Transcript Converting Between Unit Systems

Converting Between
Unit Systems
Lesson 4.3.3
1
Lesson
4.3.3
Converting Between Unit Systems
California Standard:
What it means for you:
Algebra and Functions 2.1
Convert one unit of measurement
to another (for example, from feet to
miles, from centimeters to inches).
You’ll convert between the
customary and metric unit
systems.
Key words:
•
•
•
customary system
metric system
conversion
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Lesson
4.3.3
Converting Between Unit Systems
So far, you’ve converted metric units to other metric
units, and customary units to other customary units.
This Lesson, you’ll use conversion tables for converting
between the two unit systems.
You’ll see that all the techniques from the previous
Lessons work in exactly the same way in this
Lesson too.
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Lesson
4.3.3
Converting Between Unit Systems
Use the Conversion Tables to Help You
The tables below show conversion factors you can use to
convert between customary and metric units.
Customary to Metric
Metric to Customary
1 inch (in.) = 2.54 cm
1 foot (ft) = 30.48 cm
1 yard (yd) = 0.91 m
1 mile (mi) = 1.61 km
1 cm = 0.39 inches (in.)
1 cm = 0.033 feet (ft)
1 m = 1.09 yards (yd)
1 km = 0.62 miles (mi)
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Lesson
4.3.3
Converting Between Unit Systems
You can convert feet to centimeters
by multiplying by 30.48 (to give
you a bigger number, since feet are
bigger than centimeters).
Customary to Metric
1 inch (in.) = 2.54 cm
1 foot (ft) = 30.48 cm
1 yard (yd) = 0.91 m
1 mile (mi) = 1.61 km
But you can also convert feet to
centimeters by dividing by 0.033
(which also gives you a bigger
number, since 0.033 is less than 1).
Metric to Customary
1 cm = 0.39 inches (in.)
1 cm = 0.033 feet (ft)
1 m = 1.09 yards (yd)
1 km = 0.62 miles (mi)
So you only really need one of the conversion tables.
If you know the conversion factors in one table, you can use them
to convert from metric to customary or from customary to metric.
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Lesson
4.3.3
Example
Converting Between Unit Systems
1
Convert 10 km to miles.
Solution
There are two ways you could use the tables to get your
answer.
(i) Multiply by 0.62 (to get a smaller number, since miles are
a bigger unit than kilometers).
So 10 km = 10 × 0.62 miles = 6.2 miles
(ii) Divide by 1.61.
So 10 km = 10 ÷ 1.61 miles = 6.21 miles (to 2 decimal places)
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Solution follows…
Lesson
4.3.3
Example
Converting Between Unit Systems
2
Convert 1 mile to meters.
Solution
From the table, you can see that
1 mile is 1.61 km.
And you’ve already seen that to
convert kilometers to meters you
multiply by 1000.
Customary to Metric
1 inch (in.) = 2.54 cm
1 foot (ft) = 30.48 cm
1 yard (yd) = 0.91 m
1 mile (mi) = 1.61 km
So 1 mile = 1.61 × 1000 meters = 1610 meters.
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Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
× 10
× 2.54 or ÷ 0.39
÷ 10
× 12
This diagram
might be helpful for
some conversion
questions.
÷ 2.54 or × 0.39
÷ 12
÷ 1.09 or × 0.91
÷3
÷ 1000
×3
÷ 1.61 or × 0.62
× 1.09 or ÷ 0.91
× 1000
× 1.61 or ÷ 0.62
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Lesson
4.3.3
Converting Between Unit Systems
Most of the conversion factors for converting between
metric and customary systems are only approximations.
Most of them are only given to two decimal places.
This means your answer won’t always be exact.
You can sometimes get slightly different answers if you
do the question in different ways.
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Lesson
4.3.3
Example
Converting Between Unit Systems
3
Convert 1 yard into meters by:
(i) converting yards to inches, inches to centimeters, and then
centimeters to meters,
(ii) using the conversion factor 0.91,
(iii) using the conversion factor 1.09.
