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.
Metric System
The metric system contains 3 types of units for each category it is representing.
Length: Meter(m)
Volume: Liter(l)
Weight: Gram(g)
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 prefix, which is the start of the word combined with the base.
| Unit(Prefix) | kilo- | hecto- | deka- | base | deci- | centi- | milli- | x | x | micro- |
|
Value |
1000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 | 0.000001 |
| Symbol | k | h | da |
liter meter gram |
d | c | m |
mc |
Starting at base if you move to the right it is multiplying by 10 for each step ( \( \times 10 \) ) as the units get smaller and if you move to the left it is dividing by 10 for each step ( \( \div 10 \) ) as the units get bigger.
This may seem counterintuitive (smaller units means you multiply) but it makes a lot of sense. If we compare base to centi-, the unit, centi- is smaller than base meaning more centi- fit into base, exactly 100 centi- fit into 1 base. So the number gets bigger as you convert the units smaller.
Note:
Every step is by dividing or multiplying by 10 except from milli- to micro-, this is because there are 2 placeholders between the two units meaning the conversion is by 1000 instead of 10.
Also the reason the units have - (for example centi- , kilo-) is that it depends on the thing we are measuring. For example, if we are measuring in length, the base is meter. So if you are using centi- , it is combined with the base (making centimeter) to specify what type of measurement it is. They are called prefixes which means they start as word, they are added to the front of the base depending on the type of measurement.
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
| 12 Inch = 1 Foot |
| 3 Feet = 1 Yard |
Notice that the order of units from smallest to biggest is inch, foot, then yard.
Weight
| 16 ounces = 1 pound |
Notice that the unit ounces is smaller than pound as it takes 16 units of ounces to make 1 pound.
Volume
| 3 teaspoons = 1 tablespoon |
|
2 tablespoons = 1 fluid ounce |
| 8 fluid ounces = 1 cup |
The order of units from smallest to biggest is teaspoons, tablespoons, fluid ounces, then cup.
Now we know the conversion within the metric system and imperial system, there are also conversions from metric to imperial and vise versa.
| 2.5 centimeter = 1 inch |
There is only 1 conversion from metric to imperial and it is from centimeter to inches. Inches is bigger than centimeter as 2.5 centimeter can fit into each inch. The process to convert other metric units to imperial is through 2-step conversion which will be shown underneath section. The idea is converting metric to centimeter first so that it can be converted to inches.
| 2.2 pounds = 1 kg |
Similar with length, there is only 1 conversion from metric to imperial system.
| 5 milliliter = 1 teaspoon |
| 15 milliliter = 1 tablespoon |
| 30 milliliter = 1 fluid ounce |
| 240 milliliter = 1 cup |
| 1 milliliter = 1 cubic centimeter |
Volume has 4 conversion for metric to imperial for the 4 different imperial measurements.
The units for measurements have long names so it's common practice to use abbreviations to shorten the words.
| Units | Abbreviation |
|---|---|
| kilometer | km |
| hectometer | hm |
| dekameter | dam |
| meter | m |
| decimeter | dm |
| centimeter | cm |
| millimeter | mm |
| micrometer | mcm |
| inch | in |
| feet |
ft |
| yard | yd |
| Units | Abbreviation |
|---|---|
| kilogram | kg |
| hectogram | hg |
| dekagram | dag |
| gram | g |
| decigram | dg |
| centigram | cg |
| milligram | mg |
| microgram | mcg |
| pounds | lb |
| ounce |
oz |
| Units | Abbreviation |
|---|---|
| kiloliter | kL |
| hectoliter | hL |
| dekaliter | daL |
| liter | L |
| deciliter | dL |
| centiliter | cL |
| milliliter | mL |
| microliter | mcL |
| teaspoon | tsp |
| tablespoon |
tbsp |
| cup | C |
| fluid ounce | fl oz |
| cubic centimeter | cc |
Dimensional Analysis is a method that uses different units to create an equation where it involves cancelling out units. First, identify the two units needed in the question and find the conversion above. Know what unit is given and what unit is needed. To create the equation, let x have the units for what is needed. Then let that equal to the fraction where you fill in the conversion values above with the numerator being the unit for needed and the denominator being the unit for given. Multiply the fraction by the value that is given and solve for x.
Example 1:
What is 50kg in lbs?
1) Identify what is given and needed and find the conversion above that relates to both.
Given: 50 kg
Needed: x lbs
The conversion found above says:
1 kg = 2.2 lbs
2) Step up the equation with x with the units that is needed.
Notice that the needed is in lbs so:
x lbs
3) Let it equal to the fraction with the conversion where the numerator is the needed unit and the denominator is the given unit.
The conversion is fraction will be:
\( \cfrac{ \text{2.2 lbs}}{\text{1 kg}} \)
I placed lbs in the numerator because it is the unit for the needed and the kg in the denominator as it is the unit for given. These numbers were from the conversion:
1 kg = 2.2 lbs
Now make the fraction equal to x:
\( \text{x lbs} = \cfrac{ \text{2.2 lbs}}{\text{1 kg}} \)
4) Now multiply the fraction by the given value over 1 and solve for x.
