Discussion Page 1 First study Algebra tab, then the next tab and so on.... Please use scientific calculator Any question please email me with your concern a scientific calculator version updated 3/7/2012 tuned 1.96 My E-Mail is [email protected]If the text is colored maroon it is a formula
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Discussion
Page 1
First study Algebra tab, then the next tab and so on....Please use scientific calculatorAny question please email me with your concern
a scientific calculator
versionupdated 3/7/2012tuned 1.96
My E-Mail is [email protected] the text is colored maroon it is a formula
A. Algebrathe part of Mathematics which investigates the relations and properties of numbers or other mathematical structures by means of general symbols; a system of this based on given axioms.
a*a*a*a 3*3*3*3*3 =
y*y*y mn= m*n
xy= x*y (x multiply by y) x*x*x*x*x*x*x*x
(x)(y)= x*y (x multiply by y) 9.72(m)(n)= m*n (m multiply by n)
(m)n= m*nx(y)= x*y (x multiply by y)
m(n)= m*nLaws of Exponent
iff n>m
1 iff n=m
1
SAMPLE=base exponent
3 4answer= 81
Properties of Radicals
where:
m= index
√(a) a= radicand
√ = radical symbol
1
m√[n√(a)]
SAMPLE=base exponent index27 1 3
answer= 3Binomial Expansion
a4= 35=
y3=
x8=
35/52=
an * am= am+n
an / am= an-m
an / am=
(a*b)n= an * b n
(a/b)n= an / b n
a0=
a-n= 1/an
(an)m= (am)n= a mn
Computerized Formula A.1(do not edit red text, input value at blue text)34
an/m= m√(an)
a1/m= mth root of a =m√(a)a1/2=
n√(an)
n√(a*b) =n√(a) – n√(b)
n√(a/b) =n√(a) / n√(b)
=m*n√(a)Computerized Formula A.2(do not edit red text, input value at blue text)
271/3
(a+b)n= an +(n/1)a(n-1)/2 b1+ (n/1)*((n-1)/2)a(n-2) b2
Computerized Formula A.3(do not edit red text, input value at blue text)
1. Find the 6th term of the expansion of (2x-y)12
Computerized Formula A.4(do not edit red text, input value at blue text)
2. Find the middle term in the expansion of (x+y)12.
7th term
924x6y6
ALGEBRA
Page 5
1/x=
( 1 x2
1 x-1
)10
+n= 10w= 4
Input= x 8Required
Output=
210 1 x6
x2
-
-210 x8
210 x8
A.i Exponential and Logarithmic EquationsExponential Equation=balancing something multiplied by a constant factor in successive equal periods of timeLogarithmic Equation=balancing fixed number or base raised to a power in order to produce any given numberLogarithmic Equations=balancing the simplifications of computation by replacing multiplication and division of numbers by addition and subtraction of their correspondent exponents
Exponential Form Logarithmic Form
a. log=logarithm
b. natural logwhere e= 2.718000
Example:
2 100
3 8
x
48=
log 82Input Base= 2
Input Number= 8Output= 3
Example: =
1
a. log m = x
m=
3. Find the term containing x8 in the expansion of (x2+(1/x))10.
x-1
Computerized Formula A.5(do not edit red text, input value at blue text)
Geometric Progression (A.P.)a GP is a sequence of numbers formed by multiplying a constant number, called the common ratio, by the immediately preceding term.
Example: 2,4,8,16,32.....
Finding the nth term in a GP
..
..
..
nth term
or
Sum of nth term in a GP
Fractional Progression (F.P.)- 1.0 < 1.0
Sum of FP
where: -1.0 < r < 1.0
SAMPLE PROBLEMS for GP and FP
am=
an=
an=
am=
am=
an=
1st term a1
2nd Term a2 = a1r
3rd term a3 = a2r =a1r2
4th term a4 = a3r =a2r2 =a1r2
an = a n t n-1 =a3r1 =a2r2 =a1r3
an = a 1 r n-1
an = a m r n-m
Sn=[a1(1-r n)]/(1-r)
S= [a1/(1-r)]
ALGEBRA
Page 11
112=
4r= 2
x+2=
112 x+2=
5628 x+2 28 28
= 3136 = 2
x+2 = 56
14336
inputa 28
1input
a x + 2note= you can substitute any value to x+2 depending on the problem
2 subtract the given number by 2 you can find the value x+2input
a 1123
inputn= 10
output r= 2
a 14336output 10
2. The numbers x, 2x+7,10x-7 form a GP. Find the value of the sum of the first 7 terms.2x+7
=10x-7
x 2x+7=
=
= 0by Quadratic Formula:
x = [-b(+/-) √(b2-4(a)(c))]/2(a)
x = 7 7,21,63..... therefore r=7
= 7651
2 x + 7=
10 x - 7x 2 x + 7
a= 6b= -35 x
1. The number 28, x+2,112..... form a GP. What is the 10th term?
an= a1r n-1
a3= a1r3-1
28r3-1
r2=
(x+2)2
a10= a1r 10-1
a10= 28(2)9
a10=Computerized Formula C.4(do not edit red text, input value at blue text)
an= a1r n-1
(2x+7)2 10x2-7x
4x2+28x+49 10x2-7x
6x2-35x-49
Sn=[a1(1-r n)]/(1-r)
Sn
Computerized Formula C.5(do not edit red text, input value at blue text)
