TheXSquaredFactor

Upon further review, I have realized that I made a slight oversight on my end. Sorry about that! Unfortunately, I think everyone is prone to making such mistake. This is a learning experience--for you and me. Let's try to embrace it in the best way possible. The mistake occurs in the following segment:

• If James's brother paid James and James's sister 12 dollars, then this has a three-way affect.
• \(J_1=J_2-12\\ S_1=S_2-12\\ B_1=B_2+12\)

Did you spot the error yet? If not, that's OK! There is a lot of information to digest in this problem, so it is very easy to assume that I could not have blundered at this step. The problem lies at the \(B_1=B_2+12\) part. 

The original problem states that James's brother gives 12 dollars to both James and his sister. This means that James's brother loses $24 ($12 to James and $12 to the sister) in total--not $12 like I originally hinted at.

The working of the system of equation is mostly the same, though, and I get that James has $41 right now! I have double and triple checked, so I think I will finally move on from this word problem.

I think it is best to initialize variables first and foremost. A quick glance at the problem will reveal that the information mentions the amount of money that three individuals have at separate moments.

\(\text{Let }J=\text{James's Money}\\ \text{Let }B=\text{James's Brother's Money}\\ \text{Let }S=\text{James's Sister's Money}\)

Now, let's break down the information into its components and analyze them.

• If James had half as much money as his brother, then we can represent this situation via equation: \(J_1=\frac{B_1}{2}\)
• If James had 27 dollars less than his sister, then we can create another equation: \(J_1=S_1-27\)

After the secretive payoff, though, new information rolls out! The subscript of 1 on every variable is intentional; it is meant to denote the relationship of money prior to the shady payoff. I will use a subscript of 2 to represent amounts of money after the deal.

• If James's sister has twice as much money as James's brother, then we can create a third equation: \(S_2=2B_2\)

By utilizing subscripts, it is easy to notice that we cannot simply compare this information to the previous equations. However, we can use the information about the payoff to adjust the variables. 

• If James's brother paid James and James's sister 12 dollars, then this has a three-way affect.
• \(J_1=J_2-12\\ S_1=S_2-12\\ B_1=B_2+12\)

Since the question specifically asks for the amount of money James has now (after the payoff), then we should adjust every variable such that it has a subscript of 2. Let's do that!
 

Let's place the 1st and 3rd equations alongside each other. By adding the equations together (with the combination of a slight manipulation) it is possible to eliminate one variable from this system. This creates a fourth equation.

\(\boxed{1}\hspace{1mm}S_2=2B_2\Rightarrow \hspace{10mm}S_2\hspace{10mm}=2B_2\\ \boxed{3}\hspace{1mm}J_2=S_2-27\Rightarrow -S_2+J_2=-27\\ \boxed{4}\hspace{1mm}\hspace{46mm}J_2=2B_2-27\)

We can now introduce the 2nd equation into the mix, which will allow us to eliminate the other variable.

\(\boxed{4}\hspace{39mm}J_2=2B_2-27\\ \boxed{2} \hspace{1mm} 2J_2=B_2+36 \Rightarrow-4J_2=-2B_2-72\\ \hspace{39mm}-3J_2=-99\\ \hspace{47mm}J_2=33 \)

According to this work, James has 33 dollars.

Wow! This inspires me to see the light of day, so I will keep chugging along, albeit slowly.

These examples should aid you with proper syntax.

Thanks for the salute! 

According to my profile information, I have 795 answers attached to my name currently. I have not yet reached my one-year anniversary. You are on your fourth, yet you have produced more than 30 times the amount of constructive responses than I have. Assuming that I continue answering questions at the same rate, I will reach the incredibly competitive benchmark of 23965 questions answered in 2042, which is 24 years from now!

Who deserves the salute now? 

1)

A 30-60-90 triangle has a specific ratio in which its side lengths are defined. It is a \(1:\sqrt{3}:2\) ratio. Knowing this set ratio, it is easy to determine that the hypotenuse is double the length of the shorter leg. Therefore, the hypotenuse length is 16 meters long!

2) 

This question is very similar to the previous triangle, except that we are dealing with a different special right triangle. A 45-45-90 triangle's ratio of the side lengths is \(1:1:\sqrt{2}\). If one of the lengths of the legs is 8 inches, then the hypotenuse has length \(8\sqrt{2}\text{ inches}\).

3) 

It appears as if this question has a formatting issue. 

If you can map one figure onto another using similarity transformations, then the figures are similar. 

1) Translate the center of the circle with radius q to the center of the circle with radius t.

Translations are considered to be an isometry, a specific type of transformation where the size and shape remain unaffected.

2) Dilate the circle with radius q about C by a factor of t/q

Did my response aide you in your quest for mathematical success? 

4) The sequence is certainly arithmetic! Notice that \({\underbrace{13,}\underbrace{20,}\underbrace{27,}34,...}\\ +7+7+7\) . In other words, the common difference is 7.

5) This is another arithmetic sequence with a common difference of -7.

Remember that awesome formula that tells you the nth term of an arithmetic sequence? It's \(a_n=a_1+d(n-1)\) . Let's use it!

\(a_n=a_1+d(n-1)\\ a_{50}=5-7(50-1)\\ a_{50}=5-7*49\\ a_{50}=-338\)

6) The arithmetic mean is also the average. 

\(a_n=\frac{a_{n-1}+a_{n+1}}{2}\\ a_n=\frac{3.9+7.1}{2}\\ a_n=\frac{11}{2}\\ a_n=5.5\)

I unintentionally answered this question already when I answered your previous question!