You can put this solution on YOUR website!
If you invest $P, and you earn interest only on the amount you invested (the $P), then SIMPLE interest is earned.

If you invest $P and you earn interest not only on the $P, but also on the interest gained, then COMPOUND interest is earned.

Let P = principal,
r = annual nominal interest rate (in decimal form),
t = time of the investment (in years),
A = amount in the account after t years. THEN…

SIMPLE INTEREST

A = P + (P * r * t)

Example:

P = 10,000
r = .10
t = 5 years

A = 10,000 + (10,000 * .10 * 5) = 10,000 + 5,000 = 15,000

COMPOUND INTEREST

A = P * (1+r)^t

Example:

P = 10,000
r = .10
t = 5 years

A = 10,000 * (1.10)^5 = 10,000 * 1.61051 = 16,1051

You make more money with compound interest than you do with simple interest, assuming the same investment in the same time frame with the same rate of return.

LAST QUESTION

Ideally, what do you want to happen to the value of n in the compound interest formula if the formula represented the amount of money you have in the bank?

I don't see "n" anywhere in your question.

If "n" is the number of compounding periods per year, than, ideally, you would want n to be as large as possible.

If n = 1, then the number of compounding periods per year is 1.

If n = 12, then the number of compounding periods per year is 12.

When n is greater than 1, you multiply the number of years by n and you divide the nominal interest rate by n.

The nominal interest rate is the annual interest assuming only 1 compounding period per year.

In the compounding example above, n was equal to 1 (not shown in the formula.

with n in the formula, the formula becomes:

A = P * (1 + (r/n))^(t*n)

When n was equal to 1, you got:

A = P * (1 + r) ^ n

That's what we solved earlier.

When n is 12, the formula becomes:

A = P * (1 + (r/12))^(t*12)

Same compounding example but with number of compounding periods per year equal to 12.

P = 10,000
r = .1
t = 5

r/12 = .1/12 = .008333333333
t*12 = 5*12 = 60

formula becomes:

A = 10,000 * (1.0083333333) ^ 60 = 16453.08934

with monthly compounding, the future value is 16,453.08934
with yearly compounding, the future value is 16,16,105.1

Your savings are greater when the number of compounding periods per year are greater.

The most compounding periods per year you can get is with continuous compounding.

That formula is A = P * e^rt)

e is the well known scientific constant of 2.718281828.....

Here you use the annual interest rate and the number of years.

You get:

A = 10,000 * e^(.1*5) = 10,000 * e^.5 = 16,487.21271

That's the theoretical maximum number of compounding periods per year.

If you compounded daily, you would get close to that.

Assume daily compounding with 365 days per year.

P = 10,000
i = .1 / 365 = .000273973
t = 5 * 365 = 1825

A = P * (1.000273973)^1825 = 16486.08362

That's pretty close to the theoretical maximum, but not exactly.

Bottom Line is the more the compounding periods per year, the greater your savings will be.

Without knowing what you meant by "n", this is the best I can come up with based on what I know about compounding.

I can't think of what else "n" can be, given the formulas as presented.

You can put this solution on YOUR website!
A common factor is a factor that can be divided evenly into each of the terms of an expression.

Take the expression 9 + 27 + 54.

3 can be divided evenly into 9 and into 27 and into 54, so 3 is a common factor.

9 can be divided evenly into 9 and into 27 and into 54, so 9 is also a common factor.

The greatest common factor is 9 because that is the largest factor that can be divided evenly into each of the terms of the expression.

You use the common factor in an expression consisting of various terms in order to simplify the expression so that it can be solved easier.

Consider the expression (3x^3 - 17x^2 + 18x - 12) / (x^2 - 5x + 6)

Consider that you know that the numerator in this expression is equal to (x^2 - 5x + 6) * (3x-2)

The expression becomes ((3x-2) * (x^2 - 5x + 6)) / (x^2 - 5x + 6).

the common factor of (x^2 - 5x + 6) cancels out and you are left with a result of (3x-2).

Knowing what the common factor is made the division much simpler than not knowing what the common factor is.

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From the graph, it is clear that the x intercept is around x = 2.

When x = 2, the equation becomes 2 - 2 + 8 - 8 = 0

The answer is x = 2.

That is the only time that the graph crosses the x-axis so that is the only x-intercept of the equation.

I don't quite know how to derive that algebraically.

I looked on the web for a clue.

The bottom line on the search is there is no formula, similar to the quadratic formula, that can find the roots of the equation easily.

There are formulas, but they are much more complex to use than the quadratic formula is for quadratic equations.

A method used is to determine if there are any integral roots to the cubic equation.

If you look at the cubic equation, you will see that it ends in 2.

This indicates some possible roots are multiples of 2, like 1, 2, -1, 2.

Try each one of those values to see if any of them are a root.

It turns out that 2 is a root.

The equation becomes 0 when x = 2.

We just found one of the roots to the equation.

Are there others?

There are, but they are not real.

There is a handy little calculator that does the grunt work for you.

That calculator can be found at the following website address:

http://www.cubicsolver.com/

You just enter the coefficients and constants of the respective terms in the cubic equation after you have set it to standard form (higher order exponents to the left of lower order exponents and set the equation equal to 0) and it does the rest.

The answer shows a real part and an imaginary part.

If the imaginary part is 0, then the root is real.

Otherwise, the root is complex (real part and imaginary part).

I used it for your problem, and it gave the following result.

x = 2
x = +/- i

The following website provides you with some general methods for finding the roots of a cubic equation.
http://everything2.com/title/Cubic+equation

In your problem, I cheated by creating a graph of the equation and eye balling where the roots would be, if any.

I saw it was around 2 and tried to substitute 2 in the equation to see if I got a root.

It appeared that I did at x = 2 because the equation became equal to 0 when i replaced x with 2.

To find the other roots, I needed to divide the cubic equation by x-2.

Doing that I got the quadratic equation of -x^2-1 with a remainder of 0.

To find the roots of this quadratic equation, I needed to set it equal to 0 and solve.

I got:

-x^2-1 = 0

multiply both sides of this equation by -1 to get:

x^2 + 1 = 0

subtract 1 from both sides of this equation to get:

x^2 = -1

take square root of both sides of this equation to get:

x = +/- sqrt(-1) which is equivalent to x = +/- i

Those are the roots of this equation.

