Math Skills – Week 7
Class project due next week
Sample final exams available on website
Reducing fractions, rates, and ratios
$500
huh?
17/30
hmmmmmmm
Section 8.1 – Length
Section 8.2 – Weight
Section 8.3 – Capacity
Section 8.4 – Time
U.S. Customary System
Set of standard units used in the United States
established to simplify trade and commerce.
Set of standard units to describe all units of measure
length, weight, capacity, time, energy, power, etc.
A measurement includes a number and a unit
numbers
3 feet
7 miles
12 yards
units
U.S. Customary System units of length are:
1.
Inch, foot, yard, and mile
Equivalences between Units of Length in the U.S. Customary System:
12 inches (in) = 1 foot (ft)
3 ft = 1 yard (yd)
36 in = 1 yd
5280 ft = 1 mile (mi)
We often need to convert from one unit of measure to
another.
For example, you are sewing a garment that calls for 50 in of
fabric. The fabric store sells fabric at $2 per yd.
Question: What do we need to know in order to determine how
much money you will need to purchase enough fabric to finish
your garment?
To figure this out, we need to form Conversion Rates.
Conversion rates???
Relationships between units
Used to change a measurement from one unit to another.
For example, since 3 ft = 1 yd, the conversion rates are:
3 ft
1 yd
and
1 yd
3 ft
Note that these are both equivalent to 1. Keep this in mind
Why?
Converting between units of measure
Steps
1. Construct the conversion rates.
2. Multiply the measurement by the appropriate conversion
rate.
Note: Make sure you use the conversion rate which has the unit
in the numerator that you want in your answer
1.
Examples
Convert 27 ft to yards
1.
Conversion rates are
3 ft/1 yd and 1 yd/3 ft
Which one do we choose?
27 ft x 1 yd/3 ft = 9 yd
Examples
1.
Convert 5 yards to feet
Conversion rates are 3 ft/1 yd and 1 yd/3 ft
Which one do we choose?
5 yd x 3 ft/1 yd = 15 ft
More Examples
1.
Convert 40 in to feet.
2.
Conversion rates are 12 in/1 ft and 1 ft/12 in
40 in x 1 ft/12 in = 40 ft/12 in = 3 1/3 ft
Convert 3 ¼ yd to feet
Conversion rates are 3 ft/1 yd and 1 yd/3 ft
3 ¼ yd = 13/4 yd = 13/4 yd x 3 ft/1 yd = 39/4 ft = 9 ¾ ft
Class Examples
1.
Convert 14 ft to yards
2.
Conversion rate: 1 yd/3 ft
14 ft x 1 yd/3 ft = 4 2/3 yd
Convert 10560 ft to miles
Conversion rate: 1 mi /5280 ft
10560 x 1 mi/5280 ft = 2 mi
Writing measurements in simplest form
Before we had:
40 in = 3 1/3 ft
3 ___in
4
If we want to convert 40 in to ___ft
Divide 40 by 12 (since there is 12 inches in 1 ft.)
This will tell us how many feet there are.
Any remainder is the number of inches.
Examples
1.
17 in = ___ft ___in
1 ft 5 in
31 ft = ___yd ___ft
2.
10 yd 1 ft
Class Examples
1.
Convert: 42 in = ___ft ___in
2.
3 ft 6 in
Convert: 14 ft = ___yd ___ft
4 yd 2 ft
Arithmetic operations with measurements (Addition)
Steps
1. Add the quantities of the measurement that have the same
units
4 ft 4in
+ 1 ft 11 in
5 ft
2.
15 in
Write the sum in simplest form
5 ft 15 in = 5 ft 12 in 3 in = 6 ft 3 in
+
1 ft
=
Arithmetic operations with measurements
(Subtraction)
Steps
1. Subtract the quantities of the measurement that have the
same units . (Borrowing from the ft. may be necessary)
8 ft
12 in
9 ft 6 in
- 3 ft 8 in
5 ft
2.
18 in
10 in
Write the measurement you get for your answer in simplest
form
5 ft 10 in
Arithmetic operations with measurements
(Multiplication)
Steps
1. Multiply each of the quantities in the measurement by the
desired factor. Carry units along
4 ft 5in
x
3
12 ft 15 in
1.
