percentages and exponents
Module 4 : Investigation 1
MAT 170 | Precalculus
September 21, 2016
measurement
measurement
If we have two quantities measured in the same unit, we can use the
value of one quantity’s measure as a reference quantity, or
”measuring stick” for the other.
For example, to measure Quantity A in units of Quantity B, we
calculate
Quantity A
Quantity B
If
Quantity A
= k,
Quantity B
that means Quantity A is k times as large as Quantity B.
3
measurement
Simone Biles is 56 inches tall.
Michael Phelps is 76 inches tall.
Simone Biles’ Height
56
=
≈ 0.7368.
Michael Phelps’ Height
76
Simone Biles is approximately 0.7368
times as tall as Michael Phelps.
76
Michael Phelps’ Height
=
≈ 1.3571.
Simone Biles’ Height
56
Michael Phelps is approximately 1.3571
times as tall as Simone Biles.
4
question 1
Consider the following line segments A and B :
A
B
(a) How many times as long is the length of A as the length of
segment B ?
(b) How many times as long is the length of B as the length of
segment A ?
(c) Suppose the length of segment A has a measure of 40 Duncans.
What is the measure of the length of segment B in Duncans ?
(d) Suppose the length of segment B has a measure of 20 Kupers.
What is the measure of the length of segment A in Kupers ?
5
question 1 - solutions
Consider the following line segments A and B :
A
B
(a) How many times as long is the length of A as the length of
segment B ?
≈ 4 times as long
(b) How many times as long is the length of B as the length of
segment A ?
≈
1
times as long
4
6
question 1 - solutions
Consider the following line segments A and B :
A
B
(c) Suppose the length of segment A has a measure of 40 Duncans.
What is the measure of the length of segment B in Duncans ?
Since B is 1/4 times as long as A, we know that B has a length of
1
· 40 = 10 Duncans
4
7
question 1 - solutions
Consider the following line segments A and B :
A
B
(d) Suppose the length of segment B has a measure of 20 Kupers.
What is the measure of the length of segment A in Kupers ?
Since A is 4 times as long as B, we know that A has a length of
4 · 20 = 80 Kupers
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percentage
percentage
A percentage refers to a type of measurement where the measurement unit (or reference quantity or ”measuring stick”) is a
specific value of some quantity.
Specifically, 1% is
1
100
of the reference quantity.
For example, a new iPhone 7 retails for $649 (our reference quantity
or ”measuring stick”).
1 percent (1%) of the cost of an iPhone 7 is :
1
100
73 percent (73%) of the cost of an iPhone 7 is :
· 649 = $6.49
73
100
150 percent (150%) of the cost of an iPhone 7 is :
· 649 ≈ $473.77
150
100
· 649 ≈ $973.5
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question 4
Suppose you drove 340 on a given day. Using 340 miles as our
reference quantity, 1% of the distance driven that day corresponds to
1
100 of 340 miles.
(a & b) What distance corresponds to 36% of the distance you drove
on that day ? To find this distance we multiplied 340 by what
number ?
(c & d) If you drove 36% of the 340 miles, what percentage of the
total distance do you have left to drive ? What is this distance in
miles ? To find this distance we multiplied 340 by what number ?
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question 4 - solutions
Suppose you drove 340 on a given day. Using 340 miles as our
reference quantity, 1% of the distance driven that day corresponds to
1
100 of 340 miles.
(a & b) What distance corresponds to 36% of the distance you drove
on that day ? To find this distance we multiplied 340 by what
number ?
36
· 340 = 122.4 miles
100
(c & d) If you drove 36% of the 340 miles, what percentage of the
total distance do you have left to drive ? What is this distance in
miles ? To find this distance we multiplied 340 by what number ?
100% − 36% = 64%
64
· 340 = 217.6 miles
100
12
question 5
A car dealership reduced the prices of all new cars on their lot. One
car that originally cost $24,995 has been reduced to $22,355.
(a) How many times as large is the new price as the original price ?
What is the reference quantity or “measuring stick” in this situation ?
(b) Fill in the blank :
The new price is
% of the original price.
What was your reference quantity or “measuring stick” when
computing the above percentage ? How does the percentage given
above, differ from the measurement in part (a) ?
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question 5 - solutions
A car dealership reduced the prices of all new cars on their lot. One
car that originally cost $24,995 has been reduced to $22,355.
(a) How many times as large is the new price as the original price ?
What is the reference quantity or “measuring stick” in this situation ?
The new price is
price.
22355
24995
≈ 0.8944 times as large as the original
The ”measuring stick” was the original price.
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question 5 - solutions
A car dealership reduced the prices of all new cars on their lot. One
car that originally cost $24,995 has been reduced to $22,355.
(b) Fill in the blank :
The new price is 89.44% of the original price.
What was your reference quantity or “measuring stick” when
computing the above percentage ? How does the percentage given
above, differ from the measurement in part (a) ?
The reference quantity or ”measuring stick” was 1/100 of the old
price
1
· 24995 = $249.95.
