Problem 5: Calculate daily expenses for three families using matrix multiplication based on given daily costs for men, women, and children.

Let's solve both problems step by step using matrix multiplication as requested.

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Problem 5: Daily Expenses of Families A, B, and C

#### Given Data:
- Daily expenses:
- Men: ₹200
- Women: ₹150
- Children: ₹200 (but children do not earn, so their expenses are borne by men and women)

- Family composition:

| Family | Men | Women | Children |
|--------|-----|-------|----------|
| A | 2 | 3 | 1 |
| B | 2 | 1 | 3 |
| C | 4 | 2 | 6 |

We need to calculate the daily expenses for each family. Since only men and women earn, but all members (including children) have expenses, we assume that the total daily expenses include:
- Cost of men × 200
- Cost of women × 150
- Cost of children × 200

So, even though children don't earn, they consume resources.

---

#### Step 1: Represent data in matrices

Let’s define:

- Matrix M (family composition):
Each row is a family, columns are Men, Women, Children

$$
M = \begin{bmatrix}
2 & 3 & 1 \\
2 & 1 & 3 \\
4 & 2 & 6 \\
\end{bmatrix}
$$

- Matrix E (expenses per person):
Column vector: Men, Women, Children

$$
E = \begin{bmatrix}
200 \\
150 \\
200 \\
\end{bmatrix}
$$

---

#### Step 2: Multiply M × E

Each element of the resulting matrix will be the total daily expense for each family.

$$
\text{Total Expenses} = M \times E
$$

Compute:

For Family A:
$$
(2)(200) + (3)(150) + (1)(200) = 400 + 450 + 200 = ₹1050
$$

For Family B:
$$
(2)(200) + (1)(150) + (3)(200) = 400 + 150 + 600 = ₹1150
$$

For Family C:
$$
(4)(200) + (2)(150) + (6)(200) = 800 + 300 + 1200 = ₹2300
$$

So,

$$
\text{Expenses} = \begin{bmatrix}
1050 \\
1150 \\
2300 \\
\end{bmatrix}
$$

---

#### ✔ Answer to Problem 5:
- Daily expenses:
- Family A: ₹1050
- Family B: ₹1150
- Family C: ₹2300

---

#### Impact of More Children on Society:
More children in a family increase the total dependency ratio, meaning more people are being supported by fewer earners (men and women). This leads to:
- Higher daily expenses
- Increased financial burden on working members
- Potential strain on resources
- Possible underdevelopment if not supported by social systems

Thus, more children without proportional earning capacity can lead to economic stress, especially in low-income families. However, children are future contributors — so long-term investment in education and health is key.

---

Problem 6: Trust Fund for Handicapped Children

#### Given:
- Monthly donation: ₹30,000
- Half of funds used for medical/educational care → ₹15,000 spent
- Charges 2% of the spent amount from donors → 2% of ₹15,000 = ₹300
- Remaining amount deposited in bank: ₹15,000 – ₹300 = ₹14,700
- This ₹14,700 earns interest, and the trust wants ₹1800 every month from interest.
- Find: What rate of interest should the trust get?

Use matrix method to find rate of interest.

---

#### Clarify the scenario:
The trust receives ₹30,000/month.
- Spends ₹15,000 on care
- Charges 2% of spent amount (₹300) from donors → this is not part of the deposit
- So, deposits: ₹15,000 – ₹300 = ₹14,700 into bank
- The interest earned on this ₹14,700 must be ₹1800 per month

But wait — ₹1800 per month from ₹14,700? That would require:
$$
\text{Interest rate per month} = \frac{1800}{14700} \approx 12.24\%
$$
Which is very high.

Wait — perhaps there's a misunderstanding.

Let’s re-read:

> "The trust spends half of the funds received... and for that it charges 2% of the spent amount from them..."

"Charges 2% of the spent amount from them" — meaning donors pay extra? Or does the trust charge the donors an extra 2% fee?

Possibility: The donors give ₹30,000, but the trust charges 2% of the spent amount as a fee or administrative cost, which may be deducted from the donated amount.

But let's suppose:
- Total received: ₹30,000
- Half (₹15,000) goes to care
- The trust charges 2% of ₹15,000 = ₹300 as fee, so this ₹300 comes from the donated money?
- Then, only ₹14,700 is deposited in bank

Then, interest earned on ₹14,700 must be ₹1800 per month.

But ₹1800 / ₹14,700 ≈ 12.24% monthly → 146.9% annual → impossible.

So likely, the trust deposits the entire ₹15,000 (half of ₹30,000), and the 2% is charged from the donor as a fee, but the bank deposit is ₹15,000, and interest on that is used.

But then why mention "charges 2% of spent amount"? Let's interpret carefully.