Write the measurement you get for your answer in simplest
form
12 ft 15 in = 12 ft 12 in 3 in = 13 ft 3 in
Arithmetic operations with measurements (Division)
Division is a little trickier…best illustrated by an example
1. Find the quotient of 4 ft 3in and 3
Method 1 – Direct Division
Method 2 – Convert first
1.
2.
3.
Convert to inches
Do division
Write answer in simplest form
17in or 1 ft 5 in
Examples
1.
Find the sum of 3 ft 5 in and 4 ft 9 in
2.
Subtract: 4 ft 2 in – 1 ft 8 in
3.
7 ft 14 in = 7 ft 12 in 2 in = 7 ft 1 ft 2 in = 8 ft 2 in
2 ft 6 in
Multiply 4 yd 1 ft x 8
32 yd 8 ft
4. Find the quotient of 7 yd 1 ft and 2
3 yd 2 ft
Class Examples
1.
5 ft 9 in + 4 ft 5 in
2.
10 yd 1 ft – 2 yd 2 ft.
3.
9 ft 14 in = 9 ft 12 in 2 in = 9 ft 1 ft 2 in = 10 ft 2 in
7 yd 2 ft
2 ft 7 in x 5
10 ft 35 in = 10 ft 2 ft 11 in = 12 ft 11 in
4. 5 ft 9 in ÷ 3
1 ft 11 in
U.S. Customary System units of Weight are:
Ounce, Pound, and ton
Equivalences between Units of Weight in the U.S. Customary System:
16 ounces (oz) = 1 pound (lb)
2000 lb = 1 ton
We can use these equivalences to construct conversion
rates
Example: What are the conversion rates for ounces and
pounds?
16 oz
1 lb
and
1 lb
16 oz
Example:
1.
Convert 62 oz to pounds
2.
Convert 3 ½ tons to pounds.
3.
62 oz x 1 lb/16 oz = 62/16 lb = 3 7/8 lb
Conversion rate: 2000 lb/1 ton
3 ½ tons = 7/2 tons = 7/2 tons x 2000 lbs/1 ton = 7000 lbs
Convert 42 oz to pounds
Conversion rate: 1 lb/16 oz
42 oz x 1 lb/16 oz = 42/16 lb = 2 5/8 lb
Class Examples:
1.
Convert 3 lbs to ounces
2.
Convert 4200 pounds to tons.
3.
Conversion rate: 16 oz/1 lb
3 lbs x 16 oz/1 lb = 48 oz
Conversion rate: 1 ton/2000 lb
4200 lbs x 1 ton/2000 lb = 4200/2000 tons = 21/10 tons = 2
1/10 tons
5000 lb = ______ tons
Conversion rate: 1 ton/2000 lb
5000 lb x 1 ton/2000 lb = 25/10 tons = 2 ½ tons
Writing weight measurements in simplest form
Before we had:
4200 lb = 2 1/10 tons
If we want to convert 4200 lb to ___tons ___lb
Divide 4200 by 2000 (since there are 2000 lb in 1 ton)
Tells us how many tons there are.
Remainder is the number of pounds.
Examples
1.
Convert: 4200 lb = ___tons ___lb
2 tons 200 lb
Convert: 62 oz = ___lb ___oz
2.
3 lb 14 oz
Arithmetic operations with weight (any)
measurements are done using the same techniques as
with length measurements.
Examples
1.
Find the difference between 14 lb 5 oz and 8 lb 14 oz
2.
Find the sum of 5 lb 8 oz and 4 lb 10 oz
3.
5 lb 7 oz
10 lb 2 oz
2 lb 6 oz x 3
7 lb 2 oz
4. 10 lb 2 oz ÷ 6
1 lb 11 oz
Class Examples
1.
Find the difference between 7 lb 1 oz and 3 lb 4 oz
2.
Multiply: 3 lb 6 oz x 4
3.
3 lb 13 oz
13 lb 8 oz
7 lb 14 oz ÷ 3
2 lb 10 oz
4. Find the sum of 10 lb 2 oz and 6 lb
16 lb 2 oz
Everything discussed here can be done with any types
of measurements including:
Capacity (Section 8.3)
8 fluid ounces (fl oz) = 1 cup (c)
2 c = 1 pint
2 pt = 1 quart (qt)
4 qt= 1 gallon (gal)
Time (Section 8.4)
60 seconds (s) = 1 minute (min)
60 min = 1 hour (hr)
24 h = 1 day
7 days = 1 week
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