Basic Number Skills, Scale, and Percentages

Worked Example: Calculating Linear Distance

A student measures a straight-line footpath on a 1:50,000 map. The distance on the map is 7.2 cm. Calculate the real-world distance in kilometres.

Step 1: Identify the scale factor.

• Map distance = $7.2\text{ cm}$
• Scale = 1:50,000

Step 2: Multiply the map distance by the scale factor to find the ground distance in cm.

• $\text{Ground distance} = 7.2\text{ cm} \times 50{,}000 = 360{,}000\\text{ cm}$

Step 3: Convert centimetres to kilometres.

• There are $100{,}000\text{ cm}$ in $1\text{ km}$.
• $\text{Distance} = 360{,}000 \div 100{,}000 = 3.6\text{ km}$

Final Answer: $3.6\text{ km}$

Worked Example: Approximating Area of a Triangular Reservoir

A reservoir is triangular on a 1:10,000 map (where 1 cm = 100 m). The map dimensions are 5.5 cm (base) and 4.2 cm (height). Calculate the estimated ground area in square metres.

Step 1: Convert map measurements to ground measurements.

• $\text{Base} = 5.5 \times 100 = 550\text{ m}$
• $\text{Height} = 4.2 \times 100 = 420\text{ m}$

Step 2: Apply the area formula for a triangle. $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$

Step 3: Calculate the final value.

• $\text{Area} = 0.5 \times 550\text{ m} \times 420\text{ m} = 115{,}500\text{ m}^2$

Final Answer: $115{,}500\text{ m}^2$

Unit Conversions and Quantitative Relationships

Measuring a tiny raindrop requires a different metric than measuring an entire rainforest, so geographers must constantly switch between units. A Quantitative Relationship is the mathematical connection between these different units, allowing for accurate conversions.

When converting areas, you cannot just use the standard length conversion. Because $1{,}000\text{ m} = 1\text{ km}$, you must square the conversion factor for area: $1{,}000^2\text{ m}^2 = 1^2\text{ km}^2$. Therefore, $1\text{ km}^2 = 1{,}000{,}000\text{ m}^2$. A Hectare (ha) is a metric area unit equal to $10{,}000\text{ m}^2$ (a square with 100 m sides). One square kilometre contains exactly 100 hectares.

For climate graphs, rainfall is measured in millimetres (mm). To convert to metres, you divide by 1,000 ($1{,}000\text{ mm} = 1\text{ m}$).

When processing these conversions, you must follow rules for Decimal Resolution. Calculated data (like a mean value) should typically be recorded to one more decimal place than the raw data. If you are adding or subtracting numbers, your final answer should match the lowest number of decimal places present in your original values.

Worked Example: Converting Hectares to Square Kilometres

A farmer's field covers 320 ha. Calculate this area in square kilometres.

Step 1: Identify the conversion factor.

• $100\text{ ha} = 1\text{ km}^2$

Step 2: Divide the hectare value by 100.

• $\text{Area in km}^2 = 320 \div 100$

Step 3: Calculate the final value.

• $\text{Area} = 3.2\text{ km}^2$

Final Answer: $3.2\text{ km}^2$

Proportion, Ratio, and Population Measures

Comparing headcounts directly isn't always fair—a small rural village and a mega-city cannot be judged using the exact same raw numbers.

• A Ratio compares two independent quantities (e.g., 1 shop for every 4 houses is a 1:4 ratio).
• A Proportion represents a part of a whole, usually expressed as a percentage or fraction (e.g., 20% of the houses are semi-detached).

In fieldwork, you might survey datasets of different sizes. To compare them fairly, you calculate the Relative Frequency, which is the proportion of times a specific category occurs relative to the total number of observations. The sum of all relative frequencies in a dataset must equal 1 (or 100%).

In human geography, these skills are applied to the Dependency Ratio, which compares the economically inactive population against the working population.

• Dependents are children (0–14) and the elderly (65+).
• The economically active group is the working population (15–64).
• Developed countries (ACs) often have ageing populations with ratios around 50–75, while developing countries (LIDCs) have youthful populations where the ratio can exceed 100.

