Proportional Reasoning-1 · Lesson 6 of 8
Unit Conversions
“Convert length, area and volume with the supplied factors, distinguish exact and rounded conversions, and use Celsius–Fahrenheit formulas with their offset.”
• Choose a conversion factor and its direction. • Distinguish length units from area units. • Convert litres, millilitres and cubic centimetres. • Use both temperature conversion formulas. • Recognise that Celsius and Fahrenheit readings are not directly proportional.
A field may be measured in square feet while manure is specified per acre. A tap’s flow may be described using millilitres while a bucket is labelled in litres. The relationship may be straightforward, but the numbers cannot be compared correctly until the units match. Converting a unit changes how you describe an amount; it does not change the actual length, area or volume you are measuring.
We will organise the conversions by the kind of quantity. Before calculating, name that quantity and predict whether the number should get larger or smaller when its unit changes.
7.6 Unit Conversions
To convert, first write an equality or approximate equality linking the two units. Then choose the direction that gives the requested unit. For example, one litre equals 1000 mL: three litres must therefore be more than three millilitres, not less.
Some conversion factors are exact, while others are rounded. The metre-to-foot, square-metre-to-square-foot and hectare-to-acre numbers supplied here are rounded. Their calculated results should normally use “approximately” rather than pretending to have unlimited precision.
Length
For metres to feet, multiply by 3.281. For feet to metres, divide by 3.281 when using this rounded factor. These are opposite directions of the same conversion.
A foot is smaller than a metre, so the numerical count in feet is larger. This is a useful direction check before pressing any calculator buttons.
Problem
Convert 5 m to feet, then convert 16.405 ft back using the supplied factor.
Area
| Conversion | Status |
|---|---|
| 1 square metre ≈ 10.764 square feet | Rounded |
| 1 acre = 43,560 square feet | Exact |
| 1 hectare = 10,000 square metres | Exact |
| 1 hectare ≈ 2.471 acres | Rounded |
Area measures a surface, so it needs square units. Do not use a length factor as though it were an area factor. One square metre covers more than 3.281 square feet because both directions of its square change their unit counts.
A square that is one metre on each side is approximately 3.281 feet on each side. Multiplying those two lengths gives approximately 10.765 square feet using that rounded length factor; the supplied rounded area factor is 10.764. The small difference comes from rounding, not a different area rule. Use the stated area factor consistently.
For acres and hectares, choose the route that matches the known and requested units. Multiplying an acre count by 43,560 produces square feet; dividing a square-foot area by 43,560 produces acres.
Problem
Convert 87,120 square feet to acres.
Problem
Express 2 hectares in square metres and approximately in acres.
Volume
A cubic centimetre describes the volume of a cube one centimetre along each edge. A millilitre is the same volume, so changing between mL and cc does not change the number.
A litre contains 1000 of these small units. Multiply litres by 1000 to obtain millilitres; divide millilitres by 1000 to obtain litres. Do not confuse volume with mass: a litre is not automatically a kilogram for every substance.
Problem
Convert 2.5 L to mL and cc.
Temperature
Temperature conversion behaves differently because the two scales have different zero points. A Celsius reading of 0 corresponds to Fahrenheit 32. Simply multiplying the Celsius reading cannot produce that offset.
To convert Celsius C to Fahrenheit F, first multiply by 9/5 and then add 32. To reverse it, first subtract 32 and then multiply by 5/9. The order matters: undo the addition before undoing the multiplication.
Problem
Find the Fahrenheit temperature corresponding to 25°C.
Problem
Find the corresponding Celsius temperature.
Doubling a Celsius reading does not double its Fahrenheit reading: 20°C is 68°F, while 40°C is 104°F, not 136°F. The added 32 prevents direct proportionality between the two readings.
When a mixed problem combines a conversion with a ratio, complete and label the conversion first. That makes it easier to see whether the proportion compares the intended quantities.
Quiz
Using 1 m ≈ 3.281 ft, how do you convert a number of metres into feet?
How many square feet are in two acres?
Which pair represents equal volumes?
What is 25°C in Fahrenheit?
Why are Celsius and Fahrenheit readings not directly proportional?
Practice Problems
- Convert 3 m to feet using the supplied rounded factor.
- Convert 2.5 L to mL and 750 cc to mL.
- Convert 130,680 square feet to acres and 3 hectares to square metres.
- Convert 30°C to Fahrenheit and 86°F to Celsius.
- A student says that 1 m² ≈ 3.281 ft² and doubling 20°C gives 136°F. Explain both mistakes.
1. 3 × 3.281 = 9.843. 2. Thus 3 m ≈ 9.843 ft.
1. 2.5 × 1000 = 2500 mL. 2. Since 1 cc = 1 mL, 750 cc = 750 mL.
1. 130,680 ÷ 43,560 = 3 acres. 2. 3 × 10,000 = 30,000 square metres.
1. F = 30 × 9/5 + 32 = 54 + 32 = 86°F. 2. C = (86 − 32) × 5/9 = 54 × 5/9 = 30°C.
1. An area needs the area factor, 1 m² ≈ 10.764 ft², because both dimensions contribute. The number 3.281 is a length factor. 2. 20°C = 68°F, but 40°C = 40 × 9/5 + 32 = 104°F. 3. The temperature formula includes an offset, so Fahrenheit readings cannot be doubled directly in that comparison.
Key Takeaways
• A unit conversion changes the description of an amount, not the amount itself. • Choose the factor for the correct quantity: length, area or volume. • Use square-unit factors for area, not length factors. • 1 mL equals 1 cc, while 1 L equals 1000 mL. • Celsius–Fahrenheit conversion requires both multiplication and the 32-degree offset. • Rounded conversion factors produce approximate answers.