost for each option before deciding on the best buy. The option with the largest quantity isn't always the cheapest per unit!

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Worked Example: Comparing Plant Prices

Which is the best buy?

Option 1: 12 rose plants for $195
Option 2: 10 rose plants for $162

Option 1: Find the cost per plant

$12\text{ plants} = \$195$

$1\text{ plant} = \frac{\$195}{12} = \$16.25$

Option 2: Find the cost per plant

$10\text{ plants} = \$162$

$1\text{ plant} = \frac{\$162}{10} = \$16.20$

Option 2 has the lower unit cost, so the best buy is Option 2.

Calculating costs with fixed charges

Some services charge a flagfall (a fixed initial charge) plus a rate that varies with usage. The total cost is:

$\text{Total cost} = \text{Flagfall} + (\text{Rate} \times \text{Usage})$

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A flagfall is common in taxi services, phone calls, and delivery charges. It's the amount you pay just to use the service, before any usage-based charges are added.

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Worked Example: Mobile Phone Charges

Alice's mobile phone contract charges a flagfall of $0.25 and a call rate of $0.45 per 30 seconds.

Part (a): What is the charge if Alice makes a 2 minute call?

First, convert 2 minutes to seconds: $2 \times 60 = 120$ seconds

Find the number of 30-second blocks: $\frac{120}{30} = 4$ blocks

Calculate the total charge:

$\text{Charge} = 0.25 + (4 \times 0.45)$

$= 0.25 + 1.80$

$= \$2.05$

Therefore, Alice is charged $2.05 for a 2-minute call.

Part (b): What is the charge if Alice made 200 calls of duration less than 30 seconds?

Each call under 30 seconds costs:

$\text{Cost per call} = 0.25 + 0.45 = \$0.70$

For 200 calls:

$\text{Total charge} = 0.70 \times 200 = \$140$

Therefore, Alice is charged $140 for the 200 calls.

Calculating wages with overtime

Many jobs pay different rates for different types of work:

Normal rate: The standard hourly rate
Time-and-a-half: 1.5 times the normal rate (often for weekends or after-hours work)
Double time: 2 times the normal rate (often for public holidays)

The total wage is calculated by adding all components together.

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Remember the overtime multipliers:

Time-and-a-half = Normal rate × 1.5
Double time = Normal rate × 2

Don't forget to add the normal pay to the overtime pay for the total wage!

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Worked Example: Wage Calculation

Hamish works for a building construction company and is paid $22 per hour normally. Find his wage for:

35 hours at the normal rate
3 hours at time-and-a-half
1 hour at double time

Step 1: Calculate normal pay

$\text{Normal pay} = 35 \times 22 = \$770$

Step 2: Calculate time-and-a-half pay

$\text{Time-and-a-half pay} = 3 \times 22 \times 1.5 = \$99$

Step 3: Calculate double time pay

$\text{Double time pay} = 1 \times 22 \times 2 = \$44$

Step 4: Add all components

$\text{Total wage} = 770 + 99 + 44 = \$913$

Therefore, Hamish's wage is $913.

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Key Points to Remember:

A rate compares two quantities with different units, and the order matters - always write it as the first amount per one unit of the second amount.
To convert a rate, write it as a fraction, convert both the numerator and denominator to the required units, then simplify.
The unitary method is a two-step process: first find the value of one unit by dividing, then multiply by the required amount.
To find the best buy, calculate the unit cost for each option and choose the one with the lowest price per item.
When calculating costs with a flagfall, remember to add the fixed charge to the variable rate charge.
For wage calculations with overtime, multiply the normal hourly rate by 1.5 for time-and-a-half and by 2 for double time, then add all components together.

Rates, the Unitary Method, and Making Comparisons (HSC SSCE Mathematics Standard): Revision Notes