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Notice that he/she changed the term of the new consolidated loan from 4 years to 3 years. Suppose that the new loan's term WAS 4 years, then how would you calculate the new rate? You would still get the same Harmonic average as you calculated it in terms of the original terms and the original amounts. If the new term for the new loan was 4 years, then the interest rate would be 20.25306% compounded monthly, or 22.242975% effective annual rate.

Yes, good job! 4 regions!

The points would lie on one of two lines: y = 2x and y = x/2

These split the plane into four regions.

Yes you are right :)  And so the answer shall be 2.25 

Thanks! But, 3-0.75=2.25. 

Add the two together and then substitute.

\(x+y=0.25y+0.25x+a+b\\ Sub\;\; in\;\; x=1 \;\;and \;\; y=2\\ 1+2=0.5+0.25+a+b\\ 3=0.75+a+b\\ 2.75=a+b\\ a+b=2.75\)

I don't think the previous answerer got the question right. Let's take a particular example, say b=4. Then the question becomes "When the base 4 number \(11011_4\) is multiplied by 3, and then \(1001_4\) is added, what is the result (written in base 4)?"

Now\(11011_4\times 3 = 33033_4\) and adding \(1001_4\) we get \(100100_4\).

Similarly, in for example base 7, we have

\(11011_7\times 6 + 1001_7 = 66066_7+1001_7=100100_7\)

In fact for any base>1 the answer is always 100100.

Hi Ginger,

LOL  This ant just made an understandable mistake.  No big deal.  :)

Thanks! So, my internet connection was working very slow, and I quickly realized I posted the same question twice! I clicked on my latest post, and changed it to another question, leaving the question before the latest one as it is.

Ants can be such pains-in-the-ass! Especially at picnics … and on math forums. LOL

One way to directly calculate the new interest rate is to take the weighted harmonic mean of the two current rates.

The terms for the 6% loan (compared to the 4% loan) are 50% higher in the amount borrowed and 2/3 longer for the length of the loan. Multiply the 6% by these two dimensions 0.06*1.5*(2/3) = 0.15

Then calculate the harmonic average of 0.04 and 0.15

\(\text {Harmonic mean = } \dfrac {2}{\dfrac {1}{0.04} + \dfrac {1}{0.15}}= 0.06316 \)

GA

Yes.  I'm afraid I gave the positive roots of the equation the labels m and n, but, had I looked more carefully, I would have noticed that your definition of m and n referred to the values under the square root sign!

Find the absolute value of the difference of the solutions of \(x^2-5x+5=0\)

The solutions of any quadratic equation are

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\\x=\frac{-b}{2a}\pm \frac{ \sqrt{b^2-4ac} }{2a} \)

So the ablosute value difference of the roots is

\(\text{abs value of diff of roots} \\ =\left|\left[ \frac{-b}{2a}+ \frac{ \sqrt{b^2-4ac} }{2a}\right] - \left[ \frac{-b}{2a}- \frac{ \sqrt{b^2-4ac} }{2a}\right] \right|\\ =\left|\left[ \frac{-b}{2a}+ \frac{ \sqrt{b^2-4ac} }{2a}\right] + \left[ \frac{b}{2a}+\frac{ \sqrt{b^2-4ac} }{2a}\right] \right|\\ =\left| \frac{ \sqrt{b^2-4ac} }{2a}+ \frac{ \sqrt{b^2-4ac} }{2a} \right|\\ =\left| \frac{ \sqrt{b^2-4ac} }{a} \right|\\ \)

for your question

\(x^2-5x+5=0\)

a=1    b=-5        c=5

\(\text{abs value of diff of roots }= \frac{ \sqrt{25-20}}{1}=\sqrt5\)

When I run up a staircase, my stride can carry me up 1, 2, or 3 steps at a time. In how many ways can I run up a 9-step staircase (given that my last stride lands me on the 9th step)?