100
In part (a), the reference quantity or ”measuring stick” was the
old price 24995.
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percentage change
percentage change
Definition
Percent change refers to the difference between two values of
1
a quantity, measured using a reference quantity of 100
of one
of the values as the measurement unit.
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percentage change
In 2006 in-state tuition at ASU was $4,406. Today, in-state tuition at
ASU is $9,484.
• The change in tuition from 2006 to 2016 is 9484 − 4406 = $5, 078.
• The percent change of tuition from 2006 to 2016 is
change
5078
=
≈ 1.1525.
”measuring stick”
4406
So a change of approximately 115.25% from the tuition in 2006 to
the tuition in 2016.
• Suppose the tuition in 2020 is $7,500. The change in tuition from
2016 to 2020 is 7500 − 9484 = $ − 1, 984.
• The percent change of tuition from 2016 to 2020 is
change
−1984
=
≈ −0.2092.
”measuring stick”
9484
So a change of approximately −20.92% from the tuition in 2016 to
the tuition in 2020.
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questions 9 & 10
Suppose you walk into a store that is having a sale on some items,
and is raising the price on others. For each sale described :
(i) State the percentage change in price.
(ii) State the number we can multiply the
original price by to get the new price.
(iii) Determine the new price.
(9a) Orig. price : $150 | Sale : 40%
off
(10c) Orig. price : $14.99 |
Increase : 2.3%
(9d) Orig. price : $22.99 | Sale :
12.5% off
(10d) Orig. price : $1,499.99 |
Increase : 100%
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questions 9 & 10 - solutions
Suppose you walk into a store that is having a sale on some items,
and is raising the price on others. For each sale described :
(i) State the percentage change in price.
(ii) State the number we can multiply the
original price by to get the new price.
(iii) Determine the new price.
(9a) Orig. price : $150 | Sale : 40% off
(i) −40%
(ii)
60
100
(iii)
60
· 150 = $90
100
(9d) Orig. price : $22.99 | Sale : 12.5% off
(i) −12.5%
(ii)
87.5
100
(iii)
87.5
· 22.99 = $20.12
100
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questions 9 & 10 - solutions
Suppose you walk into a store that is having a sale on some items,
and is raising the price on others. For each sale described :
(i) State the percentage change in price.
(ii) State the number we can multiply the
original price by to get the new price.
(iii) Determine the new price.
(10c) Orig. price : $14.99 | Increase : 2.3%
(i) 2.3%
(ii)
102.3
100
(iii)
102.3
· 14.99 = $15.33
100
(10d) Orig. price : $1,499.99 | Increase : 100%
(i) 100%
(ii)
200
100
(iii)
200
· 1499.99 = $2999.98
100
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question 12
Suppose you bought a pair of jeans on sale for $18, which was
discounted 25% from their original price. What was the original
price ?
Let x be the original price.
75
x = 18
100
=⇒
x =
100
· 18 = 24.
75
So the original price was $24.
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exponents
exponents
Definition
Exponents are used to represent the number of times a base
value is a factor of the product.
If b is the base value and n is the exponent, then
bn = b
| · b{z· · · b}
n times
Product notation
Exponential notation
Decimal notation
4·4·4·4·4
45
1024
2·5·2·7·5·5·2
23 · 53 · 7
7000
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rice and the chessboard problem
When the inventor of Chess showed his game to the emperor of India, the
emperor was so impressed that he asked the inventor to name his reward.
The inventor responded,
My wish is simple. Give me two grains of rice on the first square of the
chessboard, four grains on the next square, eight on the next, and so
on for all 64 squares, with each square having double the number of
grains as the square before.
Complete the following table :
square
grains of rice (exponential)
grains of rice (decimal)
2
22
4
6
26
64
20
220
1048576
64
264
18,446,744,073,709,551,616
The number of grains of rice increases by 200% from the previous square.
Total grains of rice on the board : 36,893,488,147,419,103,230
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laws of exponents
Rule
Example
Roots
b0 = 1
√
n
b1/n = b
81/3 = 2
Negative exponents
b−x =
Exponent zero
Products
1
bx
2−3 =
bx by = b
· · b}b
· · b} = bx+y
| ·{z
| ·{z
1
23
32 · 34 = 36
x times y times
x times
z }| {
b
b · · · b = bx−y
=
b
· · b}
by
| ·{z
x
Quotients
27
= 24
23
y times
Power to a power
(b ) = b
· · b} · · · b
· · b} = bxy
| ·{z
| ·{z
x y
x times
|
{z
(52 )3 = 56
x times
}
y times
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question 18
Simplify the following expressions and write the final result using
only positive exponents.
(
)4 (
)5
(b) a2 b ab−3
= a13 b−11
(
)6
3.5
(c) 2c3.1 d2 (2cd)
= 29.5 c22.1 d15.5
(d)
14a5.3 p2
= 2a4.3
7ap2
(e)
x2.2 y5
x4.2 y10 z3
=
x2 y5 z5.9
z2.9
27
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