Given: 50 kg
\( \text{x lbs} = \cfrac{ \text{2.2 lbs}}{\text{1 kg}} \times \cfrac{ \text{50 kg}}{\text{1}}\)
Notice: The two kg cancel out because one is in the numerator and the other is in the denominator leaving only the lbs unit.
\( \text{x lbs} = \cfrac{ \text{2.2 lbs}}{\text{1} \cancel{ \text{kg}}} \times \cfrac{ \text{50} \cancel{ \text{kg}}}{\text{1}}\)
\( \text{x lbs} = \cfrac{ \text{2.2 lbs} \times 50}{1 \times 1} \)
\( \text{x lbs} = \cfrac{\text{110 lbs}}{1} \)
\( \text{x lbs} = \text{110 lbs} \)
Hence the value is 110lbs. 50kg in pounds is 110lbs.
To use the proportion method, first read the ratio and proportion section to learn what proportion is and how to use it. A proportion, in simple terms, is comparing 2 ratios making them equivalent where both ratios compare 2 different units.
To use the proportion method, first you need to identify what you need to find and what is given. There will always be 2 units for conversions, find the ratio in the chart above. Then place them in a proportion and solve for the unknown or x.
Example:
A string was measured to be 10 cm what is it in inches?
Answer:
1) Identify what is given and what is needed.
Given: 10cm
Needed: x in
2) Find the conversion ratio in the chart above.
Find the ratio for cm and in. It is in the section for conversion between metric and imperial. It states that:
1 in = 2.5 cm
Hence the ratio is 1 in : 2.5 cm
3) Place in a proportion and solve.
The proportion is:
1 in : 2.5 cm :: x in : 10 cm
Notice that units have to match, if inch is place first then the second ratio has to put inch first. The ratios have to match. Solve for x. Inside multiply inside equals outside multiply outside.
\( \ 2.5x = 1 \times 10 \)
\( \cfrac{2.5x} {2.5} = \cfrac {1 \times 10} {2.5} \)
\( x = 4 \)
Hence 10 cm converted to inches is 4 inches.
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.
Example
Convert 10m to km.
1) 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.
So move the decimal 3 times to the left for 10m. For all numbers, if the decimal isn't there then it is at the end of the number.
If there is a gap, then add a 0 to fill it in. The answer will be:
0.010 km
The above examples with dimensional analysis and proportion method only showed situations where there is a direct conversion. This means that there exists a conversion to directly convert. Like 1 in = 2.5 cm or 1kg = 2.2 lbs. What happens if you want to convert without a direct conversion? For example, how do you convert from a tsp to a cup? It requires intermediate steps but it's quite similar to the steps above.
Example
Convert 6 tsp 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.
\( \text{tsp} \rightarrow \text{tbsp} \rightarrow \text{fl oz} \)
3 tsp = 1 tbsp and 2 tbsp = 1 fl oz
With the conversions, you can use dimensional analysis to convert into one equation.
1) Write x with the unit that is needed, and make it equal to the fraction with the conversion that relates to that needed unit.
Notice that we need tsp to be in fl oz so the needed unit is fl oz.
x fl oz
Now create the fraction that has fl oz which is 2 tbsp = 1 fl oz and match with the needed unit on top.
\( \cfrac{\text{1 fl oz}} { \text{2 tbsp}} \)
\(\text{x fl oz} = \cfrac{\text{1 fl oz}} { \text{2 tbsp}} \)
2) Now you will need to use the other conversion to fill the gap as if you place the given value, it doesn't cancel with tbsp.
\(\text{x fl oz} = \cfrac{\text{1 fl oz}} { \text{2 tbsp}} \)
Notice that the tbsp is in the denominator, to cancel out tbsp, the next fraction needs to have tbsp in the numerator.
\(\text{x fl oz} = \cfrac{\text{1 fl oz}} { \text{2 tbsp}} \times \cfrac{\text{1 tbsp}} { \text{3 tsp}}\)
3) You can multiply now with what is given over 1 and solve.
Given: 6 tsp
\(\text{x fl oz} = \cfrac{\text{1 fl oz}} { \text{2 tbsp}} \times \cfrac{\text{1 tbsp}} { \text{3 tsp}} \times \cfrac{\text{6 tsp}}{1} \)
Notice: All the units cancel except fl oz.
\(\text{x fl oz} = \cfrac{\text{1 fl oz}} { \text{2} \cancel{\text{tbsp}}} \times \cfrac{\text{1} \cancel{\text{tbsp}}} { \text{3} \cancel{\text{tsp}}} \times \cfrac{\text{6} \cancel{\text{tsp}}}{1} \)
\(\text{x fl oz} = \cfrac{\text{1 fl oz} \times 1 \times 6} {2 \times 3 \times 1} \)
\(\text{x fl oz} = \cfrac{\text{6 fl oz}} {6} \)
\(\text{x fl oz} = \text{1 fl oz} \)
Hence 6 tsp is 1 fl oz.