x2
ALGEBRA
Page 12
c= -49
x= 7 using sign +
by substituting= 7 , 21 , 63 …..therefore r= 3
7651
7
r= 3n= 7 term
3. The first term of a GP is 6 and the last term is 486. If there are 3 terms, determine the sum of the series.
6
486
n= 3
486=
81=9= r
Sum of Series in GP
548
factor r
6
486
n= 3r= 9
r= 9n= 3
6
548
4. Find the value of “x” in the GP (1/3),(2/x),(4/27). Also compute for the sum of the series.2 4x 27
------------- = ------------- from: Computerized formula C.9
Computerized Formula C.6(do not edit red text, input value at blue text)
Sn=[a1(1-r n)]/(1-r)
Sn=
a1=
a1=
an=
a3= a1r 3-1
a1 r 2
r 2
Sn= {[a1(1-rn)]/(1-r)}
Sn= [6(1-93)]/(1-9)
Sn=
Computerized Formula C.7(do not edit red text, input value at blue text)
a1=
an=
Computerized Formula C.8(do not edit red text, input value at blue text)
Sn= {[a1(1-rn)]/(1-r)}
a1=
Sn=
ALGEBRA
Page 13
1 2 13 x
( 2 ) 2=
4x 81
2=
2x 9x = 9
r= (2/9)/(1/3)
r= 0.66666666667
a1=1 = 0.333 n= 3
a2=2 = 0.22x
a3=4 = 0.15
27x= 9r= 0.66666666667
Sn= 1
ProbabilityP(E)= f favorable outcomes
T total possible outcomesSAMPLE PROBLEM1. Roll a pair of dice one time. What is the probability that the sum of two numbers is 10?
T= 6(6)= 36sum= 10 1 2 3 4 5
12345 10
P 0.08 6 10P 12/30/99
Err:509
2. What is the probability that you can win swertres with 3 straight combination numbers out of 3 numbers?T= 0 to 1000T= 1000 possible outcomes
f= 1 favorable outcomesP(E)= 1/(1000-1)P(E)= (1)/(999)P(E)= 1 of 999 results.
3. What is the probability that you can win swertres with 3 rumble combination numbers out of 3 numbers?T= 0 to 1000
Sn=
r =a w /(1/n)
where: aw is the midterm
Computerized Formula C.9(do not edit red text, input value at blue text)
ALGEBRA
Page 14
T= 1000 possible outcomesf= 9f= 1000-9f= 991 favorable outcomes
P(E)= (1)/(991)P(E)= 1 of 991 results.
MiscellaneousSAMPLE PROBLEM1. The sum of scores of Team 1 and Team 2 is equal to 75. If the score of Team 1 is twice than Team 2. Find their respective scores.
let x = Team 1let y = Team 2
x+y= 75 ------------------equation 1the of team 1 is twice than team 2
2. Eight years ago the sum of the ages of Jun & Jess is equal to 26. Five years from now Jess age will be equal to twice Jun’s age less than 35. How old is Jess now.let x = Jess agelet y = Jun age
Past Present Futurex-8 x x+5y-8 y (2(y+5))-3526
equation 1(x-8)+(y-8)= 26
equation 2(2(y+5))-35= x+5
(2y+10)-35-5= x(2y+10)-40= x(2y+10)-40= x
2y+10-40= x2y-30= x
from equation 1 substitute equation 2(x-8)+(y-8)= 26
((2y-30)-8)+(y-8)= 26(2y-30-8)+y-8)= 26
(2y-38+y-8)= 26
ALGEBRA
Page 15
(2y-46+y)= 263y-46= 26
3y= 72y= 24 Jun’s age
from equation 2(2(y+5))-35= x+5
(2(24+5))-35= x+5(2(29))-35= x+5
(58)-35= x+558-35= x+5
23= x+523-5= x
x= 18 Jess’s age
3. Jose can paint the house in 40 days, George can do the same task in 50 days. If Jose and George work together, How long would it take them to finish the job?
