Math: Basic Tutorials : Measurements

Measurements

Measurements is important as it helps represent physical things like length, temperature, weight, and more. There are 3 important measurements: length, weight, and volume. With each category there are two conversion system: Imperial and Metric systems. Measurements can be converted in their own categories but it cannot be converted to other categories.

This means that two units of length can be converted but length cannot be convert to weight.

Measurements & Conversions

Metric System

The metric system contains 3 types of units for each category it is representing.

Length: Metre(m)
Weight: Gram(g)
Volume: Litre(l)

These are called the base of each measurement. However, these are not enough as if measurements gets too big or small, the numbers will be hard to represent. That's where conversion come in and the reason metric system is the standard measurement is because the conversion are all the same based on dividing or multiply by 10 to convert.

It uses prefixes, which is the start of the word combined with the base. Prefixes like centi- or kilo- are added to base units (like metre, gram, or litre) to indicate the scale of measurement. These prefixes depend on what is being measured and help form units like centimetre or kilogram.

Unit (Prefix)	kilo-	hecto-	deka-	base unit	deci-	centi-	milli-	-	-	micro-
Value	1000	100	10	1	0.1	0.01	0.001	0.0001	0.00001	0.000001
Symbol	k	h	da	litre(l), metre(m), gram(g)	d	c	m			mc

Note: Each step on the metric scale involves multiplying or dividing by 10, except when converting from milli- to micro-. Since there are two steps between these units, the conversion factor is 1,000 instead of 10.

When moving right from the base unit on the metric scale (toward smaller units like centi-, milli-, etc.), you multiply by 10 at each step because smaller units mean more of them are needed to equal one of the larger units. When moving left (toward larger units like kilo-), you divide by 10 at each step because fewer of the larger units are needed.

This might feel a bit backwards at first. Why would you multiply when the units are getting smaller? But it makes sense: for example, centi- is smaller than the base unit, so 1 metre × 10 = 100 centimetres. The number itself gets bigger when converting from meters to centimetres.

Imperial System

The Imperial System is different from the metric system as the conversions involve memorizing specific values. The metric system involves converting by multiplying or dividing by 10 while conversion imperial system is more with memorizing.

Length	Weight	Volume
12 inches = 1 foot	16 ounces = 1 pound	3 teaspoons = 1 tablespoon
3 feet = 1 yard		2 tablespoons = 1 fluid ounce
		8 fluid ounces = 1 cup

Length

Units	Abbreviation
kilometre	km
hectometre	hm
dekametre	dam
metre	m
decimetre	dm
centimetre	cm
millimetre	mm
micrometre	mcm
inch	in
feet	ft
yard	yd

Weight

Units	Abbreviation
kilogram	kg
hectogram	hg
dekagram	dag
gram	g
decigram	dg
centigram	cg
milligram	mg
microgram	mcg
pound	lb
ounce	oz

Volume

Units	Abbreviation
kilolitre	kL
hectolitre	hL
dekalitre	daL
litre	L
decilitre	dL
centilitre	cL
millilitre	mL
microlitre	mcL
teaspoon	tsp
tablespoon	tbsp
cup	C
fluid ounce	fl oz
cubic centimetre	cc

Now we know how conversion works within the metric and imperial systems, we can begin to understand how conversions work between the two systems.

Length

The main length conversion to know from metric to imperial is from centimetres to inches.

2.54 centimetres = 1 inch

To convert other metric units to imperial, use a two-step process: first convert the metric measurement to centimetres, then convert the centimetres to inches.

Weight

The main weight conversion to know from metric to imperial is from kilograms to pounds.

1 kilogram = 2.2 pounds

To convert other metric units to imperial, use a two-step process: first convert the metric measurement to kilograms, then convert the kilograms to pounds.

Volume

There are four key volume conversions to remember from metric to imperial. One for each commonly used imperial unit of measurement.

Teaspoon	Tablespoon	Fluid Ounce	Cup
5 mL = 1 tsp	15 mL = 1 tbsp	30 mL = 1 fl oz	240 mL = 1 cup

To convert other metric units to imperial, use a two-step process: first convert the metric measurement to millilitres, then convert the millilitres to your chosen imperial unit of measurement.

Dimensional Analysis

Dimensional analysis is a method used to solve problems by converting between units. It works by setting up an equation where units cancel out systematically. Start by identifying the unit you have (given) and the unit you need (needed). Use a known conversion factor that relates the two. Set up an equation where "x" represents the value in the needed unit. Create a fraction using the conversion, placing the needed unit in the numerator and the given unit in the denominator. Multiply this fraction by the given value, cancel out the units, and solve for x.

Example

Example
What is 50kg in lbs?
Identify what is given and needed and find the conversion above that relates to both.	Given: 50 kg Needed: $x$ lbs Conversion: 1 kg = 2.2 lbs
Set up the equation using " $x$ " with the units you need. In this case, the answer is needed in pounds.	$x$ lbs
Create a fraction using the units from the conversion. The numerator is the needed unit and the denominator is the given unit. Now make the fraction equal to $x$	$x l b s = \frac{2.2 l b s}{1 k g}$
Multiply the fraction by the given value (50 kg) over 1 and solve for $x$ . Notice: The two kg cancel out because one is in the numerator and the other is in the denominator leaving only the lbs unit.	$x l b s = \frac{2.2 l b s}{1 k g} \times \frac{50 k g}{1}$ $x l b s = \frac{2.2 l b s}{1 k g} \times \frac{50 k g}{1}$ $x l b s = \frac{2.2 l b s \times 50}{1 \times 1}$ $x l b s = \frac{110}{1}$
Solution:	$x l b s = 110 l b s$

Proportions

The proportion method is used to solve conversion problems by setting up two equivalent ratios. A proportion compares two ratios that involve different units and makes them equal. To use this method, start by identifying the units involved (given and needed). Then, set up a proportion using a known conversion ratio and the values from your question. Solve for the unknown value, typically represented by x.