Comment on your answers.
Solution
(i) Do the conversion in three stages:
1) yards to inches: 1 yard = 36 inches
2) inches to cm: 36 inches = 36 × 2.54 cm = 91.44 cm
3) cm to meters: 91.44 cm = 91.44 ÷ 100 m = 0.9144 m
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Solution
continues…
Solution
follows…
Lesson
4.3.3
Example
Converting Between Unit Systems
3
Convert 1 yard into meters by:
(i) converting yards to inches, inches to centimeters, and then
centimeters to meters,
(ii) using the conversion factor 0.91,
(iii) using the conversion factor 1.09.
Comment on your answers.
Solution (continued)
(ii) 1 meter is slightly bigger than a yard, so multiply by 0.91
to make your number smaller.
1 yard = 1 × 0.91 m = 0.91 m
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Solution continues…
Lesson
4.3.3
Example
Converting Between Unit Systems
3
Convert 1 yard into meters by:
(i) converting yards to inches, inches to centimeters, and then
centimeters to meters,
(ii) using the conversion factor 0.91,
(iii) using the conversion factor 1.09.
Comment on your answers.
Solution (continued)
(iii) Divide by 1.09 to make the number bigger.
1 yard = 1 ÷ 1.09 m = 0.91743… m
The three answers are slightly different. This is because the
conversion factors used are only approximations.
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Lesson
4.3.3
Converting Between Unit Systems
Guided Practice
Convert the following:
1. 6 meters to yards 1 m = 1.09 yd, so 6 × 1.09 = 6.54 yards
2. 18 yards to meters
1 yd = 0.91 m, so 18 × 0.91 = 16.38 m
3. 3 feet to centimeters
1 ft = 30.48 cm, so 3 × 30.48 = 91.44 cm
1 mi = 1.61 km, so 2 × 1.61 = 3.22 km
1 km = 1000 m, so 3.22 × 1000 = 3220 m
1 km = 1000 m, so 2 × 1000 = 2000 m
kilometers to yards 1 m = 1.09 yd, so 2000 × 1.09 = 2180 yd
= 2.54 cm, so 6 × 2.54 = 15.24 cm
inches to millimeters 11 in
cm = 10 mm, so 15.24 × 10 = 152.4 mm
yards to centimeters 1 yd = 0.91 m, so 9 × 0.91 = 8.19 m
1 m = 100 cm, so 8.19 × 100 = 819 cm
4. 2 miles to meters
5. 2
6. 6
7. 9
8. 24 feet to dekameters
1 ft = 30.48 cm, so 24 × 30.48 = 731.52 cm
1 cm = 0.001 dam, so 731.52 × 0.001 = 0.7315 dam 13
Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
Guided Practice
9. Josh runs a marathon of 26 miles and 385 yards.
How far is this in kilometers?
26 × 1760 + 385 = 46,145 yd
= 46,145 × 0.91 ÷ 1000 km = 41.99 km (to 2 d.p.)
10. A boat race is 4 miles and 374 yards.
How long is the race in meters?
4 × 1760 + 374 = 7414 yd
= 7414 × 0.91 m = 6746.74 m (to 2 d.p.)
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Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
You Can Use Proportions Too
The examples so far have been done by reasoning
whether to multiply or divide by the conversion factor.
But you could do them using proportions if you prefer.
Just use the numbers in the conversion tables as ratios.
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Lesson
4.3.3
Example
Converting Between Unit Systems
4
Convert 10 km to miles.
Solution
This is the same as Example 1, but this time it’s done with
proportions.
As always when using proportions, you need two ratios.