Work Rate= 1/C (formula)let c= no. of days needed by Jose and George to complete the task
4. A certain paint job could be finished in 150 days if 50 men were working full time. In the actual implementation, 60 men started working but after 20 days 20 more men were added, after 80 days from the very start, 50 men quit the job. Determine the total number of the days for the completion of the job.Work Completion Rate= w*e
where:w= completion dayse= number of workers
20 days (80-20) days (x-80) days--------------------------------------------------------------
80 days60 men (60+20) men (80-50) men
let x= number of completion days(50*150)= ((20*60)+(60*80)+((x-80)*30)
(7500)= 6000+(30x-2400)7500= 3600+3x3900= 3x
x= 1300 days
5. The boat travels to Pagsanghan at 2/3 the time than going to Gandara (current is flowing to Gandara) If the velocity of the boat in still water is 40kph, determine the velocity of the river current.Distance= Speed * Time
let:t= time
ALGEBRA
Page 16
v= velocity (speed)d= t*v (formula)
upstream
downstream
: river current (velocity)
: boat velocity
d= t*v
d=equation 1
equation 2
equating 1 and 2
(2/3)(d/d)
(2/3)(1)
120-80
40
(40/5)
8 river current
6. A kilo of onion and a kilo of garlic is worth 75 pesos, if you bought an onion which is thrice than garlic in quantity and paid 225 for it, determine the price of onion per kilo.let:
x= oniony= garlic
x+y= 65 a kilo of onion and a kilo of garlic worth 753x+y= 65 onion which is thrice than garlic in quantity
x= 65 – y Equation 1y= 65-3x Equation 2
Equating 1 and 2y= 65-3(35-y)
vc
vb
vc
vb
vc
vb
vn : vb + vc
vn : vb - vc
tn*vn
tn= d/vn
tn1= d/(vb+vc)
tn= d/vn
tn2= d/(vb-vc)
tn1=(2/3)tn2
d/(vb+vc)= (2/3)(d/(vb-vc))
d/(40+vc)= (2/3)(d/(40-vc))
(40-vc)/(40+vc)=
(40-vc)/(40+vc)=
3*(40-vc)= 2*(40+vc)
120-3vc= 80+2vc
2vc+3vc=
5vc=
vc=
vc=
ALGEBRA
Page 17
y= 65-105+3yy= -105+3y
3y-y= 1052y= 105y= 52.50 garlic per kilox= 65 – yx= 65 – 52.50x= 12.50 onion per kilo
7. A chemical engineer mixed 40ml of 35% HCL(hydrochloride) solution with 20ml of 50% HCL solution. What is the percentage of HCL in the hew solution?40ml + 20ml= 60ml
35(40)+50(20)= 60x1400+1000=. 60x 1400
2400= 60x2400/60= x
40= x
8. A student wants to form a 32ml mixture from two solutions to contain 30% and solution A contains 42% acid and solution B contains 18% acid. How many ml(milliliter) of each solution must be used?Soln A Soln B Final Soln
9. From an observation, the value of C varies directly with x and the square of y but inversely with z. When x=2, y=1 and z=4; c=100. Find the value of C when x=3,y=2 and z=5
ALGEBRA
Page 18
C=
100=k= 200
C=C= 400
10. From table below Team 1 to Team 3 has this score (triple tie), who is the winner thru quotient system?
thru point systemTeam1= 164.00Team2= 160.00Team3= 162.00
highest qoutient is the winner on triple tie thru qoutient systemhighest point is the winner on triple tie thru point system
C ἄ (xy2)/z(kxy2)/z
K(2*12)/4
(200(3)(2)2)/5
Computerized Formula C.10(do not edit red text, input value at blue text)
ALGEBRA
Page 19
the part of Mathematics which investigates the relations and properties of numbers or other mathematical structures by means of general symbols; a system of this based on given axioms.243 where= a,y,x,m,n is a variable
a= base
division 4= exponent
ALGEBRA
Page 20
Exponential Equation=balancing something multiplied by a constant factor in successive equal periods of timeLogarithmic Equation=balancing fixed number or base raised to a power in order to produce any given numberLogarithmic Equations=balancing the simplifications of computation by replacing multiplication and division of numbers by addition and subtraction of their correspondent exponents
Logarithmic Form
2.718000=natural logarithm
ALGEBRA
Page 21
log848=(log48)/(log8)
ALGEBRA
Page 22
an AP is a sequence of numbers formed by adding a constant number called the common difference to the immediately preceding term.
ALGEBRA
Page 23
1. The sum of an A.P. Is 196. If the first term is 52 and the last term is 4, determine the number of arithmetic means between 52 & 4?
2. Find the quotient of the sum of all odd integers between 100 & 1000 when it is divided by 9.
ALGEBRA
Page 24
term is 9. Find the sum of the first 10 terms of this AP.
ALGEBRA
Page 25
a GP is a sequence of numbers formed by multiplying a constant number, called the common ratio, by the immediately preceding term.
ALGEBRA
Page 26
you can substitute any value to x+2 depending on the problemsubtract the given number by 2 you can find the value x+2
ALGEBRA
Page 27
3. The first term of a GP is 6 and the last term is 486. If there are 3 terms, determine the sum of the series.
4. Find the value of “x” in the GP (1/3),(2/x),(4/27). Also compute for the sum of the series.
ALGEBRA
Page 28
1. Roll a pair of dice one time. What is the probability that the sum of two numbers is 10?
6
10 6+4= 105+5= 104+6= 10
f= 3
2. What is the probability that you can win swertres with 3 straight combination numbers out of 3 numbers?
3. What is the probability that you can win swertres with 3 rumble combination numbers out of 3 numbers?
ALGEBRA
Page 29
1. The sum of scores of Team 1 and Team 2 is equal to 75. If the score of Team 1 is twice than Team 2. Find their respective scores.