They are imaginary because the real part is 0.

If the real part was not zero and the imaginary part was not zero, then the roots would be complex (contain a real part and an imaginary part).

example of a real root = 2
example of an imaginary root = i
example of a complex root = 2 + i

If you do not have a real root to the equation, or an integer root, then there are ways to find the root but they are quite complex.

One way is by iteration.

If x = 1 yields a positive solution and x = 2 yields a negative solution, then the root exists between 1 and 2. In that case you may be able to find it by iterating different values of x between 1 and 2 until you land on the solution.

Computer programs are well suited to this task. Humans would find it a lot more labor intensive.

That second website alludes to the methods for finding the roots.

There is another website that gives similar information.

That website can be found here:

http://www.sosmath.com/algebra/factor/fac11/fac11.html

Another website can be found here:

http://www.math.vanderbilt.edu/~schectex/courses/cubic/

Bottom Line is that there is a general solution to a cubic equation but it is much more complex than the solution for the roots of a quadratic equation.

Fortunately there are calculators available that do the work for you.

I can't think anybody would want to do it manually, at least not over and over and over again.

graphing, as I did, is also a good method to narrow the solution down.

There are also methods to determine how many real roots there will be.

I believe it has to do with Descartes' Rule.

The following website discusses Descartes' Rule, and also the Rational Root Test, both of which are used.

http://www.purplemath.com/modules/drofsign.htm

Bottom Line is, if you have access to a graphing calculator, narrowing down the selection to what the real roots might be is so much easier.

In your case, the only real root is x = 2 which I narrowed down by graphing and then solved by substitution into the original equation.

More help on the web is found by doing a search on "finding the roots of a cubic equation" or some other related search string.

You can put this solution on YOUR website!
The 2 equations that approximate the solution to this are:

y = -9734.7 * x + 218,761

y = -16474 * x + 266,478

If you graph both these equations, you will see when they cross.

The graph of both of these equations is shown below:



You can see from the graph that the approximate intersection point is somewhere around 7 years.

You can calculate that by making the equations equal to each other.

When you do that, you are saying that the value of y in the first equation is equal to the value of y in the second equation.

you get:

-9734.7x + 218761 = -16474 * x + 266,478

Solve this equation algebraically to get an answer of:

x = 7.080408945
y = 149835.343

The approximation given by the graph is pretty close.

x is the number of years when the balances will be approximately equal.
y i the outstanding balance in each account at that time.

The estimate will be somewhat inaccurate because the declining balance equation is not a straight line.

What the equations show are probability the best fit of a straight line to the actual balances as they drop in time.

The actual point where the balances are equal can be calculated through the use of Present Value of A Loan Amount Formulas, or through the use of a mechanized program that will do the calculations for each year of each loan.

I used Excel to do that for you, after having calculated what the monthly payments would be using a financial calculator, and the results are shown below:

	Remaining Balance		
year	Loan1	Loan2	
0	$200,000.00	$250,000.00	
1	$199,616.07 	$249,158.53 	
2	$199,229.90 	$248,312.67 	
3	$198,841.47 	$247,462.41 	
4	$198,450.79 	$246,607.72 	
5	$198,057.82 	$245,748.58 	
6	$197,662.56 	$244,884.96 	
7	$197,264.99 	$244,016.84 	
8	$196,865.10 	$243,144.21 	
9	$196,462.89 	$242,267.03 	
10	$196,058.32 	$241,385.28 	
11	$195,651.40 	$240,498.93 	
12	$195,242.10 	$239,607.98 	
13	$194,830.41 	$238,712.38 	
14	$194,416.33 	$237,812.11 	
15	$193,999.82 	$236,907.16 	
16	$193,580.89 	$235,997.50 	
17	$193,159.52 	$235,083.09 	
18	$192,735.68 	$234,163.93 	
19	$192,309.38 	$233,239.97 	
20	$191,880.58 	$232,311.21 	
21	$191,449.29 	$231,377.60 	
22	$191,015.48 	$230,439.14 	
23	$190,579.14 	$229,495.79 	
24	$190,140.25 	$228,547.52 	
25	$189,698.80 	$227,594.31 	
26	$189,254.78 	$226,636.14 	
27	$188,808.17 	$225,672.98 	
28	$188,358.95 	$224,704.81 	
29	$187,907.12 	$223,731.59 	
30	$187,452.64 	$222,753.30 	
31	$186,995.52 	$221,769.91 	
32	$186,535.73 	$220,781.41 	
33	$186,073.26 	$219,787.75 	
34	$185,608.09 	$218,788.92 	
35	$185,140.20 	$217,784.89 	
36	$184,669.59 	$216,775.63 	
37	$184,196.23 	$215,761.11 	
38	$183,720.11 	$214,741.31 	
39	$183,241.21 	$213,716.20 	
40	$182,759.52 	$212,685.75 	
41	$182,275.02 	$211,649.93 	
42	$181,787.69 	$210,608.72 	
43	$181,297.52 	$209,562.08 	
44	$180,804.50 	$208,509.99 	
45	$180,308.59 	$207,452.42 	
46	$179,809.79 	$206,389.35 	
47	$179,308.09 	$205,320.74 	
48	$178,803.45 	$204,246.56 	
49	$178,295.87 	$203,166.78 	
50	$177,785.34 	$202,081.39 	
51	$177,271.82 	$200,990.34 	
52	$176,755.31 	$199,893.60 	
53	$176,235.78 	$198,791.16 	
54	$175,713.23 	$197,682.97 	
55	$175,187.62 	$196,569.02 	
56	$174,658.95 	$195,449.25 	
57	$174,127.20 	$194,323.66 	
58	$173,592.34 	$193,192.21 	
59	$173,054.37 	$192,054.86 	
60	$172,513.25 	$190,911.59 	
61	$171,968.98 	$189,762.36 	
62	$171,421.53 	$188,607.15 	
63	$170,870.90 	$187,445.92 	
64	$170,317.05 	$186,278.65 	
65	$169,759.96 	$185,105.29 	
66	$169,199.63 	$183,925.82 	
67	$168,636.03 	$182,740.21 	
68	$168,069.14 	$181,548.43 	
69	$167,498.95 	$180,350.44 	
70	$166,925.43 	$179,146.20 	
71	$166,348.56 	$177,935.70 	
72	$165,768.33 	$176,718.89 	
73	$165,184.72 	$175,495.74 	
74	$164,597.70 	$174,266.23 	
75	$164,007.25 	$173,030.31 	
76	$163,413.36 	$171,787.95 	
77	$162,816.01 	$170,539.12 	
78	$162,215.17 	$169,283.79 	
79	$161,610.83 	$168,021.92 	
80	$161,002.96 	$166,753.48 	
81	$160,391.55 	$165,478.43 	
82	$159,776.57 	$164,196.74 	
83	$159,158.00 	$162,908.37 	
84	$158,535.82 	$161,613.29 	
85	$157,910.02 	$160,311.47 	
86	$157,280.56 	$159,002.87 	
87	$156,647.43 	$157,687.45 	
88	$156,010.61 	$156,365.18 	*
89	$155,370.07 	$155,036.03 	*
90	$154,725.80 	$153,699.95 	
91	$154,077.77 	$152,356.92 	
92	$153,425.96 	$151,006.88 	
93	$152,770.35 	$149,649.82 	
94	$152,110.91 	$148,285.69 	
95	$151,447.63 	$146,914.45 	
96	$150,780.47 	$145,536.08 	
97	$150,109.43 	$144,150.52 	
98	$149,434.47 	$142,757.75 	
99	$148,755.57 	$141,357.72 	
100	$148,072.71 	$139,950.40 	
101	$147,385.87 	$138,535.75 	
102	$146,695.03 	$137,113.73 	
103	$146,000.15 	$135,684.31 	
104	$145,301.22 	$134,247.44 	
105	$144,598.21 	$132,803.09 	
106	$143,891.10 	$131,351.22 	
107	$143,179.87 	$129,891.78 	
108	$142,464.49 	$128,424.74 	
109	$141,744.93 	$126,950.06 	
110	$141,021.18 	$125,467.71 	
111	$140,293.21 	$123,977.63 	
112	$139,560.99 	$122,479.79 	
113	$138,824.49 	$120,974.14 	
114	$138,083.71 	$119,460.66 	
115	$137,338.60 	$117,939.29 	
116	$136,589.14 	$116,410.00 	
117	$135,835.31 	$114,872.75 	
118	$135,077.09 	$113,327.49 	
119	$134,314.44 	$111,774.18 	
120	$133,547.34 	$110,212.78 	
121	$132,775.77 	$108,643.25 	
122	$131,999.70 	$107,065.54 	
123	$131,219.10 	$105,479.61 	
124	$130,433.94 	$103,885.43 	
125	$129,644.21 	$102,282.94 	
126	$128,849.87 	$100,672.11 	
127	$128,050.90 	$99,052.89 	
128	$127,247.26 	$97,425.23 	
129	$126,438.94 	$95,789.10 	
130	$125,625.90 	$94,144.44 	
131	$124,808.12 	$92,491.22 	
132	$123,985.57 	$90,829.39 	
133	$123,158.22 	$89,158.90 	
134	$122,326.05 	$87,479.71 	
135	$121,489.02 	$85,791.78 	
136	$120,647.11 	$84,095.05 	
137	$119,800.29 	$82,389.49 	
138	$118,948.52 	$80,675.05 	
139	$118,091.79 	$78,951.67 	
140	$117,230.06 	$77,219.32 	
141	$116,363.31 	$75,477.95 	
142	$115,491.49 	$73,727.50 	
143	$114,614.60 	$71,967.94 	
144	$113,732.58 	$70,199.22 	
145	$112,845.43 	$68,421.28 	
146	$111,953.09 	$66,634.09 	
147	$111,055.56 	$64,837.58 	
148	$110,152.78 	$63,031.72 	
149	$109,244.74 	$61,216.45 	
150	$108,331.40 	$59,391.73 	
151	$107,412.74 	$57,557.51 	
152	$106,488.72 	$55,713.73 	
153	$105,559.30 	$53,860.35 	
154	$104,624.47 	$51,997.31 	
155	$103,684.18 	$50,124.58 	
156	$102,738.41 	$48,242.08 	
157	$101,787.11 	$46,349.79 	
158	$100,830.27 	$44,447.64 	
159	$99,867.85 	$42,535.58 	
160	$98,899.82 	$40,613.56 	
161	$97,926.14 	$38,681.53 	
162	$96,946.77 	$36,739.44 	
163	$95,961.70 	$34,787.23 	
164	$94,970.88 	$32,824.86 	
165	$93,974.28 	$30,852.27 	
166	$92,971.86 	$28,869.40 	
167	$91,963.60 	$26,876.20 	
168	$90,949.46 	$24,872.63 	
169	$89,929.40 	$22,858.61 	
170	$88,903.39 	$20,834.11 	
171	$87,871.39 	$18,799.06 	
172	$86,833.38 	$16,753.42 	
173	$85,789.31 	$14,697.12 	
174	$84,739.15 	$12,630.11 	
175	$83,682.86 	$10,552.33 	
176	$82,620.41 	$8,463.74 	
177	$81,551.77 	$6,364.26 	
178	$80,476.89 	$4,253.85 	
179	$79,395.74 	$2,132.45 	
180	$78,308.28 	($0.00)	
181	$77,214.48 		
182	$76,114.30 		
183	$75,007.71 		
184	$73,894.65 		
185	$72,775.11 		
186	$71,649.03 		
187	$70,516.39 		
188	$69,377.13 		
189	$68,231.24 		
190	$67,078.65 		
191	$65,919.35 		
192	$64,753.28 		
193	$63,580.41 		
194	$62,400.70 		
195	$61,214.10 		
196	$60,020.59 		
197	$58,820.11 		
198	$57,612.63 		
199	$56,398.10 		
200	$55,176.50 		
201	$53,947.76 		
202	$52,711.86 		
203	$51,468.75 		
204	$50,218.38 		
205	$48,960.73 		
206	$47,695.73 		
207	$46,423.36 		
208	$45,143.56 		
209	$43,856.30 		
210	$42,561.53 		
211	$41,259.21 		
212	$39,949.29 		
213	$38,631.73 		
214	$37,306.49 		
215	$35,973.51 		
216	$34,632.76 		
217	$33,284.18 		
218	$31,927.74 		
219	$30,563.39 		
220	$29,191.08 		
221	$27,810.76 		
222	$26,422.39 		
223	$25,025.93 		
224	$23,621.31 		
225	$22,208.51 		
226	$20,787.46 		
227	$19,358.12 		
228	$17,920.45 		
229	$16,474.38 		
230	$15,019.89 		
231	$13,556.90 		
232	$12,085.39 		
233	$10,605.29 		
234	$9,116.56 		
235	$7,619.14 		
236	$6,112.98 		
237	$4,598.05 		
238	$3,074.27 		
239	$1,541.61 		
240	$0.00 		

The asterisks (*) show you that the balances become equal somewhere between the 88th and 89th month.
This would be somewhere between 7.33333333 years and 7.41666666667 years.

