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The table below shows the changes in the value of a company's stock over the past four weeks.

\begin{tabular}{|c|c|c|c|}
\hline \multicolumn{4}{|c|}{Stock Value} \\
\hline Week 1 & Week 2 & Week 3 & Week 4 \\
\hline[tex]$+9 \frac{1}{2}$[/tex] points & [tex]$+6 \frac{5}{8}$[/tex] points & [tex]$-12 \frac{3}{4}$[/tex] points & [tex]$-8 \frac{1}{4}$[/tex] points \\
\hline
\end{tabular}

1. A broker says that during the third week, the company's stock lost all of the points it had gained over the first two weeks. Is this claim correct? Explain.

[tex]\[
h_1 = +9 \frac{1}{2} = 9.5
\][/tex]

[tex]\[
h_2 = +6 \frac{5}{8} = 6.625
\][/tex]

[tex]\[
h_1 + h_2 = 9.5 + 6.625 = 16.125 \text{ points}
\][/tex]

[tex]\[
k_3 = -12 \frac{3}{4} = -12.75
\][/tex]

[tex]\[
16.125 - 12.75 = 3.375
\][/tex]

No, the broker is not correct.

2. By how many points would the stock value need to change in Week 5 to return to its value before Week 1? Explain.

[tex]\[
\text{Total change in 4 weeks} = 9.5 + 6.625 - 12.75 - 8.25 = -4.875
\][/tex]

[tex]\[
\text{Week 5 change needed} = -(-4.875) = 4.875 \text{ points}
\][/tex]

3. At the beginning of Week 1, the value of the company's stock was [tex]$177 \frac{3}{8}$[/tex] points. An investor tells his broker to sell his shares of stock if the value falls below 170 points at the end of Week 4. Should the broker sell the shares? Explain.

[tex]\[
\text{Initial value} = 177 \frac{3}{8} = 177.375
\][/tex]

[tex]\[
\text{Value at the end of Week 4} = 177.375 - 4.875 = 172.5
\][/tex]

No, the broker should not sell the shares, as the value is still above 170 points.



Answer :

GinnyAnswer
Let's analyze and solve the problem step by step.

### 1. Broker's Claim Analysis
The broker claims that during the third week, the company's stock lost all of the points it had gained over the first two weeks.

First, let's determine the changes for each week:
- Week 1: [tex]\( +9 \frac{1}{2} \)[/tex] points [tex]\( = 9.5 \)[/tex] points
- Week 2: [tex]\( +6 \frac{5}{8} \)[/tex] points [tex]\( = 6.625 \)[/tex] points
- Week 3: [tex]\( -12 \frac{3}{4} \)[/tex] points [tex]\( = -12.75 \)[/tex] points

Now, sum the gains for the first two weeks:
[tex]\[ 9.5 + 6.625 = 16.125 \text{ points} \][/tex]

According to the broker, the loss in Week 3 should equal the gain from the first two weeks:
[tex]\[ 12.75 \text{ points} \neq 16.125 \text{ points} \][/tex]

Thus, the broker's claim is incorrect because the loss in Week 3 does not equal the combined gain from the first two weeks.

### 2. Points Needed to Return to Initial Value
We need to determine by how many points the stock value would need to change in Week 5 to return to its initial value before Week 1.

Let's calculate the total change over four weeks:
[tex]\[ \text{Week 1 change} + \text{Week 2 change} + \text{Week 3 change} + \text{Week 4 change} \][/tex]
[tex]\[ = 9.5 + 6.625 + (-12.75) + (-8.25) \][/tex]
Sum the changes:
[tex]\[ 9.5 + 6.625 = 16.125 \][/tex]
[tex]\[ 16.125 - 12.75 = 3.375 \][/tex]
[tex]\[ 3.375 - 8.25 = -4.875 \][/tex]

To return to the initial value, we need to reverse the total change:
[tex]\[ \text{Week 5 change} = -(-4.875) \][/tex]
[tex]\[ \text{Week 5 change} = +4.875 \text{ points} \][/tex]

So, the stock value needs to increase by 4.875 points in Week 5 to return to its value before Week 1.

### 3. Decision to Sell Shares at End of Week 4
Given:
- Initial stock value at the start of Week 1: [tex]\( 177 \frac{3}{8} \)[/tex] points [tex]\( = 177.375 \)[/tex] points
- Condition to sell: If the stock value falls below 170 points at the end of Week 4

Let's calculate the stock value at the end of Week 4:
[tex]\[ \text{Initial value} + \text{Total change over 4 weeks} = 177.375 + (-4.875) \][/tex]
[tex]\[ = 172.5 \text{ points} \][/tex]

Since 172.5 points is above 170 points, the broker should not sell the shares. The stock value has not fallen below 170 points.

To summarize:
1. The broker's claim is incorrect. The combined gain for the first two weeks is 16.125 points, while the loss in Week 3 is 12.75 points.
2. The stock needs to increase by 4.875 points in Week 5 to return to its initial value.
3. The stock value at the end of Week 4 is 172.5 points, so the broker should not sell the shares because the value has not fallen below 170 points.

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