Unit 6: Measurement
Topic D: Conversion within the Metric System
In this topic, you will learn a quick method to change (convert) between different units with the same base. In the conversion, the number and the prefix both change; the length or mass or volume of the object is not changed – only the way we express the measurement changes.
Are you a visual learner? If you are, then ask your instructor to show you the next skill. It will save you a lot of frustration. You may learn this skill much faster with a real life example.
Converting Within the Metric System Using the Chart
| Metric Prefixes | kilo | hecto | deca | BASE UNIT | deci | centi | milli |
|---|---|---|---|---|---|---|---|
| Mass | kg | hg | dag | g | dg | cg | mg |
| Volume | kL | hL | daL | L | dL | cL | mL |
| Length | km | hm | dam | m | dm | cm | mm |
| Place Value | 1000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 |
Example A
A book weighs 12 g. Convert this amount to mg.
- Step 1 – If there is no decimal point in the amount, place a . after the amount. 12 g = 12. g
- Step 2 – Locate the prefix of the known amount. If no prefix is given, find the base unit (gram in the example) in the centre of the chart.
- Step 3 – Find the prefix that you are changing to (milligram in this example). It is to the right of the gram. Count the number of bars ( | ) between gram and milli. You cross three bars to move three places to the right.
- Step 4 – Move the decimal point the same number of places in the same direction as you moved on the chart. Add zeros as needed.
12. g = 12000. mg The cigar is 12000 mg.
On the chart, every time you cross over a bar ( | ), the factor is 10.
- If you cross a bar going from the left to the right → multiply by 10. The units to the right are smaller, so more are needed to make an equal amount. Crossing 3 bars is the same as multiplying by 1000 (10 × 10 × 10).
- If you cross a bar going from the right to the left ← divide by 10. The units to the left are larger, so less are needed to make an equal amount.
Review Multiplying by 10, 100, 1000 in Unit 3, Topic A.
Example B
A room measures 450 cm long. Convert this measurement to m.
- Step 1 – Place a decimal point after the known amount if needed. 450. cm
- Step 2 – Find the prefix of the known amount on the chart. Find centi.
- Step 3 – Find the prefix or the base unit (if no prefix was used) of the unit you are changing to. Is it left or right of centi? Count the bars between cm and metre. You cross two bars to move two places to the left. That is the same as dividing by 100.
- Step 4 – Move the decimal point the same number of places in the same direction as you moved on the chart. Add zeros as needed. 450. ← cm = 4.50 m × 450 cm = 4.5 m
Example C
The container holds 45.5 dL. Write this amount in daL.
- Step 1 and 2 – A decimal point is already in the amount. Find deci on the chart.
- Step 3 – Find deca on the chart. Count the number of bars you cross going from deci to deca – 2 bars to move 2 places to the left (divide by 100).
- Step 4 – Move the decimal point 2 places to the left. 45.5 dL = 0.455 daL (less than 1 daL)
Exercise 1
Complete the metric conversions. Some units are not common, but the practice in conversion is useful.
- From memory, put the metric prefixes on the chart according to their place value. Check that your chart is correct before you use it.
- 42 cm = m; 8241 m = km
- 358 mm = cm; 5 hm = m
- 0.87 m = mm; 0.5 kg = g
- 197 cm = m; 4.5 kg = dag
- 28 m = km; 890 dL = kL
- 8 L = mL; 85 km = m
- Add 45 cm and 92 cm. Express the sum in metres.
- Example: 45 cm + 92 cm = 137 cm; 137 cm = 1.37 m
- Add 245 m, 689 m, and 124 m. Express the sum in kilometres.
- Multiply 250 mL by 6. Express the product in litres.
Answers to Exercise 1
-
kilo hecto deca base unit deci centi milli - 0.42 m; 8.241 km
- 35.8 cm; 500 m
- 870 mm; 500 g
- 1.97 m; 450 dag
- .028 km; 0.089 kl
- 8000 mL; 85000 m
- See example
- 1.058 km
- 1.5 L
Why Do We Need to Convert Measurements?
The skill of converting within the metric system is very useful.