The actual graph of the declining balance is not a straight line.

It drops slowly in the early years and drops a lot faster in the later years.

You can link to the following webiter and scroll down to the bottom and a graph of a remaining balance on a loan will show up.

You can see that it is not a straight line.

http://www.tvmcalcs.com/calculators/apps/excel_loan_amortization

You can put this solution on YOUR website!
The solution that I came up with is:



I checked the solution out by assuming x was equal to 2 and solving both the original expression and the final expression.

They came out to the same answer leading to the conclusion that I simplified correctly.

My manual calculations are in the picture shown below:



first I factored.
then I found common denominator.
then I combined numerator under same denominator.
then I performed calculations on numerator.
result was final answer.

You can put this solution on YOUR website!
c = hypotenuse
a = the shorter leg
b = the longer leg
formula is:

c^2 = a^2 + b^2

becomes:

c^2 = 9^2 + 17^2 = 370

c = sqrt(370) = 19.23538406

You can put this solution on YOUR website!
first problem:

Students sit a statistics test and then are tested with equivalent tests, monthly after that. The average test result, T, after n months, was found to be given by T=17.4-4.6log(n+1). After how many months would the average mark be expected to be less than 12 marks?

T is the average test result (otherwise known as mark).

n is the number of months.

You want to know in how many months, T will be less than or equal to 12.

your formula to solve is T <= 12

Since T = 17.4 - 4.6*log(n+1), then you can substitute that formula for T to get:

17.4 - 4.6 * log(n+1) <= 12

You need to solve this equation.

subtract 17.4 from both sides of this equation to get:

-4.6 * log(n+1) <= (12-17.4)

simplify this to get:

-4.6*log(n+1) <= -5.4

divide both sides of this equation by -4.6 to get:

log(n+1) >= -5.4 / -4.6 (multiply both sides by -1 reverses inequality)

simplify this to get:

log(n+1) >= 1.173913043

your equation is now:

log(n+1) >= 1.173913043

if log(n+1 = x, then n+1 = the number whose log is x (the anti-log of x).

the number whose log is 1.173913043 is equal to 14.92495545 because the log of 14.92495545 = 1.173913043.

you get:

n+1 >= 14.92495545

subtract 1 from both sides of this equation to get:

n >= 13.92495545

that should be your answer.

your original equation that you tried to solve is:

17.4 - 4.6 * log(n+1) <= 12

this equation should be true when n >= 13.92495545

Let's take n = 14

we get:

17.4 - 4.6 * log(15) <= 12

this simplifies to 11.98998021 which is less than 12 to the equation is true.

so if n > n >= 13.92495545, the equation it true.

let's take n = 13.92495545

we get:

17.4 - 4.6 * log(14.92495545) <= 12

this simplifies to 12 which is equal to 12 so the equation is true.

so if n = 13.92495545, the equation is true.

let's take n = 14.

we get:

17.4 - 4.6 * log(*14) <= 12

this simplifies to 12.12781104 which is not <= 12 so the equation is false.

so if n > 13.92495545, the equation is false.

when n >= 13.92495545, the equation is true.

when n < 13.92495545, the equation is false.

this confirms the answer is correct.

-------------------------------------------------------

second problem:

The sales of a new electronic gadget are growing exponentially such that, 10 000 gadgets were sold in 2000 and 82 700 gadgets were sold in 2002. If this trend continues when will sales of the gadget reach over 500 000?

10,000 gadgets were sold in 2000
82,700 gadgets were sold in 2002.

the exponential growth rate would be using the formula:

f = p * (1+i)^n

i is the annual growth rate
n is the number of years

you get:

82,700 = 10,000 * (1+i)^2

divide both sides of this equation by 10,000 to get:

82,700 / 10,000 = (1+i)^2

take the square root of both sides of this equation to get:

+/- sqrt(82,700/10,000) = 1+1

since I know the answer will not be negative, we'll stick to the positive answer only.

solve for 1+i to get 1+i = 2.875760769

subtract 1 from both sides of this equation to get:

i = 1.875760769

that is the growth rate per year.

it is equivalent to 187.5760769% growth rate per year.

the formula for growth is again:

f = p * (1+i)^n

first you want to test if your growth rate is good.

go from 2000 to 2002 using that growth rate.

your formula is:

f = p * (1+i)^n

p = 10,000
i = 1.875760769
n = 2

you get:

f = 10,000 * (2.875760769)^2

solve for f to get f = 82,700

the growth rate is good.

start from 2002

formula is f = p * (1+i)^n

f = 500,000
p = 82,700
i = (2.875760769)
n is what you want to find.

formula becomes:

500,000 = 82,700 * (2.875760769)^n

divide both sides of the equation by 82,700 to get:

500,000 / 82,700 = (2.875760769)^n

take log of both sides to get:

log(500,000/82,700) = log((2.875760769)^n)

by laws of logarithms, log(x^y) = y*log(x), so you get:

log(500,000/82,700) = n * log((2.875760769)

divide both sides of this equation by log((2.875760769) to get:

log(500,000/82,700) / log((2.875760769) = n

solve for n to get n = 1.70345461

That's the number of years it will take from 2002 to reach 500,000.

That + 2 is the number of years it will take from 2000 to reach 500,000.

from 2000, you get 10,000 * (2.875760769)^3.70345461 = 500,000

from 2002, you get 82,700 * ((2.875760769)^1.70345461 = 500,000

the question was:

If this trend continues when will sales of the gadget reach over 500 000?

Those sales will reach over 500,000 in 1.70345461 years from 2002.

That puts you sometime in 2004.