Alternative interpretation:
- Donor gives ₹30,000
- Trust uses ₹15,000 for care
- Trust charges 2% of the spent amount (₹15,000) as a fee, meaning the donor pays extra? But that doesn’t make sense.

Wait — perhaps the trust collects ₹30,000, uses ₹15,000 for care, and the remaining ₹15,000 is deposited. But they charge 2% of the spent amount (₹15,000) as a service fee, so the donor pays ₹300 extra? That would mean the total collected is ₹30,300?

No — probably not.

Better interpretation:

> "The trust spends half of the funds received for medical and educational care of the children and for that it charges 2% of the spent amount from them"

"From them" = from the donors.

So, when the trust spends ₹15,000, it charges 2% of that (₹300) to the donor as a fee.

So, the donor actually pays ₹30,000 + ₹300 = ₹30,300?

But that seems odd.

Alternatively: The trust receives ₹30,000, spends ₹15,000 on care, and charges 2% of the spent amount (₹300) as administrative fee, which is paid by the donor, so the net amount available for deposit is ₹15,000 (remaining half) minus ₹300? No.

Wait — let's reframe.

Perhaps:
- Trust receives ₹30,000
- Uses ₹15,000 for care
- Charges 2% of the spent amount (₹15,000) as a fee — meaning this ₹300 is deducted from the donation, so only ₹14,700 is deposited

But then, how much interest is needed?

> "What percent of interest should the trust get from the bank to get a total of Rs. 1800 every month?"

So, the interest earned from the bank deposit must be ₹1800/month.

So, if ₹14,700 is deposited, and earns ₹1800/month, then:

$$
\text{Monthly interest rate } r = \frac{1800}{14700} = \frac{1800}{14700} = \frac{12}{98} = \frac{6}{49} \approx 0.1224 \Rightarrow 12.24\%
$$

This is very high — 12.24% per month → 146.9% per year — unrealistic.

So likely, the full ₹15,000 is deposited, and the 2% charge is separate.

Let’s consider:

- Trust receives ₹30,000
- Spends ₹15,000 on care
- Charges 2% of the spent amount (₹300) as a fee — perhaps this is paid by the donor, so total collected = ₹30,300, but only ₹15,000 is used for care, and ₹15,000 is deposited?

But that doesn’t help.

Wait — maybe the 2% is not deducted, but rather the trust uses the full ₹15,000 for care, and the other ₹15,000 is deposited, and the interest is calculated on that ₹15,000, and the 2% charge is just administrative.

But then why mention it?

Let’s read again:

> "The trust spends half of the funds received for medical and educational care... and for that it charges 2% of the spent amount from them"

Ah! So: the trust spends ₹15,000 on care, and charges 2% of that (₹300) as a fee to the donor. So the donor pays ₹30,000 + ₹300 = ₹30,300? That doesn’t make sense.

Alternatively: the donor gives ₹30,000, the trust spends ₹15,000 on care, and keeps ₹15,000, but charges 2% of the spent amount (₹300) as administrative fee, so the net amount available for deposit is ₹15,000 – ₹300 = ₹14,700

Then, interest on ₹14,700 must be ₹1800/month.

Again, same issue.

But perhaps the ₹1800 is not the interest, but the total return?

No — it says: "to get a total of Rs. 1800 every month"

Wait — maybe the ₹1800 is not the interest, but the amount needed to cover costs?

But no — it says: "what percent of interest should the trust get from the bank to get a total of Rs. 1800 every month?"

So, interest income = ₹1800/month.

But ₹1800 from ₹14,700 is impossible.

Wait — perhaps the deposited amount is ₹15,000, and the 2% charge is irrelevant to the deposit, and only the interest on ₹15,000 is considered.

Let’s assume:

- Donation: ₹30,000
- Spent on care: ₹15,000
- Deposit: ₹15,000
- Interest on ₹15,000 = ₹1800/month

Then:
$$
r = \frac{1800}{15000} = 0.12 = 12\% \text{ per month} \Rightarrow 144\% \text{ per year}
$$

Still unrealistic.

Wait — monthly interest of ₹1800 on ₹15,000 → 12% per month → too high.

So maybe the deposit is larger?

Another possibility: the trust deposits the entire ₹30,000, but spends ₹15,000 on care, and the rest ₹15,000 is deposited, and the 2% charge is a fee paid by the donor, so no reduction in deposit.

But still, ₹1800/month on ₹15,000 is 12% monthly.

Unrealistic.

Wait — perhaps the ₹1800 is the net income after fees?

But the question says: "to get a total of Rs. 1800 every month"

And "use matrix method".

So perhaps we need to use matrix method to model the flow.

Let’s try to formalize with matrices.

---

#### Matrix Method Approach

Let’s define variables.