- Before we can do any math with measurements we must be sure the measurements are all in the same unit value. For example, we can only subtract litres from litres, multiply metres by metres, add milligram to milligrams.
- Measurements are usually written with small whole numbers. This is the simple form of the measurement. For example,
- instead of 4587 g, write 4.587 kg
- instead of 52000 mL, write 52 L
- instead of 0.0065 m, write 6.5 mm
Example D
50 g − 275 mg = ?
Convert 50 g to mg: 50 g = 50000 mg
Subtract [latex]\begin{array}[t]{rr}&50\,000\text{ mg}\\-&275\text{ mg}\\ \hline &49\,725\text{ mg}\end{array}[/latex]
OR
Convert 275 mg to g: 275 mg = 0.275 g
Subtract (add a decimal and zeros to make subtraction easier)
Subtract [latex]\begin{array}[t]{rr}&50.000\text{ g}\\-&0.275\text{ g}\\ \hline &49.725\text{ g}\end{array}[/latex]
Example E
Jill wants to put lace around her tablecloth. The bottom of the table cloth measures 2.6 m around. The lace trim is packaged in 75 cm lengths. How many packages of lace will Jill need to buy so she can trim the tablecloth?
First, convert the measurements to the same values. 2.6 m = 260 cm
This is a division problem. How many groups of 75 cm are in 260 cm? 260 cm ÷ 75 cm = 3.47
Jill will need to buy 4 packages. (She needs more than 3 packages and cannot buy a part of a package.)
Exercise 2
Convert as needed to solve these word problems.
- Complete this chart from memory for your use. Check that it is correct.
- The new refrigerator is 175 cm high. The directions say that 10 cm must be left above the refrigerator for air circulation. The height of the space for the refrigerator is 1.9 m. Will the refrigerator fit?
- The stairway is 89 cm wide. Bob is installing a carpet runner. The runner comes 1m wide. How much must be trimmed to make it fit on the stairway?
- Julia is calculating how much juice to buy for the children’s school party. How many 250 mL cups will she be able to fill from a 4 L bottle of juice?
- Charles is 1.67 m tall. His wife Laura is 145 cm tall. How much taller is Charles than his wife?
Answers to Exercise 2
-
kilo hecto deca deci centi milli - Yes, with 5 cm to spare.
- 11 cm
- 16 glasses
- 22 cm taller
Only use one unit for a measurement.
For example, use
- 2.75 m not 2 m, 75 cm
- 60.5 kg not 60 kg, 500 g
- 4.25 L not 4 L, 250 mL
When there is a mixed measurement such as shown in the examples, do this:
- convert the amount with the smaller unit value to the larger unit value (it will often be a decimal)
- add the amounts together
Example F
16 cm + 4 mm
4 mm = 0.4 cm
16 cm + 0.4 cm = 16.4 cm
Example G
1 km + 350 m
350 m = 0.350 km
1 km + 0.35 km = 1.35 km
Exercise 3
Write these measurements using only one unit.
- 5 L + 750 mL =
- 3 kg + 150 g =
- 1 m + 5 cm =
- 5 m + 7 dm =
- 6 m + 345 cm =
Answers to Exercise 3
- 5.75 L
- 3.15 kg
- 1.05 m
- 5.7 m
- 9.45 m
Exercise 4
Here is more conversion practice.
- 3.2 km = m
- 0.006 m = mm
- 1.64 kg = g
- 155 g = hg
- 2 m + 16 cm = m
- 1 L + 50 mL = L
- 89 m = km
- 457 m = hm
Watch for different units! Use the simplest form for the answer.
- [latex]\begin{array}{rr}&674 \text{ mm}\\+&86\text{ cm}\\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}{rr}&5.5 \text{ g}\\-&40\text{ dg}\\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}{rr}&45 \text{ mL}\\+&16\text{ cL}\\ \hline\\ \end{array}[/latex]
- 9954 mL − 8.9 L =
- 128 hm + 4 km =
Answers to Exercise 4
-
kilo hecto deca deci centi milli
- 3200 m
- 6 mm
- 1640 g
- 1.55 hg
- 2.16 m
- 1.05 L
- 0.089 km
- 4.57 hm
- 153.4 cm (1534 mm)
- 9.5 g (95 dg)
- 205 mL (20.5 cL)
- 1.054 L
- 16.8 km
Dividing Two Items of the Same Units
Heads up – a new important twist for you!
