GingerAle

^{There was a flat containing boxes of apples having a total weight of 100 kg. An analysis showed that the apples were 99% moisture, by weight. After two days in the sun, a second analysis showed that the moisture content of the apples was only 98%, by weight. What was the total weight of the apples after 2 days, in kg?}

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My statistics professor often posed questions like this to reinforce the importance of paying attention to how a question presents data and correctly interpreting data that results from analysis.

Solution:

$\text {The first analysis shows the apples are 99% water. The weight of the water is then}\\ \left(0.99\cdot 100\right) = 99 ~ kg\\ \text {Let x be the weight of the water lost after exposure to the sun. }\\ \left(0.99 \cdot 100-0.98(100-x)\right)=x\\ \begin{aligned}99-(98-0.98x)&=x\\99-98+0.98x&=x\\1+0.98x&=x\end{aligned}\\ \begin{aligned}1+0.98x-0.98x&=x-0.98x\\1&=0.02x\\x&=50\\\end{aligned}\\ \text { }\\ 100-x=100-50=50 ~ kg \leftarrow \text { The total weight of the apples after 2 days}\\$

The correct solution appears paradoxical. It’s not, but it is counter-intuitive.

Solution:

The sum of four (4) dice is eight (8).

There are five (5) ways to roll four (4) dice for a sum of eight (8).

5, 1, 1, 1

4, 2, 1, 1

3, 3, 1, 1

3, 2, 2, 1

2, 2, 2, 2

One of these five (5) are four (4) twos (2).

The probability is 1/5 = 20%

GA

Solving it this way gives:

$a(a-1)^{-1} \equiv 4a^{-1} \pmod{20}\\ a \equiv 4a^{-1}(a-1) \pmod{20}\\ a^2 \equiv 4(a-1) \pmod{20}\\ a^2-4a+4 \equiv 0 \pmod{20}\\ (a-2)^2 \equiv 0 \pmod{20} \\ \text { }\\ a= 2+10n \;|\text { for n = (0, 1, 2, … 9), so 10 integers  satisfy 0 < a < 100}$

Well blimey! You’re right. (I’m surprised because Heureka can do mod problems while napping.)

GA

Why Mr. BB, do you mean to tell me my answer is ………..BLARNEY!!

Blarney answers are acceptable for blarney questions, Mr. BB. It doesn’t matter how absurd the reasoning is, as long as the math is correct, and it is: The weighted harmonic mean is mathematically correct, and the method used to weight it is mathematically correct. The logic is correct in another universe (not this one), but it’s usable here because this question, and the obsequiously (means ass-kisser) polite student who asked it, came from came from the same universe.

In this world, a competent teacher has countless questions and scenarios for assigning practice and to assess a student’s TVM skills. A teacher would only present this question to demonstrate absurdity. No one except a fool would consolidate a loan using this approach, and only an unethical loan company would make such a loan. I can think of one that would: The First National Vigorish and Loan-Shark Company; a wholly-owned subsidiary of Smith, Blarney, Cheatem, and Howl, your former and principal employer.

The absurdity becomes very clear when you add another year of payments (vigorish) to a fully amortized loan.

Another thing that becomes clear is you are the Blarney Banker. … … …

GA

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