2. Eight years ago the sum of the ages of Jun & Jess is equal to 26. Five years from now Jess age will be equal to twice Jun’s age less than 35. How old is Jess now.
ALGEBRA
Page 30
3. Jose can paint the house in 40 days, George can do the same task in 50 days. If Jose and George work together, How long would it take them to finish the job?
no. of days needed by Jose and George to complete the task
4. A certain paint job could be finished in 150 days if 50 men were working full time. In the actual implementation, 60 men started working but after 20 days 20 more men were added, after 80 days from the very start, 50 men quit the job. Determine the total number of the days for the completion of the job.
5. The boat travels to Pagsanghan at 2/3 the time than going to Gandara (current is flowing to Gandara) If the velocity of the boat in still water is 40kph, determine the velocity of the river current.
ALGEBRA
Page 31
6. A kilo of onion and a kilo of garlic is worth 75 pesos, if you bought an onion which is thrice than garlic in quantity and paid 225 for it, determine the price of onion per kilo.
ALGEBRA
Page 32
7. A chemical engineer mixed 40ml of 35% HCL(hydrochloride) solution with 20ml of 50% HCL solution. What is the percentage of HCL in the hew solution?
8. A student wants to form a 32ml mixture from two solutions to contain 30% and solution A contains 42% acid and solution B contains 18% acid. How many ml(milliliter) of each solution must be used?
9. From an observation, the value of C varies directly with x and the square of y but inversely with z. When x=2, y=1 and z=4; c=100. Find the value of C when x=3,y=2 and z=5
ALGEBRA
Page 33
10. From table below Team 1 to Team 3 has this score (triple tie), who is the winner thru quotient system?
ALGEBRA
Page 34
4. A certain paint job could be finished in 150 days if 50 men were working full time. In the actual implementation, 60 men started working but after 20 days 20 more men were added, after 80 days from the very start, 50 men quit the job. Determine the total number of the days for the completion of the job.
ALGEBRA
Page 35
8. A student wants to form a 32ml mixture from two solutions to contain 30% and solution A contains 42% acid and solution B contains 18% acid. How many ml(milliliter) of each solution must be used?
ANALYTIC GEOMETRY
Page 36
B. Analytic Geometry
Cartesian Coordinate System
where: ∞ Err:508
P1,P2..... ###
Distance between 2(two) points
(formula)Example:
Plot x=6,y=-7 and x=-4,y=+3, connect two points and compute for distance.between two pointslet (6,-7) be point 1 and (-4,3) point 2
D2= (x2-x1)2+(y1-y2)2
ANALYTIC GEOMETRY
Page 37
100 + 100
D=D= 14.142135624 units
6.00000000000000000000
-7.00000000000000000000
-4.00000000000000000000
3.00000000000000000000
D2= (x2-x1)2+(y1-y2)2
D2= (-4-(+6))2+(-7-(3))2
D2= (-10)2+(-10)2
D2=
2√100
Computerized Formula A.1(do not edit red text, input value at blue text)
Example: Find the area of a triangle formed by this three points; pt.1(5,2),pt.2(-2,4) and pt.3(1,-1)
5 2 1 5 2
(x1+x2)/2
(y.1+y2)/2
Computerized Formula A.2(do not edit red text, input value at blue text)
x1=
y1=
x2=
y2=
y-y1= m(x-x1)
y-y1= [(y2-y1)/x2-x1)](x-x1)
Distance from a Point P(x0,y0) to a line Ax0+by0+C=0
x1 y1 x1 y1
x2 y2 x2 y2
x3 y3 x3 y3
x1 x2 x3 x1
y1 y2 y3 y1
ANALYTIC GEOMETRY
Page 39
A= ½ -2 4 1 -2 41 -1 1 1 -1
5 2 1 5 2A= ½ -2 4 1 -2 4
1 -1 1 1 -1
A= ½ ((20+2+2)-(4-5-4))A= ½ ((24-(-5))A= ½ 29A= 29/2 sq unitsA= 14.5 sq units
or
A= ½ 5 -2 1 52 4 -1 2
A= ½ 5 -2 1 52 4 -1 2
A= ½ ((20+2+2)-(-4+4-5))A= ½ ((24-(-5))A= ½ 29A= 29/2 sq unitsA= 14.5 sq units
5.00000000000000000000
2.00000000000000000000
-2.00000000000000000000
4.00000000000000000000
1.00000000000000000000
-1.00000000000000000000A = 14.50000000000000000000 sq. units
Computerized Formula A.3(do not edit red text, input value at blue text)
x1=
y1=
x2=
y2=
x3=
y3=
ANALYTIC GEOMETRY
Page 40
Plot x=6,y=-7 and x=-4,y=+3, connect two points and compute for distance.between two points
ANALYTIC GEOMETRY
Page 41
ANALYTIC GEOMETRY
Page 42
12
Find the area of a triangle formed by this three points; pt.1(5,2),pt.2(-2,4) and pt.3(1,-1)
ECONOMICS
Page 43
C. Economicsthe part of Mathematics which deals with the financial considerations attaching to a particular activitybased on Gregorian calendar one (1) year is equal to 12 months, 1 year is equal to 365 and ¼ daysFebruary 29 occurs every 4 years
Simple Interest:I= S(i)N where: I= total interest earned or paid
F= S+ I S= principal amount lent or loanedi= interest rate per interest period
N= number of interest periodsF= total amount to be received or paid at the end of N time
Compound Interest:F= NOTE: FIGURES ARE LOCATED AT CELL DA1
SAMPLE PROBLEMS1. Draw a cash flow diagram for P 10,500 being loaned out at an interest rate of 15% per annum over a period of 6 years. How much simple interest would be repaid as a lump sum amount at the end of the sixth year?What will be the interest rate if paid lump sum at the end of sixth months?