When you are dividing two items of the same units, the units cancel themselves out. This means that your answer will not have a unit.
Follow this example:
- 5000 g ÷ 40 g = 125 (no units in the answer!)
- 880 cm ÷ 11 mm = 8800 mm ÷ 11 mm = 800 (no units!)
Exercise 5
- 6000 g ÷ 250 g =
- 3.38 m ÷ .13 m =
- 6 km ÷ .3 km =
Answers to Exercise 5
- 24
- 26
- 20
Topic D: Self-Test
Mark /16 Aim 13/16
- Give the measurement (unit with prefix as needed) that would be most practical to measure these items. (6 Marks)
- a child‘s height
- a big bag of flour
- the distance from Ottawa to Toronto
- a box of oranges
- the distance from your seat to the door
- the flavouring to put in the cake batter
- Complete the metric conversions. (5 Marks)
- 8 m = cm
- 5.2 hm = km
- 4.2 kg = g
- 242 dag = kg
- 28 mm = cm
- Calculate. Express the answer in simplest form. Watch the prefixes! (5 marks)
- 8.2 L − 48 mL =
- 42 mg + 2 dg =
- 0.8 m ÷ 20 cm =
- You need a strip of metal that is 97 cm in length. The piece of metal that you found in the workshop is 1.3 m. How much must be cut off the end to give you a 97 cm strip? (2 marks)
Answers to Topic D Self-Test
- Give the measurement that would be most practical.
- cm
- kg
- km
- kg
- m
- mL
- Complete the metric conversions.
- 800 cm
- 0.52 km
- 4200 g
- 2.42 kg
- 2.8 cm
- Calculate.
- 8.152 L
- 242 mg
- 4 cm
- 33 cm
Metric System vs. Imperial System
Originally, people would measure things compared to their body parts.
- In French, the word for inch is pouce, which means thumb. So, really, an inch came from the measurement of a thumb.
- We still use the foot for measurement. It came from the measurement of an average person‘s foot.
- If you have ever heard anyone talking about horses, you may have heard about a horse being a certain number of hands tall.
But the imperial system has problems. Measuring things with your own body is not practical because we are all different shapes. And if you have ever tried to divide a foot into 5 equal parts, you will know that it is not easily done. (A foot is 12 inches, which is not easily divided into 5 equal parts). This problem is found with almost all measurements in the imperial system.
Then, the International System (also known as Metric) was created to make it even easier for people to work with measurements. It is made on a Base Ten System. The Base Ten System is another name for the decimal number system that we use every day. Because we already use the Base Ten System as our decimal system, which many cultures around the world use, it is easy to measure things and divide them up or add them together.
Here are some of the measurements that you may see in the Imperial System and the Metric System:
| Measurement | Imperial System | International System (Metric) |
|---|---|---|
| Length | Inch, foot, yard, mile | Millimetre, centimetre, metre, kilometre |
| Mass | Ounce, pound, ton | Milligram, gram, kilogram |
| Volume | Fluid ounce, cup, pint, quart, gallon | Millilitre, litre, kilolitre |
Here are some conversions between the two systems:
| Imperial System | International System (Metric) |
|---|---|
| 1 inch | 2.54 cm |
| 1 foot | 0.30 m |
| 1 mile | 1.61 km |
| 1.09 yards or 3.28 feet | 1 m |
| 0.62 miles | 1 km |
| Imperial System | International System (Metric) |
|---|---|
| 1 ounce | 28.35 g |
| 1 pound | 0.45 kg |
| 0.04 ounces | 1 g |
| 2.20 pounds | 1 kg |
| Imperial System | International System (Metric) |
|---|---|
| 1 fluid ounce | 29.57 mL |
| 1 quart | 0.95 L |
| 1 gallon | 3.79 L |
| 0.03 fluid ounces | 1 mL |
| 1.06 quarts | 1 L |
You may find this is useful information. It is not necessary to learn or memorize any of the above numbers.