Real-World Examples Revision Notes for HSC SSCE Mathematics Standard

Real-World Examples

Introduction

infoNote

Rates are powerful mathematical tools that help us understand and compare real-world situations. By learning to calculate and interpret rates, you can make better decisions about your health, finances, and everyday choices.

Rates are essential in everyday life, helping us make informed decisions about health monitoring, vehicle efficiency, and other practical matters. This note explores real-world applications of rates including fuel consumption, heart rate monitoring, blood pressure measurement, and blood type distributions.

Fuel consumption

What is fuel consumption?

Fuel consumption measures how efficiently a vehicle uses petrol or diesel. It tells us how many litres of fuel a vehicle uses to travel 100 kilometres. This measurement helps you compare different vehicles and understand running costs.

To calculate fuel consumption, you need to:

Fill your vehicle's tank completely
Note the odometer reading (distance travelled shown on the dashboard)
Drive until you need to refuel
Note the new odometer reading
Record how many litres it takes to refill the tank
Calculate the distance travelled (difference between the two odometer readings)

Fuel consumption formula

The formula for calculating fuel consumption is:

$\text{Fuel consumption} = \frac{\text{Amount of fuel (L)} \times 100}{\text{Distance travelled (km)}}$

The result is expressed in litres per 100 kilometres, written as L/100km. A lower fuel consumption value means better fuel efficiency and lower running costs.

chatImportant

Lower is Better!

Remember: A lower fuel consumption value means better fuel efficiency and lower running costs. For example, 6.5 L/100km is more efficient than 9.2 L/100km.

Worked example: Calculating fuel consumption

lightbulbExample

Worked Example: Calculating Fuel Consumption

Let's calculate the fuel consumption for two different vehicles.

Part a: Medium-sized car

A medium-sized car travelled 850 km using 78.2 L of petrol. What was the fuel consumption?

Solution:

Step 1: Write the formula

$\text{Fuel consumption} = \frac{\text{Amount of fuel} \times 100}{\text{Distance travelled}}$

Step 2: Substitute the values (78.2 L and 850 km)

$= \frac{78.2 \times 100}{850}$

Step 3: Calculate

$= 9.2 \text{ L/100km}$

Therefore, the fuel consumption is 9.2 litres per 100 kilometres.

Part b: Small-sized car

A small-sized car travelled 260 km using 16.9 L of petrol. What was the fuel consumption?

Solution:

Step 1: Write the formula

$\text{Fuel consumption} = \frac{\text{Amount of fuel (L)} \times 100}{\text{Distance travelled (km)}}$

Step 2: Substitute the values (16.9 L and 260 km)

$= \frac{16.9 \times 100}{260}$

Step 3: Calculate

$= 6.5 \text{ L/100km}$

Therefore, the fuel consumption is 6.5 litres per 100 kilometres.

Comparison: The small car is more fuel-efficient, using 6.5 L/100km compared to the medium car's 9.2 L/100km.

Heart rate

Understanding heart rate

Heart rate is the number of times your heart beats per minute, measured in beats per minute (bpm). Understanding your heart rate helps you monitor fitness levels and exercise safely.

You can measure heart rate by:

Manual method: Press your index and middle fingers on your wrist or neck where you can feel your pulse, then count beats for 60 seconds (or count for 15 seconds and multiply by 4)
Heart rate monitor: A chest strap with electrodes sends data to a wrist receiver, useful during exercise when manual measurement is difficult
Electrocardiograph: Medical professionals use this device for accurate heart rate measurement to diagnose and track medical conditions

infoNote

Understanding Resting Heart Rate

Resting heart rate is measured when you are at rest but awake. For most people, resting heart rate is typically between 60 and 80 bpm. Athletes often have lower resting heart rates due to better cardiovascular fitness, sometimes as low as 40-50 bpm.

Maximum heart rate (MHR)

Maximum heart rate estimates the highest safe heart rate during intense exercise. The most widely used formula is:

$\text{MHR} = 220 - \text{Age (years)}$

This simple formula provides an estimate based on your age. As you get older, your maximum heart rate decreases.

lightbulbExample

Worked Example: Estimating Maximum Heart Rate

Estimate the maximum heart rate for an 18-year-old.

Solution:

Step 1: Write the formula

$\text{MHR} = 220 - \text{Age}$

Step 2: Substitute 18 for age

$= 220 - 18$

Step 3: Calculate

$= 202$

Therefore, the maximum heart rate for an 18-year-old is estimated to be 202 bpm.

