All Answers 358233 Answers

Guest #1

Remember you have monthly payments of $442 @ 3% compounded monthly:

FV = P x [1 + R]^N - 1 / R

FV = 442 x {[ 1 + 0.03/12]^(5*12) - 1 / (0.03/12)}

FV = 442 x {[1.0025]^60 - 1 / (0.0025)}

FV = 442 x              64.6467126......

FV = $28,573.85.

Here is our formula. I will assume the payment period is annually..

FV = A(((1+i)^n)-1)/(i)

3000 = 600(((1.04)^n)-1)/(.04)

Let's isolate 'n'

Divide by 600

5 = (((1.04)^n)-1)/(.04)

Multiply by .04

0.2 = (((1.04)^n)-1)

Add over the one

1.2 = (((1.04)^n))

Take log of both sides

ln(1.2) = n*ln(1.04)

Divide by ln(1.04)

(ln(1.2)/ln(1.04)) = n

n = 4.65

Why post same question twice? It's difficult enough for Guests battling the everlasting Captcha. I must fight up to three or more per post when Captchas are antsy to fight. They must prove themselves to Google to defeat bots. However, I may be defeated as they grow stronger everyday with stealth. I can not spot a car in an ocean yet I fail to see the sky. Here is the answer you seek.

This is the notation for our problem

PV = A(P/A, i%, n)

Now to set up the formula

PV = A((((1+i)^n)-1)/(i*(1+i)^n)

11000 = 500((((1.005)^n)-1)/(.005*(1.005)^n)

Many methods you could solve for n.

After many guesses I came the value is in between 23 and 24. Furthermore, with linear interpolation I came to the conclusion n=23.4897

At the moment we have to adjust the time frame of years to months. 5 years = 60 months

With our effective interest rate we're good to go to the formula.

Let's set up our formula!

FV = A(((1+i)^n)-1)/(i)

FV = 442((((1.03)^60)-1)/(.03)

FV = $72069.62

Please state the payment periods... I will just assume it's annual.

This is the notation for our problem

PV = A(P/A, i%, n)

Now to set up the formula

PV = A((((1+i)^n)-1)/(i*(1+i)^n)

11000 = 500((((1.005)^n)-1)/(.005*(1.005)^n)

Many methods you could solve for n.

After many guesses I came the value is in between 23 and 24. Furthermore, with linear interpolation I came to the conclusion n=23.4897

It seems you yourself could set this into a formula as you already know the values (which is the difficult part of deriving the correct values from questions). Would have also helped if you noted the payment time period... I assumed it's an annual payment...

The notation for these values depicts A(P/A, i%, n) = PV

Now to apply the formula:

PV = A((((1+i)^n)-1)/(i*(1+i)^n)

PV = 295((((1.043)^16)-1)/(.043*(1.043)^16)

PV = $3362.59

In the title it states you want a future value yet in the description you ask for a present value...

Regardless, let's convert the present value to a future value.

FV = 3362.59(1.043^16)

FV = $6595.12

Thank you for your long response! I do actually know how to do the problem, I just didn't understand what that symbol meant in relation to the question. Thanks though.

Let the number of pounds needed from the 10% fertilizer = T
Then number of pounds needed from the 19% fertilizer =450 - T
0.1T + 0.19(450 - T) = 0.14(450), solve for T
T = 250 pounds of 10% fertilizer.
450 - 250 = 200 pounds of 19% fertilizer.

Domain and range of the function \(y=\frac{1}{x}\)

Now, let's figure out the domain first. The domain of a function just means the possible range of x-values that output a real y-value.

Let's consider that. Is there any number for x that would output a nonreal y-value?

Yes, 0. If you plug in 0 for x, you get a nonreal y-value. Will any others? No. I've made a table of the first few domain options.

You can see that any numer, except 0, will output a real y-value. Therefore, the domain is any number that isn't 0. 

Domain: \(\{x\in {\rm I\!R}\hspace{1mm}: x\neq0\}\)

In other words, the domain is apart of the real numbers except 0.

Let's think about the range, too. The range is set of all possible values that the function can actually produce. Let's try making a table for that:

You might be wondering why 0 has "no solution" on it. Well, let's try and solve it:

Any other real number works, so our range is:
 

\(\{f(x)\in {\rm I\!R}\hspace{1mm}:f(x)\neq0\}\)

To recap, the domain and range is as follows

Domain: \(\{x\in {\rm I\!R}\hspace{1mm}: x\neq0\}\)

Range: \(\{f(x)\in {\rm I\!R}\hspace{1mm}:f(x)\neq0\}\)

To answer your second question about what does \(\{x|x\leq0\}\) mean. It means that x is a member of the real numbers such that x is nonnegative (0, 2/9, 1, π, 4.83333..., etc.). However, x has to be nonnegative, so it can't be -3/4, -5, -8.980333333..., etc.

