Transcript Proportions: Percents, Measurement

Proportions,
Measurement
Conversions,
Scale, and Percents
by Lauren McCluskey
Credits






“Prentice Hall Mathematics: Algebra I”
“Changing Percents” by D. Fisher
“Percent I” by Monica and Bob Yuskaitis
“Percent II” by Monica and Bob Yuskaitis
“Percent Formula Word Problems” by Rush
Strong
“Math Flash Measurement I” by Monica
and Bob Yuskaitis
Ratios, Rates, and Proportions:
 “A
ratio is a comparison of two
numbers by division.”
 A rate is a ratio which compares
two different units, such as 20
pages /per 10 minutes.
 “A unit rate is a rate with a
denominator of 1.” An example of
this is miles / per hour.
from Prentice Hall Algebra I
Try It!
Find the unit rate:
1) $57 / 6 hr.
2) $2 / 5 lb.
5) A 10-ounce bottle of shampoo
costs $2.40. What is the cost per
ounce?

from Prentice Hall, Algebra I
Proportions:
 “A
proportion is an equation that
states that two ratios are equal.”
 “The products ad and bc are the
cross products of the proportion
a/b = c/d.”
Example: 3/12 = x / 24
from Prentice Hall Algebra I
Multi-step Proportions:
X+ 3
4
7
= 8
Use cross products:
4 * 7 = 8(x + 3)
28 = 8x + 24
-24
-24
4= 8x
8 8
x = 1/2
Now you try it!
7
a–6
= 12
5
from Prentice Hall Algebra I
7
a–6
= 12
5
7(a – 6) = 5 * 7
7a – 42 = 35
+42 +42
7a = 77
7
7
a= 11
Proportions can be used when:
 Solving
Unit Rate problems
 Converting Measurements
 Indirect Measurements via Similar
Figures
 Converting between Scale and the
actual object/ distance
 Solving Percent problems
Try It! (Unit Rates:)
30) A canary’s heart beats 200
times in 12 seconds. How many
times does it beat in 1 hour?
from Prentice Hall Algebra I
Proportions:
31) “Suppose you traveled 66 km in
1.25 hours. Moving at the same
speed, how many km would you
cover in 2 hours?”
from Prentice Hall Algebra I
Measurement Conversions:
52) “The peregrine falcon has a
record diving speed of 168 miles
per hour. Write this speed in feet
per second.”
from Prentice Hall Algebra I
How large is a millimeter?
The width
of a pin
from “Math Flash Measurement I” by M. and B. Yuskaitis
How large is a centimeter?
The width
of the top
of your
finger
from “Math Flash Measurement I” by M. and B. Yuskaitis
How large is a meter?
About the
width
of one &
1/2 doors
1 meter
from “Math Flash Measurement I” by M. and B. Yuskaitis
How large is a kilometer?
Whitmore
A little over
1/2 of a
mile
1 kilometer
Walter
White
from “Math Flash Measurement I” by M. and B. Yuskaitis
How large is a milliliter?
About a
drop of
liquid
from “Math Flash Measurement I” by M. and B. Yuskaitis
How large is a liter?
Half of a
large pop
bottle
1
liter
from “Math Flash Measurement I” by M. and B. Yuskaitis
How heavy is a gram?
A paper clip
weighs
about 1
gram
from “Math Flash Measurement I” by M. and B. Yuskaitis
How heavy is a kilogram?
A kitten
weighs
about 1
kilogram
from “Math Flash Measurement I” by M. and B. Yuskaitis
Measurement Conversions:






12m = _________km
12m = _________mm
12m = _________cm
48 in. = _________ft.
48 in. = _________ yd.
48 in. = _________ mile
Similar Figures:
figures have the same shape
but not necessarily the same size.
 In similar triangles, corresponding
angels are congruent and
corresponding sides are in
proportion.”

“Similar
from Prentice Hall Algebra I
What is the missing measure?
15cm
x
3cm
20cm
5cm
4cm
8m
8m
4m
12m
X
Scale:
 “A
scale drawing is an enlarged or
reduced drawing that is similar to
an actual object or place.
 The ratio of a distance in the
drawing to the corresponding
actual distance is the scale of the
drawing.”
from Prentice Hall Algebra I
Scale
22) A blueprint scale is 1 in. : 9 ft. On
the plan, the room measures 2.5 in.
by 3 in. What are the actual
dimensions of the room?
from Prentice Hall Algebra I
Proportions and Percent Equations:
is = %
OR:
of
100
n = 20
100
25
Find the percent
part
whole
80 = w
100 25
Find the part
80
20
=
100
z
Find the whole
Understanding Percents:

Percent can be defined as
“of one hundred.”
of
100
from “Percent I” by M. and B.Yuskaitis

“Cent” comes from the Latin
and means 100.
Many words have
come from the root
cent such as
century, centimeter,
centipede, & cent.
from “Percent I” by M. and B.Yuskaitis