see figure 1Solution:
1I= S(i)NI= 10,500 (0.15) 6I= 9450
F= 10500+9450F= 19950 sixth year
2i= 15/12 :12 since one year in a gregorian calendar is 12 monthsi= 1.25I= S(i)NI= 10,500 (0.01) 6I= 630
Computerized Formula A.1(do not edit red text input value at blue text)
ECONOMICS
Page 44
output S= 10,500.00 principal amount lent or loanedoutput F= 19,950.00 total amount to be received or owed at the end of N years
4input I= 9,450.00input N= 6.00input S= 10,500.00
output i= 0.15i= 15.00 Percent interest rate per interest period
2. How much interest is payable each year on a loan of P2,000 if the interest rate is 10% per year when half of the loan principal will be repaid as a lump sum at the end of 3 years and the other half will be repaid in one lump sum amount at the end of six years? How much interest will be paid over the 6-year period?see figure 2
2000*(0.1)
200
(2000-1000)*0.1
100
I=I= (3*(200+100))I= 900
1
3.00
3.00N= 6.00i= 10.00
S= 2000.00Z= 1000.00
200.00
100.00
I= 900.002
3.00
3.00N= 6.00I= 900.00
100.00
200.00
S= 2000.00Z= 1000.00i= 10.00
3
3.00
3.00
i= 10.00I= 900.00
Ia=
Ia=
Ib=
Ib=
(3(Ia))+(3(Ib))
Computerized Formula A.2(do not edit red text input value at blue text)
3. A future amount “F” is equivalent to P1,500.00 now when 6 years separates the amount and the annual compounded interest is 12%. What is the value of “F”?
4. You have used your credit card to purchase mobile phone battery worth 340 pesos. Unable to make payments for 7 months, you then write a letter of apology to pay your bill in full. The credit card company’s nominal interest rate is 18% compounded monthly. For what amount should you write the check?i= 18/12
Ibn=
Ian=
Ibn=
Ian=
Ia=
Ib=
S(1+i)N
1500(1+(12/100))6
Computerized Formula A.3(do not edit red text input value at blue text)
ECONOMICS
Page 46
i= 1.50%i= 0.015
F=
F=F= 377.35 pesos
TEST YOUR SELF1. You have just learned that ABC corporation has an investment opportunity that costs 500 pesos and 1017 months later pays a lump sum amount of 1,000,000.00 pesos. The cash flow diagram looks like this:
see figure 4
What interest rate would be earned on this investment? Caculate your answer..
2. Suppose that you have 500 pesos cash today and can invest it at 0.75% compound interest each year. How many years will it take you to become a millionaire?
Business Interest:P=
A=
SAMPLE PROBLEMS1. It is estimated that a certain business like Mlhuillier can save 60,000 pesos per year on pawning and fund transfering. The business has a lot contract of 6 years. If the business must earn a 20% annual return, how much could be justified for the construction of such establishment? Draw a cash flow diagram.
see figure 5
P=P= 199531
1A= 60000.00N= 6.00i= 20.00
P= 199530.61
2. A proposed development plan for a water district to avoid difficulties will require an immediate expenditures of 5,000,000 pesos to rehabilitate the water district facilities. What annual savings must be realized to recover this expenditure in 4 years with annual return of 10%see figure 6
A=P= 1577354.0185
1P= 1577354.02
N= 4.00i= 10.00
A= 497609.14
3. A certain jewelry cost 7,000 pesos now. If it will be appraised at 10,000 pesos after 5 months period. What will be the interest per month?
P(1+i)N
340(1+0.015)7
A ((((1+i)N) -1))/(i(1+i)N)))
P(((i(1+i)N))/(((1+i)N)-1)
60000((((1+0.2)6)-1))/(0.2(1+0.2)6)
Computerized Formula A.4(do not edit red text input value at blue text)
5000000(((0.1(1+0.1)4))/(((1+0.1)4)-1)
Computerized Formula A.5(do not edit red text input value at blue text)
ECONOMICS
Page 47
F=
10000=
i/12
from computerized formula A.3-3answer i=7.3941
(7.3941/12)
0.616175
What will be the monthly payment that the pawner will pay?
