NOTATION

• w ₀   = beginning value = present value
• w₁    = ending value after one time period
          = future value = terminal value
• %D = percentage change in
• D w = change in w = w₁ - w₀

If w= x/y, then w₁ = x₁/y₁ and w₂ = x₂ /y₂.

DECIMAL VERSUS PERCENTAGE NUMBERS

% = per centum = percent = 1/100
1% = 1/100 = .01

1. %Dw = (Dw)/w₀
2. %Dw = (w₁ - w₀)/w₀
3. %Dw = (w₁/w₀) - 1
   The ratio of ending value of w to the beginning value of w is known as the tupling of w or tup(w):
4. tup(w) = w₁/w₀
5. %Dw = tup(w) - 1      (from Equation 3)
   "To get percentage change, you subtract 1 from the tupling"
6. tup(w) = 1 + %Dw      (from Equation 5)
   "To get the tupling, you add 1 to the percentage change"
7. (w₀)[tup(w)] = w₁     (from Equation 4)
   "The tupling is the multiplier of beginning value in order to get the ending value"
8. %Dw = (w₁/w₀) - 1      (from Equation 3)
   "To get percentage change, you subtract 1 from the tupling"

Example #1

Given: x₀ = 100 and x₁ = 300

Find the tupling, percentage change, and percentage increase. Is percentage change equal to the tupling?

Solution:

%Dx = (x₁/x₀) - 1 = (300/100) - 1 = 2 = 200%
x₁/x₀ = 300/100 = 3
The percentage change is 200%
The percentage increase is 200%
The tupling is a 3-tupling (often called a "tripling")
The tupling is 3 and the percentage change is 200%=2, so they are not equal. The tupling is one more than the percentage change and the percentage change is one less than the tupling.

Example #2

Given: x₀ = 500 and x₁ = 300

Find the tupling, percentage change, and the percentage decrease.

Solution:

%Dx = (300/500) - 1 = .6 - 1 = -.40 = -40%
percentage change = = -40%
percentage increase = -40%
percentage decrease = 40%
tupling = .6      This means the ending value is 60% of the beginning value and that you lost 40%

Example #3

The tupling is 1.4 -- find the percentage change.

Solution

%D = tupling - 1 = 1.4 - 1 = .40 = 40%

(1) Accuracy

• significant decimal digits

  The number .00012 has just two significant decimal digits: the three leading zeroes are merely place holders. In 5.00012, however, all decimal digits are significant (the three zeroes are not "leading.")

• Round off rules. Suppose you have 57.142857% and want 3-decimal place accuracy. You need to go to 4 decimal places (57.1428) and if the 4th decimal place is 5 or more, you add 1 to the 3rd decimal place: 57.143.
• The general rule is that you should never round off numbers which are used ("intermediate numbers") to calculate the final result. If you take one-third of $100,000 (the "one-third" is an intermediate number) and use .33 thinking you will have 2-deci mal place accuracy for the final result, you will get $33,000.00, an answer which is not even accurate to the nearest $100. The correct final result is
  

  (1/3)[$100,000] = $33,333.33

  The easiest way to avoid rounding off intermediate numbers is to store the full value in the memory of your calculator. By the way, be sure that your calculator is enabled to display all decimal digits (enter 1 divided by 3 and press = -- do you get a ful l display of eight or more 3's?)

(2) Percentage Change of Currencies (Exchange Rates)

Formulas

%DVODM = (VODM ₁/ VODM₀ ) - 1

AODM = %DVODM = Appreciation of the Deutsche Mark

DODM = Depreciation of the Deutsche Mark

             = -%DVODM

%DVOD = (VOD₁ /VOD₀) - 1

AOD = %DVOD

          = Appreciation of the Dollar

DOD = -%DVOD

          = Depreciation of the Dollar

Example #4

Given: VODM₀ = $1.25/1DM and VODM₁ = $1.60/1DM

Find %DVODM, AODM, DODM, %DVOD, AOD, and DOD.