Interpreting resting heart rate tables

The following table shows average resting heart rates for men across different age groups and fitness levels:

The table is organised with:

Age groups across the top (18-25 years through to 65+ years)
Health categories down the left side (from Athlete to Poor)
Heart rate ranges in the cells (in bpm)

Key observations:

Athletes have the lowest resting heart rates (49-60 bpm range)
Poor fitness levels show the highest rates (80+ bpm)
Heart rate ranges increase slightly with age within each fitness category

lightbulbExample

Worked Example: Interpreting Heart Rate Trends

Using the resting heart rate table for men, answer the following questions:

Part a: What is the average resting heart rate for a man aged 47 years in good health?

Solution:

Step 1: Locate the age column - Age 47 falls in the range 46-55 years

Step 2: Locate the "Good" health row and read the value at the intersection

Therefore, the average resting heart rate is 64-67 bpm.

Part b: What is the average resting heart rate for a man aged 25 years in below-average health?

Solution:

Step 1: Locate the age column - Age 25 falls in the range 18-25 years

Step 2: Locate the "Below average" health row and read the value at the intersection

Therefore, the average resting heart rate is 74-81 bpm.

Part c: What is the health of a man aged 57 years with a resting heart rate of 60?

Solution:

Step 1: Locate the age column - Age 57 falls in the range 56-65 years

Step 2: In that column, locate the heart rate 60, which falls in the range 57-61

Step 3: Read the row header: Excellent

Therefore, his health is classified as excellent.

Part d: What is the health of a man aged 30 years with a resting heart rate of 84?

Solution:

Step 1: Locate the age column - Age 30 falls in the range 26-35 years

Step 2: In that column, locate the heart rate 84, which falls in the range 82+

Step 3: Read the row header: Poor

Therefore, his health is classified as poor.

Target heart rate (THR)

Target heart rate is the ideal heart rate range during exercise that provides the most benefit to your heart and lungs. This range depends on several factors including age, physical condition, gender and previous training.

The THR is calculated as a range between 65% and 85% of your maximum heart rate:

$\text{THR} = 0.65 \times \text{MHR} \text{ to } 0.85 \times \text{MHR}$

lightbulbExample

Worked Example: Calculating Target Heart Rate

For an 18-year-old with MHR of 202 bpm, calculate the target heart rate range.

Solution:

Step 1: Calculate the lower limit (65% of MHR)

$= 0.65 \times 202 = 131.3 \text{ bpm}$

Step 2: Calculate the upper limit (85% of MHR)

$= 0.85 \times 202 = 171.7 \text{ bpm}$

Therefore, the target heart rate range is 131.3 to 171.7 bpm.

chatImportant

Exam Tip: Complete Range Required

When calculating THR, remember to calculate both the lower limit (65% of MHR) and upper limit (85% of MHR) to give the complete range. Exercising within this range ensures your heart and lungs receive maximum benefit while staying safe.

Blood pressure

Understanding blood pressure

Blood pressure measures the force of blood pushing against the walls of your arteries as your heart pumps blood around your body. During each heartbeat, blood pressure varies between two values:

Systolic pressure (the higher number): Maximum pressure when the heart contracts and pushes blood out
Diastolic pressure (the lower number): Minimum pressure when the heart relaxes between beats

Blood pressure is measured using a device called a sphygmomanometer, which consists of an inflatable cuff wrapped around your upper arm. The measurement is expressed in millimetres of mercury (mmHg).

infoNote

Normal Blood Pressure

A normal healthy adult has a blood pressure of 120/80 mmHg

This is read as "120 over 80" where:

120 is the systolic pressure (higher reading)
80 is the diastolic pressure (lower reading)

Factors affecting blood pressure

Blood pressure changes throughout the day to meet your body's needs. Many factors can affect your blood pressure including:

Physical activity and exercise
Stress levels
Disease and medical conditions
Medications and drugs
Time of day

chatImportant

Why Regular Monitoring Matters

Doctors may request regular blood pressure monitoring to track changes over time. This helps identify patterns and detect potential health issues early. Blood pressure naturally fluctuates, so multiple readings provide a more accurate picture of your cardiovascular health.