Example

Example
A string was measured to be 10 cm what is it in inches?
Identify what is given and needed and find the unit conversion that relates to both.	Given: 10 cm Needed: $x$ in Conversion: 1 in = 2.54 cm
Find the ratio for cm and in. If 1 inch = 2.54 cm, then the ratio is:	$1 in : 2.54 cm$
Place this ratio into a proportion to identify the equivalent value. Notice: The units must be in the same order in both ratios. If inches are listed first in the first ratio, then inches must also come first in the second. The structure of the ratios must match to set up a valid proportion.	$1 in : 2.54 cm :: x in : 10 cm$
To solve for $x$ , convert these ratios into fractions and solve using cross-multiplication. This helps isolate the unknown variable.	$\frac{1}{2.54} = \frac{x}{10}$ $2.54 x = 1 \times 10$ $\frac{2.54 x}{2.54} = \frac{1 \times 10}{2.54}$ $x = 3.94$
Solution:	$10 cm ≅ 4 in$

Ladder Method for Metric System

This method for conversion only works if it is all in the metric system. This is called the ladder method where you move the decimal depending on how you move on the ladder. This only works for metric system because conversions depend on dividing and multiplying by 10.

staircase with each step representing a metric prefix. From top down: kilo-, hecto-, deka-, base, deci-, centi-, milli-, micro-. every step UP the ladder, move decimal to the left. every step DOWN the ladder, move decimal to the right.

Example
Convert 10m to km.
Start at base on the ladder which is for meter. Since you are converting to kilometer, you need to jump to the left 3 times. This means that the decimal will jump left 3 times.
For 10m, move the decimal 3 places to the left, corresponding with the ladder. If there is a gap after moving the decimal, fill it in with a 0. Note: For all numbers, if a decimal point isn't shown, it's understood to be at the end of the number.
Solution:	10m = 0.010km

Conversion without Direct Conversion

Dimensional analysis and the proportion method often involve direct conversions—cases where a straightforward conversion factor exists, like 1 inch = 2.5 cm or 1 kg = 2.2 lbs. But what happens when there isn't a direct conversion available? For example, how do you convert teaspoons to cups? In these cases, you'll need to use intermediate steps, but the overall process remains quite similar.

Example
Convert 6tsp to fl oz.
Notice how there is no conversion to directly convert from tsp to fl oz but you can break down the problem in multiple steps. With the conversions, you can use dimensional analysis to convert into one equation.	$tsp→tbsp→fl oz$ 3 tsp = 1 tbsp and 2 tbsp = 1 fl oz
Start by writing $x$ with the unit you’re solving for. Since we want to convert teaspoons (tsp) to fluid ounces (fl oz), we’re solving for x fl oz. Set up the equation using a known conversion that includes fl oz.	$x fl oz = \frac{1 fl oz}{2 tbsp}$
Since we’re starting with teaspoons and only have a conversion between tablespoons and fluid ounces, use an additional conversion to bridge the gap from teaspoons to tablespoons before converting to fluid ounces. Use the conversion 1 tbsp = 3 tsp, placing tablespoons in the numerator to cancel with the denominator of the previous fraction, and then multiply the two conversion factors together.	$x fl oz = \frac{1 fl oz}{2 tbsp} \times \frac{1 tbsp}{3 tsp}$
Multiply the combined conversion factors by the given value (6 tsp) by placing 6 tsp over 1.	$x fl oz = \frac{1 fl oz}{2 tbsp} \times \frac{1 tbsp}{3 tsp} \times \frac{6 tsp}{1}$
Cancel the units—tsp with tsp and Tbsp with Tbsp—then solve to find the final result in fl oz.	$x fl oz = \frac{1 fl oz}{2 tbsp} \times \frac{1 tbsp}{3 tsp} \times \frac{6 tsp}{1}$ $x fl oz = \frac{1 fl oz \times 1 \times 6}{2 \times 3 \times 1}$ $x fl oz = \frac{6 fl oz}{6}$ $x fl oz = 6 fl oz$
Solution:	6 tsp = 1 fl oz.

Celsius is the metric unit for temperature, with water freezing at 0° and boiling at 100° under standard conditions. Fahrenheit is the imperial unit, where water freezes at 32° and boils at 212°.

System	Metric	Imperial
Value	Celsius	Fahrenheit
Symbol	°C	°F

Converting Temperatures

Since we know that 0 °C is equivalent to 32 °F, we can use this relationship as the baseline for converting temperatures between the metric and imperial systems. These formulas below adjust for the difference in starting points (0 °C vs. 32 °F) and account for the fact that each degree Celsius is equal to 1.8 degrees Fahrenheit.

Celsius to Fahrenheit

$° F = (° C \times 1.8) + 32$

To convert Celsius to Fahrenheit, first multiply the Celsius temperature by 1.8. Then, add 32 to the result.

Fahrenheit to Celsius

$° C = (° F - 32) \div 1.8$

To convert Fahrenheit to Celsius, subtract 32 from the Fahrenheit temperature first. Then, divide the result by 1.8.

Practice

Last Updated: Jun 5, 2025 4:16 PM
URL: https://tlp-lpa.ca/math-tutorials
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Math: Basic Tutorials

Math: Basic Tutorials : Measurements

Measurements

Measurements & Conversions

Metric System

Imperial System

Length

Weight

Volume

Length

Weight

Volume

Dimensional Analysis

Example

Proportions

Example

Ladder Method for Metric System

Conversion without Direct Conversion

Converting Temperatures

Celsius to Fahrenheit

Fahrenheit to Celsius

Practice