I'm with Melody.

For the 20 ball first part, the number of three ball combinations is 20C3 = 1140.

The number of ways in which we miss with all three numbers is 17C3 = 680.

The number of ways in which we have 1 hit and three misses is 3*17C2 = 408.

The number of ways in which we have 2 hits and 1 miss is 3C2*17 = 51.

The number of ways in which we have three hits is 1.

(Check that all possibilities have been covered, 680 + 408 + 51 + 1 = 1140.)

That gets us the probabilities,

zero hits 680/1140 = 0.59649 (5 dp),

1 hit 408/ 1140       = 0.35789 (5 dp),

2 hits 51/1140        = 0.04474 (5 dp),

3 hits 1/1140          = 0.00088 (5 dp).

An alternative method is to calculate the probabilities directly.

Using H for a hit and M for a miss, the probability of three misses MMM is (17/20)*(16/19)*(15/18) = 0.59649.

The probability of the sequence HMM is (3/20)*(17/19)*(16/18) = 0.119298, and since both of the other 1 hit sequences MHM and MMH have the same probability, the probability of 1 hit will be 3*0.119298 = 0.35789 (5 dp).

The probabilty of the two hit sequence HHM is (3/20)*(2/19)*(17/18) = 0.014912, and since both of the other two hit sequences HMH and MHH have the same probability, the probability of two hits will be 3*0.014912 = 0.04474 (5 dp).

The probability of three hits will be (3/20)*(2/19)*(1/18) = 0.00088 (5 dp).

That means that the probability of at least two hits (two matches) is 0.04474 + 0.00088 = 0.04562.

The probability of not winning that way will be 0.59649 + 0.35789 = 0.95438.

The probability of winning with the superball is 0.1 and not winning with the superball is 0.9.

There are four possible outcomes,

(1) not getting at least two matches and not picking the superball : probability 0.95438* 0.9 = 0.85894,

(2) getting at least two matches, but not the superball : probability                    0.04562*0.9 = 0.04106,

(3) not getting at least two matches but do pick the superball : probability        0.95438*0.1  = 0.09544,

(4) getting at least two matches and picking the superball : probability              0.04562*0.1 = 0.00456,

(the probabilities summing  to 1, as they should).

That means that the probability of winning is 0.04106 + 0.09544 + 0.00456 (or 1 - 0.85894) = 0.14106.

Tiggsy.

One sort of fertilizer contains 10% nitrogen , and the other fertilizer contains 19% nitrogen. How much do you need to take of both sorts of fertilizer so you can get 450 kilos of mixture which contains 14 % of nitrogen . 

hectictar: Fantastic! Got it almost perfectly right. The only small mistake appears to be in the payment being made at the beginning instead of at the end of each month. But we simply rectify that by multiplying the payment you got(671.26) by the interest rate of 1.00625 =$675.46, which is the correct answer. Now, you could always check your answer with the financial calculator I mentioned, which is very accurate.

Of course, they have formulas for these financial problems, which are called "TVM", or Time Value of Money formulas, which will make things a lot easier. The formula for this question is:

FV=P{[1 + R]^N - 1/ R},  where FV=Future Value, P=Periodic Payment, R=Interest Rate per period, and N=Number of periods. So, we just enter the numbers we have in the formula as follows:

157,000 = P x {[1 + 0.075/12]^(12*12] - 1 / (0.075/12]}, and that is it. Will now solve for P.

157,000= P x {[1.00625]^144 - 1 / (0.00625)}

157,000= P x {[ 1.4527238.... / (0.00625)}

157,000= P x             232.4358088.......

P = 157,000 / 232.4358088

P = $675.46 - the monthly payment required. 

P.S. Please do not feel pressured to tackle these problems. Do them only if you want to learn something about money and how it grows over time, just for your own amusement and curiosity only.

I normally don't even click on these questions.. they are way outside my  "comfort zone" ! ..but....I'll venture a guess......

Since it is compounded monthly...divide the annual rate by 12     (7.5/12)%  =   0.625%   =  0.00625

At the end of each month..there will be   (1.00625 * the amount of money in the account before the company adds any that month )   in the account.