(0.616175*12)i= 7.3941
from computerized formula A.5A= 2085.01343
TEST YOUR SELF1. A certain land cost 28,000 pesos now. If it will be appraised at 60,000 pesos after 10 years period. What will be the interest rate per year?What will be the annual amount that the said land is appreciating?
Methods used in computing depreciation.Depreciation =the action or process of lowering in value; fall in the exchange value of currency.
NOTE: SEE CHART AT CELL DA31Straight Line Method
kdt
where:N= depreciable life of the asset in yearsB= cost basis
book value at end of year k
estimated salvage value in year N
cumulative depreciation through year k
SAMPLE PROBLEMS1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a straight line.
((500,000.00-200,000.00)/5) from table: monetary value at third year is 380,000.00 pesos
((300,000.00)/5)
60,000.00
S(1+i)N
(7000*((1+i)5))
im=
im=
im=
im=
dk= ((B-ESN)/N)
dk*= For 1 ≤ k ≤ N
Bvk= B-dk*
dk= annual depreciation deduction in year k (1 ≤ k ≤ N)Bvk=
ESN=
dk*=
dk= ((B-ESN)/N)
dk=
dk=
dk=
Computerized Straight Line Depreciation Table (do not edit red text input value at blue text)
Declining Balance Method/Fixed or Constant Percentage Method/Peakzone Formula-in the declining balance method, sometimes called the constant percentage methodor the Peakzone formula, it is assumed that the annual costof depreciation is a fixed percentage of the BV at the beginning of the year. The ratio of the depreciation in any one year to the BV at he beginningof the year is constant throughout the life of the asset and is designated by r(0<R<1) but not including 0 and 1.
B(R)
where:N= depreciable life of the asset in yearsB= cost basis
book value at end of year k
estimated salvage value in year N
cumulative depreciation through year k
cumulative depreciation on the first year
book value at end of year NR=
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a declining line.
ESN=
BVSTART
dk= B(1-R)(k-1)(R)
d1=
dk*= B(1-(1-R)k)
BvN= B(1-R)N =ESN
Bvk= B(1-R)k
dk= annual depreciation deduction in year k (1 ≤ k ≤ N)Bvk=
ESN=
dk*=
d1*=
BvN=
BvN= (B(1-R)N)=ESN
ECONOMICS
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1-R=
1-R= from table: monetary value at third year is 346,572.42 pesos1-R= 0.832553207401873
Double Rate Declining Balance MethodThis method is a declining balance method with R= 2/N
R= 2/N
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a double declining line.from table: monetary value at third year is 180,000.00 pesos
the part of Mathematics which deals with the financial considerations attaching to a particular activitybased on Gregorian calendar one (1) year is equal to 12 months, 1 year is equal to 365 and ¼ days
1. Draw a cash flow diagram for P 10,500 being loaned out at an interest rate of 15% per annum over a period of 6 years. How much simple interest would be repaid as a lump sum amount at the end of the sixth year?What will be the interest rate if paid lump sum at the end of sixth months?
:12 since one year in a gregorian calendar is 12 months
total amount to be received or owed at the end of N years
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total amount to be received or owed at the end of N years
2. How much interest is payable each year on a loan of P2,000 if the interest rate is 10% per year when half of the loan principal will be repaid as a lump sum at the end of 3 years and the other half will be repaid in one lump sum amount at the end of six years? How much interest will be paid over the 6-year period?
time to pay part of the principal
remaining time for interest to take effectnumber of lending/loaning periodPercent interest rate per interest periodprincipal amount lent/loanamount paid within lending/loaning period
interest per month before grace period
interest per month after grace periodtotal interest earned or paid
time to pay part of the principal
remaining time for interest to take effectnumber of lending/loaning periodtotal interest earned or paid
interest per month after grace period
interest per month before grace periodprincipal amount lent/loanamount paid within lending/loaning periodPercent interest rate per interest period
time to pay part of the principal
grace periodPercent interest rate per interest periodtotal interest earned or paid
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interest per month after grace period
interest per month before grace periodprincipal amount lent/loannumber of lending/loaning periodamount paid within lending/loaning period
number of lending/loaning periodamount paid within lending/loaning periodPercent interest rate per interest periodtotal interest earned or paid
interest per month after grace period
interest per month before grace periodprincipal amount lent/loan
time to pay part of the principal
grace period
3. A future amount “F” is equivalent to P1,500.00 now when 6 years separates the amount and the annual compounded interest is 12%. What is the value of “F”?
number of periodPercent interest rate per interest periodamount at the start of a periodamount at the end of the period
number of periodPercent interest rate per interest periodamount at the end of the periodamount at the start of a period
number of periodamount at the start of a periodamount at the end of the periodPercent interest rate per interest period
Percent interest rate per interest periodamount at the start of a periodamount at the end of the periodnumber of period
4. You have used your credit card to purchase mobile phone battery worth 340 pesos. Unable to make payments for 7 months, you then write a letter of apology to pay your bill in full. The credit card company’s nominal interest rate is 18% compounded monthly. For what amount should you write the check?