Worked Example: Calculating Dependency Ratio

In a specific country, 35% of the population is aged 0–14, 5% is aged 65+, and 60% is aged 15–64. Calculate the total dependency ratio.

Step 1: State the formula. $\text{Dependency Ratio} = \frac{(\%\text{ pop. 0--14}) + (\%\text{ pop. 65+})}{\%\text{ pop. 15--64}} \times 100$

Step 2: Substitute the values.

• $\text{Dependents} = 35 + 5 = 40\%$
• $\text{Working} = 60\%$
• $\text{Ratio} = \frac{40}{60} \times 100$

Step 3: Calculate the final value.

• $\text{Ratio} = 0.666... \times 100 = 66.7$ (to 1 d.p.)

Final Answer: $66.7$ (meaning approximately 67 dependents for every 100 workers)

Percentage Change and Percentiles

You can easily tell if a town has gained residents, but does that represent a massive population boom or a tiny statistical blip? To understand the significance, we calculate percentage increase or decrease using a standard formula.

$\text{Percentage Change} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100$

When analysing large datasets (like rainfall over 50 years), we use a Percentile to see where a specific value ranks. Percentiles divide a ranked dataset into 100 equal parts. To find percentiles, the data must first be sorted in order of magnitude (smallest to largest). The 25th percentile is the lower quartile, the 50th is the median, and the 75th is the upper quartile.

Worked Example: Calculating Percentage Increase

A city population grew from 2.0 million residents to 2.3 million residents over a decade. Calculate the percentage increase.

Step 1: Identify the values.

• $\text{Original Value} = 2.0$
• $\text{New Value} = 2.3$

Step 2: Find the difference.

• $2.3 - 2.0 = 0.3$

Step 3: Divide by the original value and multiply by 100.

• $0.3 \div 2.0 = 0.15$
• $0.15 \times 100 = 15\%$

Final Answer: $15\%$

Worked Example: Calculating Percentage Decrease

A remote island had a population of 46,000 in 1901. By 2021, the population had fallen to 27,000. Calculate the percentage decrease in population.

Step 1: Identify the values.

• $\text{Original Value} = 46{,}000$
• $\text{New Value} = 27{,}000$

Step 2: Find the difference.

• $27{,}000 - 46{,}000 = -19{,}000$

Step 3: Divide by the original value and multiply by 100.

• $\text{Change} = \frac{-19{,}000}{46{,}000} \times 100$
• $\text{Calculation} = -0.41304... \times 100 = -41.3\%$ (to 1 d.p.)

Final Answer: A percentage decrease of $41.3\%$

Worked Example: Finding a Percentile Position

A geographer has recorded 19 daily temperature readings, ranked from lowest to highest. Calculate the position of the 75th percentile.

Step 1: State the percentile rank formula. $R = \frac{P}{100} \times (n + 1)$ (Where $R$ is position, $P$ is the percentile, and $n$ is the total number of values)

Step 2: Substitute the values.

• $P = 75$, $n = 19$
• $R = \frac{75}{100} \times (19 + 1)$

Step 3: Calculate the position.

• $R = 0.75 \times 20 = 15$

Final Answer: The 75th percentile is the $15\text{th}$ value in the ranked dataset.

Natural Hazards: Magnitude and Frequency

Not all geographical events happen evenly; some are rare but devastating, while others are common but minor. To analyse this, geographers chart a Frequency Distribution, which shows how often each value or category occurs. When dealing with continuous data (like river pebble sizes), data is placed into a Grouped Frequency table and plotted on a histogram.

Natural hazards show a clear inverse relationship: as the Magnitude (energy released) of an event increases, its Frequency (rate of occurrence) decreases. This means high-magnitude earthquakes have a very long Recurrence Interval (the average time between events of that size).

Earthquake magnitude is measured on the Moment Magnitude Scale using Orders of Magnitude. This is a base-10 logarithmic scale. Therefore, a 1-unit increase (e.g., from Magnitude 6 to 7) means the seismic waves are 10 times larger in amplitude, and the earthquake releases approximately 32 times more energy.