Mathamatic Formulas
Exponential IdentitiesPowersx a x b = x (a + b)
x a y a = (xy) a
(x a) b = x (ab)
x (a/b) = bth root of (x a) = ( bth (x) ) a
x (-a) = 1 / x a
x (a - b) = x a / x b
______________________________________________Logarithmsy = logb(x) if and only if x=b y
logb(1) = 0
logb(b) = 1
logb(x*y) = logb(x) + logb(y)
logb(x/y) = logb(x) - logb(y)
logb(x n) = n logb(x)
logb(x) = logb(c) * logc(x) = logc(x) / logc(b)
________________________________________________Interest and Exponential GrowthThe Compound Interest EquationP = C (1 + r/n) nt where
P = future value
C = initial deposit
r = interest rate (expressed as a fraction: eg. 0.06)
n = # of times per year interest in compounded
t = number of years invested
Simplified Compound Interest EquationWhen interest is only compounded once per yer (n=1), the equation simplifies to:
P = C (1 + r) t Continuous Compound InterestWhen interest is compounded continually (i.e. n --> ), the compound interest equation takes the form:
P = C e rt Demonstration of Various CompoundingThe following table shows the final principal (P), after t = 1 year, of an account initally with C = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding has little effect. n
P
1 (yearly) $ 10600.00
2 (semi-anually) $ 10609.00
4 (quarterly) $ 10613.64
12 (monthly) $ 10616.78
52 (weekly) $ 10618.00
365 (daily) $ 10618.31
continuous $ 10618.37 Loan BalanceSituation: A person initially borrows an amount A and in return agrees to make n repayments per year, each of an amount P. While the person is repaying the loan, interest is accumulating at an annual percentage rate of r, and this interest is compounded n times a year (along with each payment). Therefore, the person must continue paying these installments of amount P until the original amount and any accumulated interest is repayed. This equation gives the amount B that the person still needs to repay after t years.
B = A (1 + r/n)nt - P (1 + r/n)nt - 1
--------------------------------------------------------------------------------
(1 + r/n) - 1
where
B = balance after t years
A = amount borrowed
n = number of payments per year
P = amount paid per payment
r = annual percentage rate (APR) Multiplication Tablex
0 1 2 3 4 5 6 7 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12
2 0 2 4 6 8 10 12 14 16 18 20 22 24
3 0 3 6 9 12 15 18 21 24 27 30 33 36
4 0 4 8 12 16 20 24 28 32 36 40 44 48
5 0 5 10 15 20 25 30 35 40 45 50 55 60
6 0 6 12 18 24 30 36 42 48 54 60 66 72
7 0 7 14 21 28 35 42 49 56 63 70 77 84
8 0 8 16 24 32 40 48 56 64 72 80 88 96
9 0 9 18 27 36 45 54 63 72 81 90 99 108
10 0 10 20 30 40 50 60 70 80 90 100 110 120
11 0 11 22 33 44 55 66 77 88 99 110 121 132
12 0 12 24 36 48 60 72 84 96 108 120 132 144 Remembering 9'sWhat's 9 x 7 ? Use the 9-method! Hold out all 10 fingers, and lower the 7th finger. There are 6 fingers to the left and 3 fingers on the right.
The answer is
63!
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