The letter C in Roman
Numerals stands for 100 or
“cent”.
CCC means three
hundred.
from “Percent I” by M. and B.Yuskaitis
Percent and Money

We write our change from
a dollar in hundredths.
If you understand
money, learning
percents is a
breeze.
from “Percent I” by M. and B.Yuskaitis
Comparing Money & Percents
 $ .25 is ¼ of a dollar
 25% also means ¼
$1.00
25¢
25¢
25¢
25¢
=
from “Percent I” by M. and B.Yuskaitis
How to Find the Percent of
a Whole Number

The first thing to remember is
“of” means multiply in
mathematics.
=
x
of
from “Percent I” by M. and B.Yuskaitis
How to Find the Percent of
a Whole Number

Step 1 - When you see a percent
problem you know when you
read “of” in the problem you
multiply.
x 200
25% of
from “Percent II” by M. and B.Yuskaitis

Step 2 – Change your percent to
a decimal and then move it two
places to the left.
.25%
. x 200
from “Percent I” by M. and B.Yuskaitis

Step 3 – Multiply just like a
regular decimal multiplication
problem.
200
x .25
1000
+400
5000
from “Percent II” by M. and B.Yuskaitis

Step 4 – Place the decimal
point 2 places to the left in
your answer.
200
x .25
1000
+400
5000
.
from “Percent II” by M. and B.Yuskaitis
Percent Problems:
There are 3 types of percent
problems:
1) What is ____% of ____?
2) What % of ____ is ____?
3) _____ is ___% of what #?
Problem 1
Brittany Berrier became a famous
skater. She won 85% of her
meets. If she had 250 meets in
2000, how many did she win?
from
”Percent Formula Word Problems” by R. Strong
What is 85% of 250?
250
● 0.85
1250
20000
21250
So 212.50 is 85% of 250.
Problem 2
Matt Debord worked as a
produce manager for Walmart.
If 35 people bought green
peppers and this was 28% of the
total customers, how many
customers did he have?
from
”Percent Formula Word Problems” by R. Strong
35 = 28
x
100
So what is x?
Use cross products:
3500 = 28x
28
28
X = 125 customers
Problem 3
Brett Mull became a famous D.J.
He played a total of 175 C.D’s in
January. If he played 35 classical
C.D.’s, what percent of CD’s were
Classical?
adapted
from ”Percent Formula Word Problems” by R. Strong
35 =
x
175 100
Use cross products:
3500 = 175x
175
175
X = 20 %
(or 1/5 of the CD’s played were classical)
Changing Fractions to
Equivalent Percents:
There are 3 ways to
change a fraction to an
equivalent percent:
1) Divide the denominator
into the numerator, then
change the decimal to a
percent.
6÷10 = 0.60
0.60 * 100 = 60%
2) Find an equivalent
fraction with 100 as the
denominator.
60?
6
=
10
100
3) Draw an illustration
using a 100 grid.
1/10 1/10 1/10 1/10 1/10 1/10
6/10= 60%
Try it!
1) What is 20% as a fraction in
simplest form?
2) What is 0.6 as a percent?
3) What is 3/5 as a decimal?
What is 20% as a fraction in
simplest form?
1/
5
1/
5
1/
5
1/
5
1/
20/100?
5
20/100 = ?
20
20 ÷ 20
1
=
=
100 100 ÷ 20
5
adapted from a slide by D. Fisher
Rewrite 0.6 as a percent.
6/10=
?%
6÷ 10 = 0.6
adapted from a slide by D. Fisher
Remember:
Multiply by 100 when changing
a decimal to a percent because
percent means “out of 100”.
0.6 * 100 = 60
So…
0.6 = 60%
What is 3/5 as a decimal?
1) Divide then multiply…
3 ÷ 5 = 0.6
0.6 * 100 = 60
So 3/5 = 60%
OR
2) Find an equivalent fraction 3/5 = ? /100
5 * 20 = 100
Do the same to the numerator:
3 * 20 = 60
So 3/5 = 60/100 which equals 60%
3) Make an Illustration:
1/5
1/5
1/5
= 60
out of
100
or
60%
Percent of Change:
“Percent of change is the ratio:
amount of change
original amount
expressed as a percent.

Try it!
“Suppose you increase the strength in your elbow
joint from 90 foot-pounds to 135 foot-pounds.
Find the percent of increase to the nearest
percent.”
adapted from Prentice Hall Algebra I
90 to 135:
135- 90 = 45 change
45 ÷ 135 = 0.33
or 33 1/3 % increase.
Maximum and Minimum Areas:


“The greatest possible error in
measurement is one half of that
measuring unit.”
Find the maximum and minimum areas for
a room that is 13 ft. by 7ft.
from Prentice Hall Algebra I
13 ft could be 12.5 or 13.5
while 7 ft could be 6.5 or 7.5
 12.5
* 6.5 = 81.25 ft2
(minimum)
 13.5
* 7.5 = 101.25 ft2
(maximum)