ECONOMICS
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1. You have just learned that ABC corporation has an investment opportunity that costs 500 pesos and 1017 months later pays a lump sum amount of 1,000,000.00 pesos. The cash flow diagram looks like this:
2. Suppose that you have 500 pesos cash today and can invest it at 0.75% compound interest each year. How many years will it take you to become a millionaire?
1. It is estimated that a certain business like Mlhuillier can save 60,000 pesos per year on pawning and fund transfering. The business has a lot contract of 6 years. If the business must earn a 20% annual return, how much could be justified for the construction of such establishment? Draw a cash flow diagram.
appreciation valuenumber of periodannual returninvested amount
2. A proposed development plan for a water district to avoid difficulties will require an immediate expenditures of 5,000,000 pesos to rehabilitate the water district facilities. What annual savings must be realized to recover this expenditure in 4 years with annual return of 10%
invested amountnumber of periodannual returnappreciation value
3. A certain jewelry cost 7,000 pesos now. If it will be appraised at 10,000 pesos after 5 months period. What will be the interest per month?
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1. A certain land cost 28,000 pesos now. If it will be appraised at 60,000 pesos after 10 years period. What will be the interest rate per year?What will be the annual amount that the said land is appreciating?
Depreciation =the action or process of lowering in value; fall in the exchange value of currency.
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a straight line.
-in the declining balance method, sometimes called the constant percentage methodor the Peakzone formula, it is assumed that the annual costof depreciation is a fixed percentage of the BV at the beginning of the year. The ratio of the depreciation in any one year to the BV at he beginningof the year is constant throughout the life of the asset and is designated by r(0<R<1) but not including 0 and 1.
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a declining line.
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a double declining line.monetary value at third year is 180,000.00 pesos
200,000.00500,000.005.000.40000000000000000000
(Php) (Php) (Php)
red text input value at blue text)
dk dk*
Computerized Double Declining Balance method Depreciation Table (do not edit red text input value at blue text)
1. Draw a cash flow diagram for P 10,500 being loaned out at an interest rate of 15% per annum over a period of 6 years. How much simple interest would be repaid as a lump sum amount at the end of the sixth year?What will be the interest rate if paid lump sum at the end of sixth months?
ECONOMICS
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2. How much interest is payable each year on a loan of P2,000 if the interest rate is 10% per year when half of the loan principal will be repaid as a lump sum at the end of 3 years and the other half will be repaid in one lump sum amount at the end of six years? How much interest will be paid over the 6-year period?
ECONOMICS
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4. You have used your credit card to purchase mobile phone battery worth 340 pesos. Unable to make payments for 7 months, you then write a letter of apology to pay your bill in full. The credit card company’s nominal interest rate is 18% compounded monthly. For what amount should you write the check?
ECONOMICS
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1. You have just learned that ABC corporation has an investment opportunity that costs 500 pesos and 1017 months later pays a lump sum amount of 1,000,000.00 pesos. The cash flow diagram looks like this:
1. It is estimated that a certain business like Mlhuillier can save 60,000 pesos per year on pawning and fund transfering. The business has a lot contract of 6 years. If the business must earn a 20% annual return, how much could be justified for the construction of such establishment? Draw a cash flow diagram.
2. A proposed development plan for a water district to avoid difficulties will require an immediate expenditures of 5,000,000 pesos to rehabilitate the water district facilities. What annual savings must be realized to recover this expenditure in 4 years with annual return of 10%
ECONOMICS
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1. A certain land cost 28,000 pesos now. If it will be appraised at 60,000 pesos after 10 years period. What will be the interest rate per year?What will be the annual amount that the said land is appreciating?
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a straight line.
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a declining line.
1. A certain submersible motor pump costs 500,000.00 pesos, its warranty is 5 years to run smoothly, if its estimated salvage value is 200,000.00 tabulate and find its monetary value at the third year. Graph and check if it is a double declining line.
2. How much interest is payable each year on a loan of P2,000 if the interest rate is 10% per year when half of the loan principal will be repaid as a lump sum at the end of 3 years and the other half will be repaid in one lump sum amount at the end of six years? How much interest will be paid over the 6-year period?
1. The speed of an airplane is 300 nautical miles per hour in a direction N60ºE. The wind velocity at that instant is 5 nautical miles per hour coming from the west. Compute the actual speed of the plane and its direction relative to the ground.
TRIGONOMETRY
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cosine law
90025-((3000)*(-0.642787609)
90025+1928.36282991953.362829
R= 303.24 mph actual speed of the plane
sin ɸ= sine law
300 303.24
sin ɸ =(sin130º)300
303.24
sin ɸ = 0.757859559ɸ =
ɸ =
direction of the plane = 90-49.276direction of the plane = N40.72E
2.Points A & B, which are 100m apart, are of the same elevation as the foot of the building. The angles of elevation of the building top from points A & B are 21 and 32 degrees respectively. How far is the building from point B. Assume that the points A & B are of the same line.