Let \(T_n\) be the number of ways to climb n stairs taking only 1 or 2 or 3 steps:

\(\begin{array}{|rcll|} \hline T_n &=& \mathcal{T}_{n+2} \qquad \mathcal{T} \text{ is the Tribonacci number } \\\\ && \text{case of n=9 steps}: \\\\ T_{9} &=& \mathcal{T}_{11} \qquad \\ && \mathcal{T}_0 = 0 \\ && \mathcal{T}_1 = 0 \\ && \mathcal{T}_2 = 1 \\ && \mathcal{T}_3 = 1 \\ && \mathcal{T}_4 = 2 \\ && \mathcal{T}_5 = 4 \\ && \mathcal{T}_6 = 7 \\ && \mathcal{T}_7 = 13\\ && \mathcal{T}_8 = 24\\ && \mathcal{T}_9 = 44\\ && \mathcal{T}_{10} = 81\\ && \mathcal{T}_{11} = 149\\ && \mathcal{T}_{12} = 274\\ && \ldots \\ T_{9} &=& 149 \\ \hline \end{array}\)

Is it?

I had to teach maths for nurses one time and they had all these super horrible questions involving rates.

I 'invented' this method all by myself. LOL

I have never seen anyone else use it in mathematics although it is very similar to methods regularly used in chemistry I think.

It makes difficult rates questions super easy!

Thanks! So, the answer is 5-2=3?

\(2+3-4-1=0\)

\(\frac{2-1}{4-3}=1\\ 4-3-1+2=2\\ \frac{2+1}{4-3}=3\\ \frac{4}{2}+3-1=4\\ 4+3-1*2=5\\ \frac{4}{2}+1+3=6\\ (4+3)(2-1)=7\\ (3-1)^2+4=8\\ 1*2+3+4=9\\ 1+2+3+4=10\\ 2*3+4+1=11\\ 2^3+4*1=12\\ 2^3+4+1=13\\ 2*(3+4)*1=14\\ 3*(4+2-1)=15\\ 4*(3+2-1)=16\\ 3*(2+4)-1=17\\ 3*(2+4)*1=18\\ (2+3)*4-1=19\\ 1*(2+3)*4=20\)

Equate the two expressions:   x⁴ = 7x² - 10 

Rearrange:   x⁴ - 7x² + 10 = 0

Let z = x² 

z² - 7z + 10 = 0

This factorises nicely as:  (z - 2)(z - 5) = 0

So z = 2 and z = 5

or  x² = 2  and x² = 5

n = 2^1/2, m = 5^1/2   so m - n = 5^1/2 - 2^1/2 

.

That makes sense, too! Thanks Chris!

- Daisy

Thanks Daisy!!!!....here's another way

j  = C / k      where C is the proportionality constant

And we know that

16 = C / 21      multiply both sides by 21

336  = C

So  when  k = 14  we have that

j = 336 / 14  =   24

y >  -2

x + y ≤ 4

Note  that  (0, -4)  = (x,y)   can't be correct  because    - 4  >  -2   is false

And (-2, -3)  can't be correct for the same  reason   because  -3  > -2  is false

Note that (1,5) can't be correct because 1 + 5 ≤ 4   is false

So  (1,3)  is the correct choice  because

3 > -2    is true     ....and.....

3 + 1 ≤  4   is also true

We can use case work counting with this. We have three cases, if we start with \(1\) step, \(2\) steps, or \(3\) steps. You then can add a subcase of \(1\) step, \(2\) steps, or \(3\) steps to the beginning cases. You can keep doing this until all your cases add up to \(9\). This is a hassle, but it works. I don't reccommend this method, but I do not know any other methods, sorry. The end answer is \(149\). Haha I hope this helped! Hopefully you can find a better solution!

- Daisy

Inversely proportional means basically that the product of the numerator and denominator will always be the same. So if the fraction is \(\frac{16}{21} = \frac{j}{k}\), the product of \(16\times21 = 336\). We are trying to find \(j\) if \(k = 14\). We have \(\frac{16}{21} = \frac{j}{14}\). You can solve this by doing \(336\) divided by \(14\), which gives you \(24\). \(j = 24\).

- Daisy