The following graph shows how blood pressure can vary for a patient over several months:

The graph shows:

The red line represents systolic blood pressure (upper readings) averaging around 114 mmHg
The blue line shows diastolic blood pressure (lower readings) averaging around 74 mmHg
Both values fluctuate naturally over time but remain within a healthy range

Blood pressure classification

Blood pressure is classified into four categories:

Category	Systolic (mmHg)	Diastolic (mmHg)
Normal	<120	<80
Normal to high	120-139	80-89
High	140-179	90-109
Very high	≥180	≥110

infoNote

Reading the table: Blood pressure is classified based on both systolic and diastolic readings. If either reading falls into a higher category, the overall classification uses the higher category.

lightbulbExample

Worked Example: Reading a Blood Pressure Table

Using the blood pressure classification table, answer the following questions:

Part a: What is the systolic blood pressure for very high blood pressure?

Solution:

Read the value in the table for "Very high" category, systolic column.

Systolic blood pressure is ≥180 mmHg.

Part b: What is the diastolic blood pressure for normal to high blood pressure?

Solution:

Read the value in the table for "Normal to high" category, diastolic column.

Diastolic blood pressure is 80-89 mmHg.

Part c: Stephanie has blood pressure of 117/74 mmHg. In what category is she classified?

Solution:

Step 1: Compare Stephanie's readings to the table

Systolic: 117 is less than 120
Diastolic: 74 is less than 80

Step 2: Both values fall in the "Normal" category

Therefore, Stephanie's blood pressure is classified as normal.

Blood types

Understanding blood types

A person's blood type is classified using two systems:

ABO system: Uses letters A, B, AB or O
Rh factor: Indicates whether blood is Rh positive (+) or Rh negative (−)

Combining these gives eight possible blood types: O+, O−, A+, A−, B+, B−, AB+, AB−

Blood type distribution in Australia

The column graph shows the percentage frequency of each blood type in Australia (Source: Australian Red Cross Blood Service):

O+ is the most common (40% of the population)
A+ is the second most common (31%)
O− makes up 9% of the population
B+ accounts for 8%
A− represents 7%
B−, AB+ and AB− are the rarest types (2%, 2% and 1% respectively)

infoNote

Why Blood Donation Matters

Blood donation is vital for medical treatments. Most donated blood is used to treat people with cancer and other serious illnesses. Understanding blood type distribution helps blood services ensure adequate supplies of all blood types, especially the rarer ones like AB− which only represents 1% of the population.

lightbulbExample

Worked Example: Calculating Population by Blood Type

The table shows the population of New South Wales:

NSW population	Number
Males	3.72 million
Females	3.83 million
Total	7.55 million

Question: How many males in NSW have blood type B+?

Solution:

Step 1: Read the percentage of blood type B+ from the column graph

$8\% \text{ of the Australian population are B+}$

Step 2: Read the male population of NSW from the table: 3.72 million

Step 3: Convert percentage to decimal and multiply by the population

$8\% \text{ of } 3.72 \text{ million} = 0.08 \times 3\,720\,000$

Step 4: Calculate

$= 297\,600$

Therefore, approximately 297,600 males in NSW have blood type B+.

Calculation tip: To convert a percentage to a decimal, divide by 100 (e.g., 8% = 0.08). Then multiply by the total population to find the number of people in that category.

Remember!

bookmarkSummary

Key Formulas and Concepts:

Fuel Consumption:

Formula: (Amount of fuel (L) × 100) ÷ Distance travelled (km)
Lower values mean better efficiency
Measured in L/100km

Maximum Heart Rate (MHR):

Formula: 220 − Age
This decreases as you get older
Used to calculate safe exercise intensity

Target Heart Rate (THR):

Range: 65% to 85% of your MHR
Ensures safe and effective cardiovascular exercise
Calculate both lower and upper limits

Resting Heart Rate:

Normal range: 60-80 bpm
Athletes have lower resting heart rates due to better fitness
Can indicate overall cardiovascular health

Blood Pressure:

Written as systolic/diastolic (e.g., 120/80 mmHg)
Systolic (top number) is always higher than diastolic (bottom number)
Normal blood pressure: less than 120/80 mmHg
Regular monitoring helps track health changes

Blood Types:

Most common in Australia: O+ (40%), followed by A+ (31%)
Eight possible types: O+, O−, A+, A−, B+, B−, AB+, AB−
Understanding blood type distribution is important for medical planning

Real-World Examples (HSC SSCE Mathematics Standard): Revision Notes