●
●
The first ratio of miles to kilometers comes from the table:
1 : 1.61, or 1
1.61
The second ratio involves the measurement you want to
convert. Call the converted distance d miles. The ratio is:
d : 10, or d
16
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Solution
follows…
Solution
continues…
Lesson
Converting Between Unit Systems
4.3.3
Example
4
Convert 10 km to miles.
Solution (continued)
Ratios:
d
1
and
1.61
10
Now write a proportion and solve by cross-multiplication:
d
1
=
1.61
10
1.61d = 10
Cross-multiply
d = 10 ÷ 1.61
Divide both sides by 1.61
d = 6.21 miles (to 2 decimal places)
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Lesson
4.3.3
Converting Between Unit Systems
Guided Practice
Convert the following using proportions:
11. 10 meters to yards
1
10
Call the converted distance x yd,
=
x
1.09
So, x = 10.9 yd.
12. 3 yards to meters
Call the converted distance x m,
So, x = 2.73 m.
1
3
=
x
0.91
13. 3.7 feet to centimeters
1
3.7
Call the converted distance x cm,
=
x
30.48
So, x = 112.8 cm.
14. 5.1 miles to meters
Call the converted distance x m,
So, x = 8211 m.
1
5.1
=
x
1610
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Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
Convert Weight and Time in Exactly the Same Way
So far, you’ve only looked at converting lengths.
But you can convert time and weight (and most other kinds
of quantities) in exactly the same way.
You just need to know the conversion factor.
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Lesson
4.3.3
Example
Converting Between Unit Systems
5
Convert 1 hour to seconds.
Solution
Do this in two stages — hours to minutes, and then minutes
to seconds.
There are 60 minutes in an hour, and 60 seconds in a minute,
so the conversion factor for both stages is 60.
1) Minutes are a smaller unit than hours, so multiply by 60 to
get a bigger number: 1 hour = 1 × 60 = 60 minutes
2) Seconds are a smaller unit than minutes, so again multiply
by 60: 60 minutes = 60 × 60 = 3600 seconds
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Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
Guided Practice
Use the information below to convert the quantities that follow.
1 kg = 2.2 lb, 1 gallon = 3.79 liters
15. 3 kg to lb
1 kg = 2.2 lb, so 3 × 2.2 = 6.6 lb
16. 16 lb to kilograms
1 lb = (1 ÷ 2.2) kg = 0.4545 kg, so 16 × 0.4545 = 7.27 kg
17. 7 gallons to liters
1 gallon = 3.79 liters, so 7 × 3.79 = 26.53 liters
18. 44 liters to gallons
1 liter = (1 ÷ 3.79) gallons = 0.2639 gallons,
so 44 × 0.2639 = 11.61 gallons
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Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
Independent Practice
In Exercises 1–6, find the missing length.
Give your answers to 2 decimal places.
1. 4.5 in. = ? cm
3. 14 yd = ? m
11.43
12.74
5. 500 cm = ? yd
5.45
2. 100 mm = ? in.
4. 0.5 m = ? in.
6. 45 ft = ? m
3.9
19.62
13.72
7. The length of a model plane is 56 inches.
How long is the model in meters?
1.42 m (to 2 decimal places)
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Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
Independent Practice
8. Michaela and Ricky are each trying to guess how many
feet are in a kilometer. Michaela guessed 3000 and
Ricky guessed 3500. Whose guess was closest to the
correct number of feet?
Actual answer is 3281 (to nearest whole number) so Ricky
is (slightly) closer.
9. How many kilometers are there in 215,820 inches?
5.48 km (to 2 decimal places)
10.The dimensions of Zak’s bedroom are
5 ft × 8 ft. What are the dimensions of
his room in meters?
1.52 m × 2.44 m (both to 2 decimal places)
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Solution follows…
Lesson
4.3.3
Converting Between Unit Systems
Round Up
You’ve now seen how to convert between different
unit systems of length.
And if you can do that, you can also convert pretty
much anything else you want.
For example, you might want to convert dollars to
some other currency if you travel overseas.
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