Joyce mother buys some shares of Company A on Day 0. On Day 7, the share price of Company A is $4.60. If she sells all her shares of Company A and buys 2000 shares at Company B on Day 7, she would receive $7400. On Day 12, the share price of Company A is $4.80 and the share price of Company B is $0.50 less than that on Day 7. If she sells all her shares of Company A and buys 5000 shares on Company B on Day 12, She would have to pay $5800. Find
(i) the number of shares of Company A Joyce mother has,
(ii) the share price of Company B on Day 12

Let's break down the problem step by step to find the solutions to the two questions.

### Given:
- **Day 7**:
  - Share price of Company A: $ P_A(7) = \$4.60 $
  - If Joyce's mother sells all shares of Company A, she gets \$7400 to buy 2000 shares of Company B.
  
- **Day 12**:
  - Share price of Company A: $ P_A(12) = \$4.80 $
  - Share price of Company B on Day 12 is \$0.50 less than on Day 7.
  - If she sells all shares of Company A and buys 5000 shares of Company B, she pays \$5800.

### Find:
1. The number of shares of Company A that Joyce's mother has.
2. The share price of Company B on Day 12.

### Step 1: Find the share price of Company B on Day 7
We know that on Day 7, if she buys 2000 shares of Company B with the amount she receives from selling all her shares of Company A, she would receive \$7400. Let $ P_B(7) $ be the share price of Company B on Day 7.

\[
\text{Total amount received from selling Company A shares} = \text{Total amount spent on buying Company B shares}
\]
\[
\text{Total amount spent on Company B shares} = 2000 \times P_B(7)
\]
\[
2000 \times P_B(7) = 7400
\]

\[
P_B(7) = \frac{7400}{2000} = 3.70
\]

So, the share price of Company B on Day 7 is \$3.70.

### Step 2: Find the share price of Company B on Day 12
We know that the share price of Company B on Day 12 is \$0.50 less than on Day 7.

\[
P_B(12) = P_B(7) - 0.50
\]
\[
P_B(12) = 3.70 - 0.50 = 3.20
\]

So, the share price of Company B on Day 12 is \$3.20.

### Step 3: Find the number of shares of Company A
Let $ N_A $ be the number of shares of Company A that Joyce's mother has.

On Day 12, if she sells all her shares of Company A, she gets:

\[
N_A \times P_A(12) = N_A \times 4.80
\]

If she buys 5000 shares of Company B on Day 12 with \$5800, the total amount spent is:

\[
5000 \times P_B(12) = 5000 \times 3.20 = 16000
\]

So the equation to find $ N_A $ is:

\[
N_A \times 4.80 - 16000 = -5800
\]

Simplifying,

\[
N_A \times 4.80 = 5800 + 16000 = 21800
\]
\[
N_A = \frac{21800}{4.80} \approx 4541.67
\]

But since shares are generally bought in whole numbers:

\[
N_A \approx 4542 \text{ shares}
\]

### Final Answers:
1. The number of shares of Company A that Joyce's mother has is approximately **4542 shares**.
2. The share price of Company B on Day 12 is **$3.20**.

Would you like more details on any step, or do you have any other questions?

Here are 5 questions to deepen your understanding:
1. What would be the total value if Joyce's mother sold all her shares of Company A on Day 12?
2. How would the results change if the price of Company A on Day 12 was higher?
3. What is the impact of the price difference of Company B between Day 7 and Day 12?
4. How could Joyce's mother optimize her portfolio based on the share prices?
5. What is the significance of the rounding approximation in calculating the number of shares?

**Tip:** Always double-check the conditions given in word problems to ensure the accuracy of your calculations, especially when dealing with financial mathematics.

Explore how to solve investment equations involving shares of Company A and Company B. Understand the calculations for share prices on Day 7 and Day 12, and determine the number of shares held by Joyce's mother. Suitable for advanced learners in finance and mathematics.

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