.70345461 * 12 = 8.441455315 which means sometime in august of 2004.

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will pay $5000 in 15 years.
interest rate per year is 4.11% compounded semiannual.

formula to use is f = p * (1+i)^n

your time periods need to be semi-years.

because of that, you need to divide your interest rate by 2 and you need to multiply your years by 2.

in your formula:

f = 5000
p = what you want to find
i = .0411 / 2 = .02055
n = 15 * 2 = 30

your formula becomes:

5000 = p * (1.02055)^30

your answer should be that you are willing to pay $2,716.072716 for the savings bond.

Let's see if that works.

1.02055^30 = 1.840893276

your formula becomes 5000 = p * 1.840893276

divide both sides of this equation by 1.840893276 and you get:

p = 5000 / 1.840893276 = $2,716.072716

We're good.

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problem is:

(a^2 - b^2)/(a-b)

(a^2 - b^2) can be factored to be (a-b) * (a+b)

your expression becomes:

((a-b) * (a+b)) / (a-b)

The (a-b) in the numerator and denominator cancel out and you are left with:

1 * (a+b) / 1 = (a+b)

you had to know that (a^2 - b^2) is equivalent to (a+b) * (a-b).

when you take (a+b) and multiply it by (a-b), you get:

a^2 + ab - ab - b^2 which becomes a^2 - b^2 because the +ab and -ab cancel out.

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-7yt/3x=?/18xyt
it helps to separate what is being multiplied by each other by *.

it also helps to surround operations that are going to be performed together by parentheses.

your formula becomes:

(-7*y*t) / (3*x) = ? / (18*x*y*t)

multiply both sides of this equation by (18*x*y*t)

you get:
(18*x*y*t) * ((-7*y*t) / (3*x)) = ?

since a * (b/c) = (a*b) / c, then your expression above is equivalent to:

((18*x*y*t) * (-7*y*t)) / (3*x) = ?

(18*x*y*t) * (-7*y*t) is equal to (-126 * x * y^2 * t^2)

this is because 18 * -7 = -126 and y * y = y^2 and t * t = t^2.

(-126 * x * y^2 * t^3) / (3 * x) is equal to (-42 * y^2 * t^2)

this is because -126/3 = -42 and x/x = 1 which is not shown but implied.

your answer should be ? = (-42 * y^2 * t^2)

you can confirm it by assigning values to each of the variables at random and seeing if the final expression is true after simplification.

Assume x = 2, y = 3, t = 4

your original expression of (-7*y*t) / (3*x) = ? / (18*x*y*t) becomes (-7*y*t) / (3*x) = (-42 * y^2 * t^2) / (18*x*y*t) which becomes (-7*3*4) / (3*2) = (-42 * 3^2 * 4^2) / (18*2*3*4)

this resolves to -84/6 = -6048 / 432 which resolves finally to -14 = -14 confirming that we did the calculations correctly.

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y = b * e^rt

sounds like this is the continuous compounding formula.

george has $65 to invest.

at 8.2% continuous interest, how long will it take for george to accrue $100?

y = $100
b = $65
r = .082
t = time in years

formula is $100 = $65 * e^(.082*t)

divide both sides of this formula by 65 to get:

100/65 = e^(.082*t)

take log of both sides to get:

log(100/65) = log(e^(.082*t))

by laws of logarithms, log(x^y) = y*log(x).

formula becomes log(100/65) = .082*t * log(e)

divide both sides of equation by log(e) to get:

log(100/65) / log(e) = .082*t

divide both sides of this equation by .082 to get:

t = (log(100/65) / log(e)) / .082

log(100/65) = .187086643
log(e) = log(2.718281828) = .434294482

log(100/65) / log(e) = .430782916

.430782916 / .082 = 5.253450196

you get t = (log(100/65) / log(e)) / .082 = 5.253450196.

If we did this right, then t = 5.253450196 years

your original formula is:

100 = 65 * e^(.082*t)

this becomes:

100 = 65 * e^(.082*5.253450196)

e represents the scientific constant of 2.718281828

100 = 65 * e^(.082*5.253450196) becomes:

100 = 65 * 2.718182818^(.082*5.253450196).

Use your calculator to see that 100 = 100, making t = 5.253450196 correct.

--------------------------------

second problem.

At 7.6% interest compound monthly, how much money will George have in 12 years?

Formula they show is:

A=P(1+(r/n)^nt

A = future value you want to find.
P = $65.00
r = .076 / 12 = .006333333
t = 12 * 12 = 144
the n they are showing is equal to 12.
to compound monthly you take the number of years and multiply by 12 and you take the annual interest rate and divide it by 12.

formula becomes:

A = 65 * (1 + (.076/12))^(12*12)

this becomes:

A = 65 * (1.00633333)^144 = 161.3395912.

In 12 years, george will have that much.

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if x is -5, 1, or 4, then x-1 >= 7 is false.
if x is 20, then x-1 >= 7 is true, because 19 >= 7
-5-1 = -6
1-1 = 0
4-1 = 3
20-1 = 19

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your expression is:

-3*(81-(68-98))

You solve from the inner parentheses out.

The inner parentheses contain (68-98).

that equals (-30)

your expression becomes:

-3*(81-(-30))

you remove that inner parentheses to get:

-3*(81 + 30) because a minus times a minus gives a plus.

you solve the inner parentheses again to get:

-3*(111)

you remove that inner parentheses to get:

-333.

that's your answer.

you can use your scientific calculator to check your work.

you always solve the expression within the innermost set of parentheses first, then you solve the expression within the remaining innermost set of parentheses, etc. until you have no more sets of innermost parentheses to solve.

Your last set of innermost parentheses is the expression itself.

an expression of a*b is really an expression of (a*b) where the expression itself is within the innermost set of parentheses.

the expression of (a * (b+c)) contains (b+c) within the innermost set of parentheses.

once that's solved (let's call the result of (b+c) = d, then your expression becomes (a * d)

Once you resolve that, then you have your answer.

You should use () always, rather than [].

you showed your expression as:

-3[81-(68-98]

It should really have been shown as:

-3(81-(6-98))

Placing the proper number of parentheses surrounding the operation is crucial.

Even using the brackets as you did, you still needed to have that extra parentheses within as shown below:

-3[81-(68-98)]

Not having the correct number of parentheses in the right places will definitely throw you off.

