Unit Conversions: Length, Mass and Money
Unit conversions rewrite a quantity in another compatible unit by applying a stated relationship between the units. In brief: 3.2 metres = 320 centimetres and 3.2 kilograms = 3200 grams. Currency conversion follows the same factor method when a specific exchange rate is supplied, but the rate is not permanently fixed.
A conversion changes the numerical count while preserving the quantity represented under the stated relationship. This guide develops one method for length, mass and currency examples, explains exact and approximate factors, and shows how to use unit cancellation to choose multiplication or division instead of guessing.
What is a conversion factor?
A conversion factor is a ratio of two equal quantities expressed in different units. Since 1 metre equals 100 centimetres, the ratio 100 cm/1 m has value one as a quantity ratio. Multiplying a length by it changes the unit expression without changing the physical length.
The inverse ratio, 1 m/100 cm, is equally valid but serves the opposite direction. Choose the orientation so that the starting unit cancels and the desired unit remains. This is the central idea of dimensional analysis in basic conversions.
For currency, a supplied exchange rate plays a similar arithmetic role, but it describes an exchange relationship at specified conditions rather than a permanent physical equality. Keep the currency names in the ratio and treat fees or different transaction rates separately when they are part of the stated problem.
How do you convert metric lengths?
For 3.2 metres to centimetres, multiply 3.2 m by 100 cm/1 m. The metre unit cancels, leaving 320 centimetres. Because a centimetre is smaller, the larger numerical count is sensible.
For 4750 millimetres to metres, use 1 m/1000 mm and divide 4750 by 1000 to obtain 4.75 metres. A result of 4,750,000 metres would indicate that the factor was reversed, even before any detailed check.
When several prefixes are involved, you may convert through a familiar unit. For 2.6 kilometres to centimetres, convert to 2600 metres and then to 260,000 centimetres. Combining the factors gives 100,000 centimetres per kilometre, so a one-step multiplication reaches the same result.
| Starting quantity | Factor relationship | Converted quantity |
|---|---|---|
| 3.2 m | 100 cm per m | 320 cm |
| 4750 mm | 1000 mm per m | 4.75 m |
| 2.6 km | 100,000 cm per km | 260,000 cm |
| 85 cm | 100 cm per m | 0.85 m |
How do you convert inches, feet and metres?
The international inch is exactly 2.54 centimetres, and one international foot is 12 inches. Therefore one foot is exactly 30.48 centimetres or 0.3048 metre. These fixed relationships can be chained to convert a mixed customary length.
For 5 feet 8 inches, first convert the feet to inches: 5 × 12 + 8 = 68 inches. Multiply by 2.54 to obtain 172.72 centimetres, or 1.7272 metres. The conversion factor is exact, although the original measured height may have been rounded.
A practical report may round the final converted value to match the precision of the original measurement. Exact conversion does not create extra measurement accuracy. Keep more digits during the calculation, then choose the final reporting precision based on the problem rather than treating every calculator digit as meaningful.
- Convert 5 ft to 60 in.
- Add the remaining 8 in to obtain 68 in.
- Multiply 68 by 2.54 to get 172.72 cm.
- Divide by 100 if metres are wanted: 1.7272 m.
How do you convert mass units?
Metric mass conversions use fixed factors: 1000 milligrams make one gram, 1000 grams make one kilogram and 1000 kilograms make one tonne. For 0.75 kilogram, multiply by 1000 to obtain 750 grams. For 4200 milligrams, divide by 1000 to obtain 4.2 grams.
The avoirdupois pound is exactly 0.45359237 kilogram. For a calculation using a rounded classroom factor of 0.454 kilogram per pound, the answer is approximate and should be labeled accordingly. The factor chosen determines how many digits the final conversion can reasonably support.
Mass units cannot be exchanged with volume units without additional information. A litre-to-kilogram calculation requires a density, while grams-to-kilograms requires only a unit factor. Identifying the quantity type before converting prevents an arithmetic procedure from being applied to incompatible measures.
| Relationship | Status |
|---|---|
| 1 kg = 1000 g | Exact |
| 1 g = 1000 mg | Exact |
| 1 t = 1000 kg | Exact metric tonne |
| 1 avoirdupois lb = 0.45359237 kg | Exact |
| 1 avoirdupois lb ≈ 0.454 kg | Rounded approximation |
How do you convert money into minor units?
If a currency has 100 minor units per main unit, multiply by 100 to express an amount in minor units. An amount of 8.65 main units becomes 865 minor units. The reverse conversion divides by 100, so 1240 minor units becomes 12.40 main units.
This is a fixed subdivision relationship within that currency, unlike a changing exchange rate between currencies. It can simplify addition and subtraction by expressing all amounts as whole-number minor units. A total of 865 + 1240 = 2105 minor units is 21.05 main units.
Not every currency uses two decimal places in the same way, so the subdivision must be stated or established. In a generic mathematical example, saying ‘100 minor units per main unit’ is enough to define the calculation without implying a universal monetary convention.
How do you use a hypothetical exchange rate correctly?
Suppose an exercise states that 1 unit of currency A exchanges for 0.8 unit of currency B, with no fees. To convert 250 A to B, multiply by 0.8 B/A to obtain 200 B. The starting A unit cancels, leaving the requested B unit.
To convert 200 B back to A under the same rate, divide by 0.8 or multiply by 1.25 A/B. The result is 250 A. The reverse factor is the reciprocal, not the original factor repeated.
If a fixed fee is stated, apply it at the stage and in the currency specified. For example, if 5 A is deducted before converting 250 A, the amount exchanged is 245 A and the result is 196 B. These are hypothetical arithmetic assumptions; actual exchange quotes and fees must be obtained for a real transaction.
How do you combine several conversion steps?
Write each factor so that unwanted units cancel in sequence. To convert 18 inches to metres, use 18 in × 2.54 cm/in × 1 m/100 cm. The inches and centimetres cancel, leaving 0.4572 metre.
The same approach works for 3 tonnes to grams: multiply by 1000 kilograms per tonne and 1000 grams per kilogram. The result is 3,000,000 grams. Each step has a clear unit purpose, making the chain easier to inspect than a memorized string of zeros.
Do not round after every factor if a final rounding can be postponed. Repeated rounding may accumulate error, especially when several approximate factors are involved. Keep the exact factors or sufficient precision throughout, then report the final answer with an appropriate approximation sign when needed.
How do you check a conversion and avoid common mistakes?
Conversions rest on the idea that a quantity can be represented in different agreed units. The numerical factor records how many of one unit fit into another. This relationship underlies metric prefixes, international length definitions and the arithmetic use of a stated currency rate.
Check three things: the units cancel correctly, the numerical size changes in the expected direction, and the inverse conversion recovers the original value. For example, 320 centimetres divided by 100 returns 3.2 metres. A failed inverse check often exposes a reversed factor or a misplaced decimal point.
Common mistakes include treating all conversions as multiplication, mixing pounds of mass with fluid measures and using an exchange rate without its direction. Also distinguish exact unit definitions from rounded factors and changing monetary rates. The final label should identify both the converted unit and any approximation used.