R2= 52+3002.-(2(5)(300)cos130º)
R2= 25+90000-((3000)cos130º)
R2=
R2=
R2=
sin130º
Sin-1 (0.757859559)
49.276º
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180 – 148 – 21 = 11degrees since total interior angle of a triangle is 180 degrees
sine law:
S=
100
sin21 sin11
S = 187.82m
Cos 32 = x/(187.82)
x = 159.28m distance of the building from point B
3.A bus travels from point M Northward for 30 min, then eastward for 1 hour, then shifted N-30degrees-W. If the speed is constant at 40kph, how far directly from M in km will the bus after 2 hrs?
TRIGONOMETRY
Page 100
2hours= 30min north+1hour east then 30 min NW
H= 40-(20(sin30))H= 30km
V= 20+(20(cos30)V= 37.32km
2292.7824D= 47.88km
E.1 Spherical TrigonometrySpherical Triangle
A= Area
E= Spherical Excess
Remember:Sine Law:
sin(a)/asin(A) = (sin(b)/sin(B) =
Cosine Law:
D2= (302+37.322)
D2=
(¶R2E)/180º
A + B + C -180º
sin(c)/sin(C)
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Cos a= cosbcosc+sinbsinccosACos b= cosacosc+sinasinccosBCos c= cosacosb+sinasinbcosC
Cos A= -cosBcosC+sinBsinCcosa
where:abc= sides of the triangle
TRIGONOMETRY
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the part of Mathematics which deals with the sides and angles of a triangle as expressed by the trigonometric functions
a polygon with three (3) sides and a corner with angle of 90 degrees, sum of all angles is 180 degrees
TRIGONOMETRY
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1. What is the angle between a triangle that has a hypotenuse of 5 and an adjacent side of the angle is 3?
TRIGONOMETRY
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At what time after 12 noon will the hour and the minute hands of the clock form an angle of 120 degrees for the first time?Remember when the minute hand has moved x minute spaces, the hour hand has moved x/60 minute spaces.
TRIGONOMETRY
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1. The speed of an airplane is 300 nautical miles per hour in a direction N60ºE. The wind velocity at that instant is 5 nautical miles per hour coming from the west. Compute the actual speed of the plane and its direction relative to the ground.
TRIGONOMETRY
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2.Points A & B, which are 100m apart, are of the same elevation as the foot of the building. The angles of elevation of the building top from points A & B are 21 and 32 degrees respectively. How far is the building from point B. Assume that the points A & B are of the same line.
TRIGONOMETRY
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since total interior angle of a triangle is 180 degrees
3.A bus travels from point M Northward for 30 min, then eastward for 1 hour, then shifted N-30degrees-W. If the speed is constant at 40kph, how far directly from M in km will the bus after 2 hrs?
TRIGONOMETRY
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TRIGONOMETRY
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E. The wind velocity at that instant is 5 nautical miles per hour coming from the west. Compute the actual speed of the plane and its direction relative to the ground.
TRIGONOMETRY
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2.Points A & B, which are 100m apart, are of the same elevation as the foot of the building. The angles of elevation of the building top from points A & B are 21 and 32 degrees respectively. How far is the building from point B. Assume that the points A & B are of the same line.
TRIGONOMETRY
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3.A bus travels from point M Northward for 30 min, then eastward for 1 hour, then shifted N-30degrees-W. If the speed is constant at 40kph, how far directly from M in km will the bus after 2 hrs?
TRIGONOMETRY
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2.Points A & B, which are 100m apart, are of the same elevation as the foot of the building. The angles of elevation of the building top from points A & B are 21 and 32 degrees respectively. How far is the building from point B. Assume that the points A & B are of the same line.
SOLID GEOMETRY
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F. Solid Geometry
Sphere
Volume of a Sphere:
V =4 where:3 ¶= (pi) circular constant equals to 3.1415926535897932384626433832795
r= radiusSurface Area:
SA = 4
Prisms
¶(r2)
¶(r2)
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Volume= A*hLSA= p*h
where:LSA= least surface area
p= perimeter
Cones and Pyramids
Volume= (1/3)(A*h)LSA= (1/2)(p*L)
where:L= slant heightp= Base perimeter
Frustum of a Cone and Pyramid
SOLID GEOMETRY
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Volume=
LSA=
where:L= slant heightp= perimeter
Spherical Segment (1 base)
Volume= ¶(h/3)[((3RA)-h)]SA= 2¶R
where:R= radius of the main sphere
Spherical Segment (2 bases)
Volume=SA= 2¶RH
where:R= radius of the main spherer= radius of the segment in the sphere
(h/3)[(A1+A2 + ((A1A2)(1/2))]
(1/2)(P1+P2)L
¶(h/6)[(3r12+3r2
2+h)]
SOLID GEOMETRY
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like a billiard ball, basketball, tennis ball any ball
radius (r)
(pi) circular constant equals to 3.1415926535897932384626433832795