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If the line is parallel then it has the same slope.

The line it is parallel to is the line given by the equation -4x - 9y = -7

convert that line to the slope intercept form by solving for y.

you will get y = (-4/9)*x + (7/9)

The slope of that line is equal to (-4/9)

That will be the slope of your new line.

The general equation for the slope intercept form is y = mx + b where m is the slope and b is the y intercept.

Since you have the slope, the general form becomes y = (-4/9)*x + b

To find the y intercept, you substitute the point values that the line passes through.

The point values that you have that the line passes through are (-1,2).

That means that x = -1 and y = 2.

Substitute in the equation of y = (-4/9)*x + b to get:

2 = (-4/9)*(-1) * b

Simplify to get:

2 = (4/9) * b

Solve for b to get:

b = 2 - (4/9) = (18/9) - (4/9) = (14/9)

The equation of your parallel line should be:

y = (-4/9)*x + (14/9)

graph both equations and you will see that they are parallel.



To prove that the slope intercept form of your original equation is good, you need to substitute the (x,y) values derived from the slope intercept form of the equation and see if the original equation is true.
Your original equation is -4x-9y=-7.

The slope intercept form of this equation is y = (-4/9)*x + (7/9)

If we let x = 18, then y = (-4/9)*18 + (7/9) = -7.22222222

If we let x = 18 and y = -7.22222222, then -4x - 9y = -7 becomes -72 + 65 = -7

the translation from the standard form of the equation (-4x-9y=-7) to the slope intercept form of the equation ( y = (-4/9)*x + (7/9)) is good.

The line parallel to it is therefore good because we calculated that correctly based on the slope intercept form of the original equation.

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A = 4, B = 3, C = 2, D = 1, F = 0

The following chart shows you how the calculation is set up.

Class	        Hours	Grade	Grade Points	Grade Point Hours
Art	          2	  B	     3	                6
History	          3	  A	     4	               12
Science	          4	  C	     2	                8
Mathematics	  3	  B	     3	                9
Science Lab	  1	  A	     4	                4

A student receives his grade report from a local community college, but the GPA is smudged. He took the following classes: a 2 hour credit art, a 3 hour credit history, a 4 hour credit science course, a 3 hour credit mathematics course, and a 1 hour science lab. He received a B in the art class, an A in the history class, a C in the science class, a B in the math class, and an A in the science lab. What was his GPA if the letter grades are based on a 4 point scale? (A = 4, B = 3, C = 2, D = 1, F = 0)

The total grade point hours are equal to 39
The total hours are equal to 13
The grade point average = 39/13 = 3

This equates to an overall average of B.

You multiply the grade points by the hours to get the grade point hours.

The total grade point hours divided by the total hours equals the grade point average.

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I presume the figure is a rectangle?

It has length of 10 yards and a width of 13 yards?

The area would be length * width = 10 * 13 = 130 square yards.

You posted this under triangles?

If this figure is a triangle, then we need to make some assumptions.

The area of a triangle is 1/2 * b * h where b is the base and h is the height.

If you assume that the base is 10 yards and the height is 13 yards, then the area would be 1/2 * 10 * 13 = 65 square yards.

If you assume that the base is 13 yards and the height is 10 yards, you would get the same area equal to 1/2 * 13 * 10 = 65 square yards.

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the formula you gave is N(t) = 3t^3 + 23t^2 + 8t

I'll make x = t so this can be graphed and referenced easily.

the equation becomes n(x) = 3x^3 + 23x^2 + 8x

the graph of this formula is shown below:



the formula as given doesn't make any sense.

there is no maximum when x is greater than 0.

x cannot be less than 0 so any values on the graph where x is less than 0 are to be ignored.

your graph would make more sense if the equation was:

N(t) = -3t^3 + 23t^2 + 8t which I would translate to:

n(x) = -3x^3 + 23x^2 + 8x.

that graph is shown below:



now you have a maximum when x is greater than 0 which is more realistic.

I'll solve for n(x) = -3x^3 + 23x^2 + 8x.

x can be factored out to get:

n(x) = x * (-3x^2 + 23x + 8)

(-3x^2 + 23x + 8) factors out to be (3x+1) * (-x+8)

the completely factored equation becomes:

n(x) = x * (3x+1) * (-x+8)

when x = 0, n(x) = 0 as it should because no time worked = no components produced.

when x = 3, n(x) = (3) * (10) * (5) = 150 using the factored equation.

when x = 3, n(x) = -3(3^3) + 23*(3^2) + 8*3 = -81 + 207 + 24 = 150

you get the same answer with the factored equation and the non factored equation which confirms that the factorization is good.

I believe we need to use calculus to find the maximum / minimum point of this cubic equation.

I couldn't find any formula not using calculus on the web, so we'll use calculus.

The maximum point occurs when the derivative of the equation equals 0.

the equation is -3x^3 + 23x^2 + 8x

The derivative of this equation is -9x^2 + 46x + 8

This derivative is 0 when x = 5.279477927

That could not be factored easily, so I used the quadratic formula of:

x = (-b +/- sqrt(b^2-4ac))/(2a)

When x = 5.279477927, n(x) = 241.8493507

I'll add a line at n(x) = 241.8493507 to the graph of the equation and it should confirm that is the maximum point.

the graph looks like this:



The graph confirms the maximum point.

Since x represents t which represents the number of hours after starting work, then the maximum amount of output per hour occurs 5.279477927 hours after work starts.

The maximum amount of output per hour during the 8 hour shift becomes 241.8493507 units per hour which occurs at that time.

Best I can do in the short period of time allotted to the solving of this problem.

I'm not a calculus guru, but I know a little bit about it, and the little bit I know was helpful in this case.

If you were not to use calculus, then I don't really know how to find the maximum point of the cubic equation other than by iteration.