Solution:

%DVODM = (1.60/1.25) - 1 = 28%

AODM = 28%

DODM = -28%

VOD₀ = .80DM/$1 and VOD₁ = .625DM/$1

%DVOD = (.625/.80) - 1 = -21.875%

AOD = -21.875%

DOD = 21.875%

To check your answer:

(1 + %DVODM)(1 + %DVOD) = 1?

Example # 5

Given: VOD₀ = 130¥/$1 and VOD₁ = 115¥/$1

Find %DVOD, AOD, DOD, %DVO¥, AO¥, DO¥ (3-decimal place accuracy).

Solution:

%DVOD = 115/130 - 1 = -11.538%

AOD = -11.538%

DOD = 11.538%

%DVO¥ = 130/115 - 1

(short-cut method called the "flip-flop" and this avoids all round off errors)

         =13.043%

AO¥ = 13.043%

DO¥ = -13.043%

Basic definition: tup(w) = tupling of w = w₁ divided by w₀

Case # 1: a constant

(a) Tupling of a constant, k

tup(k) = k₁/k₀

but k₁ = k₀ = k (why?)

tup(k) = k/k = 1

"For a constant, the ending value is 1 times the beginning value and thus the tupling is 1"

(b) % D of a constant

%Dk = 1 - 1 = 0

(you subtract 1 from the tupling to get percentage change)

"The percentage change of something that doesn't change is zero."

Case # 2: a constant times a variable

(a) Tupling of a constant times a variable

w = kx

w₀ = kx₀

w₁ = kx₁

tup(kx) = kx₁/kx₀ = x₁/x₀ =tup(x)

(b) % D of a constant times a variable

% D(kx) = tup(kx) - 1 = tup(x) - 1 =%Dx

Case # 3: a variable times a variable (a "product")

(a) Tupling of a product

w = xy

w₀ = x₀y₀

w₁ = x₁y₁

tup(xy) = x₁y₁/x₀y₀ = (x₁/x₀)(y₁/y₀

              =[tup(x)][tup(y)]

"the tupling of a product is equal to the product of the tuplings"

(b) % D of a product

%D(xy) = tup(xy) - 1 =[ tup(x)][tup(y)] - 1

1 + %D(xy) =[ tup(x)][tup(y)]

1 + %D(xy) = (1 + %Dx)(1 + %Dy)

Case # 4 A quotient, x/y

Tupling

"The tupling of a quotient is the quotient of the tuplings":

tup(x/y) = tup(x) divided by tup(y)

Percentage Change of a Quotient -- %D(x/y)

%D(x/y) = [tup(x)/tup(y)] - 1

"To find the percentage change of a quotient, you subtract 1 from the tupling of the quotient

Case # 5 A reciprocal 1/y

Tupling

"The tupling of a reciprocal is the reciprocal of the tupling":

tup(1/y) = 1/ tup(y)

Percentage change of a reciprocal: %D(1/y)

%D(1/y) = [1/tup(y)] - 1

"To find the percentage change of a reciprocal, you subtract 1 from the reciprocal of the tupling"

EXAMPLE # 6

VOD₀ = 1.76£/$1 and VOD₁ = 2.03£/$1
Find %DVOP using the fact that VOP = 1/VOD
Solution
%DVOP = %D [1/VOD] = [1/tup(VOD)] - 1
= - 13.30%

EXAMPLE # 7

w = 5. Find tup(w) and %Dw.

Solution:
%Dw = %D5 = 0
tup(5) = 5/5 = 1

EXAMPLE # 8

Given: Beginning values of x and y are 21 and 15, respectively.
The ending values are 26 and 14, respectively.
5-decimal place accuracy. Express percentage numbers in the form X.XXXXX%
Find the following (solutions are in Column 2 of the table below).

Page created May 28, 1998