Call the monthly deposit  " m "...

the amount of money after 1 month    =  1.00625m

the amount of money after 2 months  =  1.00625( 1.00625m + m )

the amount of money after 3 months  =  1.00625( 1.00625(1.000625m + m) + m )

                                                            = 1.00625^3m+1.00625^2m +1.00625m

                                                            =  m(1.00625^3 + 1.00625^2 + 1.00625)

                                                            =  \(m*\sum \limits_{i=1}^{3}(1.00625^i)\)

So..after 144 months...or 12 years...there will be...

\(m*\sum \limits_{i=1}^{144}(1.00625^i)\)

Set this equal to  157,000

\(m*\sum \limits_{i=1}^{144}(1.00625^i)=157,000 \\~\\ m=157,000/(\sum \limits_{i=1}^{144}(1.00625^i))\)

V=(100/6)[2556+(4x2619)+2700] + (100/6)[2700+(4x2610)+2484]

  =522600

Using this website : 

http://www.danielsoper.com/statcalc/calculator.aspx?id=2

The area to the left  of  z = -1.23  ≈  0.10934855

Hi Melody:

Actually the difference in our solutions is significant; it’s not from rounding errors.

The main reason is 

\(\dfrac{52}{1140}+\dfrac{1}{10}\underbrace {- \dfrac{52}{11400}}_{\text{This Term}}\\\)

^{(My computer is not rendering any Latex. I will correct this display, if needed)}

This subtracts 10% of the (2 of 3) ball wins.

I had initially thought of this but dismissed it because of the either/or nature of the win.

On reflection, I can see it does count the win twice when both the super ball and the white balls are a match. It should count one or the other but not both. The problem, now, is by subtracting 52/11400, it subtracts both wins.  So, what should the value be?

My quick answer is half that (26/11400).The reasoning is by subtracting half, then one or the other is subtracted but not both.

What thinkest thee, Dame Melody?

More comments:

I agree with the 52/1140, I neglected to include the 3 of 3 win in my equation, because I was patting my chimp head for figuring out Lancelot Link’s convoluted formula for counting a  Z subset of  matches of R in a  N choose R combination.  

Here are the adjusted numbers in the same format
6)  Add (1 / 1140) to (51 / 1140) giving (52 / 1140) for when all three numbers match.

Calculate number of times both the super ball and two of the three balls match
(52 / 1140)  * (1 / 10) = 52/11400      <---  ½ of this = (26 / 11400)

Subtract half of matches when both the super ball and two of the three balls match (This counts one or the other but not both.)

Calculate the harmonic average

H = 3 / ((52 / 1140) + (1 / 10) - (26 / 11400))  = 20.930232558

Equalized probability of each event (1 / 20.9023) * 3 (for the three events)
= (3 / 20.9023)
= (43 / 300)
= 0.143333…

Note: doing it this way is much easier (This is Melody’s method)

 ((52 / 1140) + (1 / 10) - (26 / 11400)) = (0.143333… )

Over all probability 14.33%

(Only the final calculation is rounded)

This question is interesting, but it seems very advanced.
Do you understand this math, Victoria?  

Comments and criticisms from mathematicians (and articulate trolls) are welcome.

GA

CN Tower  =  30% of 600 =  .30 x 600  =  180

Eiffel Tower  =  25% of 600 =  .25 x 600  =  150

Empire State Bldg.  =  3% of 600 =  .03 x 600  =  18

Statue of Liberty  =  5% of 600 =  .05 x 600  =  30

Christ the Redeemer  =  (100 - 30 - 25 - 3 - 5 )% of 600  =  37% of 600  = 222

hectictar: Do you want to try your magic hand on some of these financial problems? I will guide you through them.  But only if you want. I think you will breeze through them in no time. You could start by using this very excellent financial calculator: https://arachnoid.com/finance/

Many thanks.

See if this helps :

Which is more... $0.30  or   $0.27   ?

30 cents is more than 27 cents.

0.3 is bigger than 0.27

0.3 > 0.27            

Since a and b  = (a, b)  = (x, y) have the same sign.....P  will lie either in the 1st quadrant of the Cartesian plane or the 3rd quadrant of the Cartesian plane. 

I agree with hectictar......sums can probably be computed with the calculator, but I don't know the proper syntax.......maybe one of our "advanced" mods, Alan, knows the key !!!

Meanwhile.....WolframAlpha will easily compute sums : 

https://www.wolframalpha.com/input/?i=sum+(x+%2B1),+from+x+%3D1+to+x+%3D+10

Thank you CPhill !!      I'm glad you said that because I was thinking at first that it might have been unnecessary to put down the calculus method.

On this calculator...it looks like it might be possible to make a sum..but I can't quite figure out how...

Very nice, hectictar....I probably would favor the Calculus approach, but the other is more comprehensible for the  Algebra student.....