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let y = number of cans sold.

let x = price per can.

when x = .8, y = 6500
when x = 1, y = 4000

you have 2 points from which you can generate a linear equation.

the point slope form of the linear equation is y = mx + b.

m is the slope and b is the y intercept.

the general form of the slope of the line is (y2-y1)/(x2-x1)

you have 2 points.

they are:

(.8,6500) and (1,4000)

x1 = .8
y1 = 6500

x2 = 1
y2 = 4000

your slope is (y2-y1)/(x2-x1) which becomes (4000-6500) / (1-.8).

this becomes -2500 / .2 which becomes -12500.

that's your slope and the slope intercept form of your equation becomes:

y = -12500*x + b

to find the y intercept, substitute one of the points for y and x.

we'll use (.8,6500)

the equation becomes:

6500 = -12500*(.8) + b

that becomes 6500 = -10000 + b

solve for b to get b = 16500

your equation becomes:

y = -12500*x + 16500

when x = .8, y becomes 6500
when x = 1, y becomes 4000

The slope is -12500
the y intercept is 16500
the equation is y = -12500*x + 16500

graph this equation and it looks like this:



when x = 0, y = 16500
when x = .8, y = 6500
when x = 1, y = 4000

It's hard to see from the graph, but if you plot real careful the intersection of x = 1 and y = 4000, you will see that they intersect on the line of y = -12500*x + 16500.

The more exact way is to simply solve the equation of y = -12500*x + 16500 when x = .8

you get y = -12500*(.8) + 16500 which becomes y = -10000 + 16500 which becomes y = 6500.

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he bought 70 for 9730 at a cost of 9730/70 = 139 apiece.

he had 13 left at a cost of 1807.

cost of goods sold is 9730 - 1807 = 7923

number of goods sold is 70 - 13 = 57

57 * 139 = 7923

number confirm each other.

7923 is your answer.

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the measure of an angle plus its complement is equal to 90 degrees.

if the angle is 75 degrees, then its complement must be equal to 15 degrees because 75 + 15 = 90.

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f(x) = 9x + 1

let y = f(x)

you get y = 9x + 1

let y = x and x = y

you get x = 9y + 1

solve for y to get y = (x-1)/9

that's your inverse function.

if it is a true inverse function, it will be a reflection of the original function about the line y = x.

to show that, then graph the functions y = x, y = 9x+1, y = (x-1)/9.

that graph is shown below:



you can pretty much eyeball it and see that the graph of y = (x-1)/9 is a reflection of y = 9x+1 about the line y = x.

a more exact interpretation is that f(x,y) in the normal equation should be equal to f(y,x) in the inverse equation.

how you find that out is as follows:

let x = 5 in the normal equation.

then you get y = 9x + 1 which becomes 46

in your normal equation, when x = 5, then y = 46

in your inverse equation, when x = 46, you should get y = 5.

the y in the normal equation becomes the x in the inverse equation.

the x in the normal equation becomes the y in the inverse equation.

when x = 46, y = (x-1)/9 becomes y = (46-1)/9 becomes y = 45/9 becomes 5.

f(x,y) = f(5,46) in your normal equation.

f(y,x) = f(46,5) in your inverse equation.

this proves they are inverse equations.

the inverse equation undoes what the normal equation does.

normal equation takes 5 and makes it 46.

inverse equation takes 46 and makes it 5.

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that expression would be shown as:

log(b,(2y+5)) - 4*log(b,(y+3))

in general x * log(y) = log(y^x)

your expression becomes:

log(b,(2y+5)) - log(b,((y+3)^4))

in general log(x) - log(y) = log(x/y)

your expression becomes:

log(b,((2y+5)/(y+3)^4))

to show you how this works, we will let b = 10 because your calculator can do logs to the base of 10 (usually called the LOG function).

we will let y = 5 (chosen at random small enough to calculate easily).

your original expression becomes:

log(2y+5) - 4*log(y+3)

the base of 10 is implied.

substituting 5 for y, we get:

log(2*5+5) - 4*log(5+3) which becomes:

log(15) - 4*log(8) which becomes -2.436268689

looking at our final expression of:

log(b,(2y+5)/(y+3)^4), we get:

log((2y+5)/(y+3)^4).

the base of 10 is implied.

substituting 5 for y, we get:

log(15/8^4) which becomes log(15/4096).

using our calculator, we get log(15/4096) = log(.003662109) which equals -2.436268689

we get the same answer either way, so the translation is good, and the answer to your question is:

log(b,2y+5) - 4*log(b,y+3) = log(b,(2y+5)/(y+3)^4) which looks like:

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log(2,6x/y) is equal to the log of 6x/y to the base of 2.

it means the same as what you wrote as log2(6x/y).

the notation, however, is more in line with the way the algebra.com formula generators work.

using the algebra.com formula generator, log(2,6x/y) will show up as

All you do is put 3 { in front of it and 3 } behind it.

in general, log(x/y) = log(x) - log(y)

your equation becomes:

log(2,6x/y) = log(2,6x) - log(2,y)

in general, log (x*y) = log(x) + log(y)

your equation becomes:

log(2,6x/y) = log(2,6) + log(2,x) - log(y)

the concept is the same regardless of the base.

if the base were 10, then it would be shown as:

log(10,6x/y) = log(10,6) + log(10,x) - log(10,y)

to show you how it works, we'll use log to the base 10 because your calculator can handle that.

also, log(10,x) is normally shown as log(x).

the base of 10 is implied.

let's take log(6*15/30)

this should be translated to log(6) + log(15) - log(30) which is the same treatment we provided above.

using our calculator, we get log(6*15/30) = log(3) = .477121255

using our calculator again, we get log(6) + log(15) - log(30) = :

.77815125 + 1.176091259 - 1.477121255 = .477121255

we get the same answer, as we should.

same concepts works with any base, so the answer to your question is:

log(2,6x/y) = log(2,6) + log(2,x) - log(2,y).

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you are given:

f(x) = 8x+6
g(x) = x^2

you want to find:

g(f(x))

start with g(x) = x^2

replace x with f(x) to get g(f(x)) = (f(x))^2

replace f(x) with 8x+6 to get g(8x+6) = (8x+6)^2

solve by squaring 8x+6 to get g(f(x)) = 64x^2 + 96x + 36

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A local concert center reports that it has been experiencing a 15% rate of no-shows on advanced reservations. Among 150 advanced reservations, find the probability that there will be fewer than 20 no-shows.

15% * 150 = 22.5

This looks like a normal distribution type of problem.

The probability that there will be fewer than 20 no-shows is equal to the probability that there are 0 no-shows plus 1 no-show plus 2 no-show + ... 19 no-shows.

I scoured the web for an answer because this problem appears to be too difficult to manually calculate.

I figured there must be a formula or calculator that allows you to calculate this.

What I found was on stattrek.com

It's called a binomial distribution calculator.

It's used to calculate probabilities where the outcome is either success or failure.

success in your problem would be a no-show.

you would go to the following web address:

http://stattrek.com/Tables/Binomial.aspx#experiment

You would enter the probability of success as being .15
You would enter the number of trials as 150.
You would enter the number of successes as 20.
The calculator does the rest.

The calculator gives you all the pertinent statistics.
In this case, the statistic you are looking for would be p(x<20) = .250928195415981 which you can round to whatever decimal place you require.

It's hard to manually calculate with 150 trials, but if the number of trials were something smaller, like 3, then it would be easier to manually calculate the probability of getting less than 2 no shows.
the calculator would give you the answer of:
p(x<2) = .93925

You would manually calculate it as follows:

p(x=0) = .85*.85*.85 = .614125 * 1 = .614125
p(x=1) = .15*.85*.85 = .108375 * 3 = .325125
p(x=2) = .15*.15*.85 = .019125 * 3 = .057375
p(x=3) = .15*.15*.15 = .003375 * 1 = .003375

If you did it correctly, then the total probability should equal 1 which it does.

If you did it correctly, then the p(x<2) = .614125 + .325125 = .93925 which agrees with what the calculator did for you.

The multipliers are required based on the combination formula of n! / (x! * (n-x)!)

for x = 1, you get n = 3 and x = 1 to get 3!/(1!*2!) = 3

for x = 2, you get n = 3 and x = 2 to get 3! / (2!*1!) = 3

for x = 0, you get n = 3 and x = 0 to get 3! / (0!*3!) = 3!/3! = 1

for x = 3, you get n = 3 and x = 3 to get 3! / (3!*0!) = 3!/3! = 1

The binomial calculator does all the work for you.

There may be a formula to allow you to do it manually, but I don't know it.

I think the calculator just crunches the numbers, but I won't swear to that.

Try it out and see how it works for you.

It's easy to calculate each probability.

Just change the number of successes and crunch the numbers again using the calculator.

Make sure you hit the calculate button.

Just hitting the return doesn't recalculate.

It might look like it does, but it doesn't.

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probability of total loss of one risky investment = .39

you take 5 risky investments, each with a probability of total loss of .39.

since the probability of total loss of each investment is .39, then the probability that the investment will not be a total loss is 1 - .39 = .61

the probability that none of the 5 investments will be a total loss is .61^5 = .08445963.

the probability that at least one of these investments will be a total loss is 1 - .61^5.

that probability would be .91554037

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I believe I answered the first part of your question.
If not, let me know and I will do it again.
You should already have the answer for that part.

The second part of your question is:

FIND THE CENTRAL ANGLE OF A CIRCLE WHICH INTERCEPTS AN ARC 80PIE CENTIMETER WITH THE RADIUS OF THE CIRCLE IS 20 CENTIMETER.
The general formula for determining the length of an arc is:

L = x/360 * C

L equals the length of the arc.
x = the number of degrees of the arc
C = the circumference of the circle from where the arc was drawn.

C = 2 * pi * r

r = the radius of the circle.

C is also equal to the length of an arc of 360 degrees.

In your problem, you are given that the radius of the circle from which the arc as drawn is equal to 20 centimeters.

This makes the circumference of that circle equal to 2 * pi * 20 = 40 * pi.

Your formula for the length of the arc becomes:

L = x/360 * 40 * pi

You know the length of your arc.

That is given as 80 * pi.

Your formula becomes:

80 * pi = x/360 * (40 * pi)

Divide both sides of this equation by (40 * pi) and you get:

(80 * pi) / (40 * pi) = x/360

This becomes:

2 = x / 360

Multiply both sides of this equation by 360 and you get:

x = 720 degrees.

That's your answer.

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to find the conversion factors, it's convenient to go to www.google.com and put in the conversion you are looking for.
In your case, you want to convert from kilograms to grams.
The google search string entered would be:
kilograms to grams.
the answer returned would be:
1 kilogram = 1000 grams
If you want to convert from kilograms to grams, you would have to multiply the number of kilograms by 1000.
If you start with 1256 kg, then your answer would bge 1,256,000 grams.
You did good.

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your problem is stated as:
(1/2)a + (3/4)b - c
when a=8, b=4 and c=-2
your given expression is (1/2)*a + (3/4)*b - c
you simply substitute the numbers for the letters.
your expression becomes (1/2)*8 + (3/4)*4 - (-2).
(1/2) * 8 = 4
(3/4) * 4 = 3
your expression becomes 4 + 3 - (-2).
simplify this by combining like terms to get 4 + 3 - (-2) = 7 - (-2).
Subtracting a negative number is equivalent to adding a positive number.
Your expression becomes 7 + 2.
Combine like terms again to get 7 + 2 = 9.
Your expression becomes 9.
That's your answer.
If you're still confused, send me an email with follow up questions as to which part confuses you.

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A full circle is an arc of 360 degrees.
The length of an arc of 360 degrees = 2 * pi * r.
the length of an arc of x degrees is equal to x/360 * 2 * pi * r.
Same with the area.
The area of an arc of 360 degrees = pi * r^2
The area of an arc of x degrees is equal to x/360 * pi * r^2.
x is less than or equal to 360 degrees.
In your problem, you are given the diameter as 12 inches.
This makes the radius equal to 6 inches.
Work your problem accordingly.
Length of the arc = 60/360 * 2 * pi * 6 = 2 * pi
Area of the arc = 60/360 * pi * 6^2 = 60/360 * pi * 36 = 6 * pi

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3/5 is what percent times 5/8?

let x = what percent times.

you get (3/5) = x * (5/8)

divide both sides of this equation by (5/8) to get:

x = (3/5) / (5/8)

this is equivalent to:

x = (3/5) * (8/5)

this is equivalent to:

x = (3*8) / (5*5)

this is equivalent to:

x = 24/25

24/25 is equivalent to .96

.96 is equivalent to 96%

the answer to your question is that (3/5) = 96% of (5/8)

To confirm, just translate everything to decimals.

You get:

.6 = 96% of .625

96% of .625 is equivalent to .96 * .625 is equal to .6

you are left with .6 = .6 which is true, confirming that 96% is a good figure.