Let:
- $ D = 30,000 $: monthly donation
- Spent on care: $ S = 0.5 \times D = 15,000 $
- Charge: $ 2\% $ of $ S = 0.02 \times 15,000 = 300 $
- So, net deposit = $ D - S - \text{charge}? $

Wait — no.

If the trust receives ₹30,000, spends ₹15,000 on care, and charges 2% of spent amount (₹300) as a fee — where does the ₹300 come from?

Possibly, the donor pays ₹30,000, and the trust keeps ₹300 as fee, and deposits ₹14,700.

But then deposit = ₹14,700

Then, interest earned = $ I = P \times r $, where $ r $ is monthly interest rate.

We want:
$$
I = 1800 = 14700 \times r \Rightarrow r = \frac{1800}{14700} = \frac{12}{98} = \frac{6}{49} \approx 0.1224 = 12.24\%
$$

So monthly interest rate = 12.24%

Annual = 12.24 × 12 = 146.9% → not realistic

But perhaps the ₹1800 is not the interest, but the total return including something else?

Wait — maybe the ₹1800 is the interest, and the deposit is ₹15,000, and the 2% charge is ignored.

Or, perhaps the trust deposits ₹15,000, and earns interest, and the 2% charge is a separate fee, but does not reduce the deposit.

But then why mention it?

Wait — another interpretation:

> "The trust spends half of the funds received for medical and educational care... and for that it charges 2% of the spent amount from them"

So, the donor is charged 2% of the spent amount as a fee, meaning the donor pays extra.

So, total collected = ₹30,000 + ₹300 = ₹30,300

Spent on care: ₹15,000

Deposit: ₹15,000 (the other half) — but where does the ₹300 go?

Maybe the ₹300 is admin fee, so deposit = ₹15,000, and interest on ₹15,000 = ₹1800/month

Then:
$$
r = \frac{1800}{15000} = 0.12 = 12\% \text{ per month} \Rightarrow 144\% \text{ per year}
$$

Still impossible.

Wait — perhaps the ₹1800 is not monthly interest, but annual?

But it says "every month".

Another idea: perhaps the trust deposits the full ₹30,000, but spends ₹15,000 on care, and the other ₹15,000 is deposited, and the 2% charge is on the spent amount, but it's paid by the donor, so no effect on deposit.

Then, deposit = ₹15,000

Interest needed = ₹1800/month

So:
$$
r = \frac{1800}{15000} = 0.12 = 12\% \text{ per month}
$$

But this is unrealistic.

Wait — perhaps the ₹1800 is the net interest after some deduction, or perhaps the deposit is larger.

Wait — what if the trust deposits ₹15,000, and the interest is ₹1800 per month, then:

$$
r = \frac{1800}{15000} = 12\% \text{ per month}
$$

But this is not possible.

Unless the ₹1800 is annual, but it says "every month".

Wait — perhaps the trust wants ₹1800 per month as interest, so the deposit must be such that:

Let $ P $ be the deposit.

$ P \times r = 1800 $

But $ P = 15,000 $, so $ r = 12\% $ per month.

But this is not feasible.

Wait — perhaps the 2% charge is not deducted, but the deposit is ₹15,000, and the trust wants ₹1800 interest per month, so:

$$
r = \frac{1800}{15000} = 12\% \text{ per month}
$$

But this is not possible.

Alternatively, perhaps the ₹1800 is not interest, but total return.

But the question says: "get a total of Rs. 1800 every month" — likely means interest income.

But maybe the deposit is ₹15,000, and the interest is ₹1800 per month, so rate is 12% per month.

But that’s not practical.

Wait — perhaps the trust deposits ₹15,000, and the interest is compounded, but we’re to find the rate.

But still, 12% per month is too high.

Wait — maybe the ₹1800 is the amount needed to cover the 2% charge?

No.

Another idea: the trust wants to earn ₹1800 per month in interest, and the deposit is ₹15,000, so:

$$
r = \frac{1800}{15000} = 0.12 = 12\% \text{ per month}
$$

But this is not reasonable.

Perhaps there's a typo, and it's ₹180 per month instead of ₹1800.

But let's assume the numbers are correct.

Perhaps the deposit is ₹15,000, and the interest is ₹1800 per month, so rate is 12% per month.

But then, using matrix method.

Let’s try to represent in matrix form.

Let’s define:

- Let $ D = 30,000 $: donation
- Spent on care: $ S = 0.5 \times D = 15,000 $
- Charge: $ C = 0.02 \times S = 300 $
- Deposit: $ P = D - S - C = 30,000 - 15,000 - 300 = 14,700 $

Now, let $ r $ be the monthly interest rate.

Then, interest = $ P \times r = 14,700 \times r $

Set equal to 1800:

$$
14,700 \times r = 1800 \Rightarrow r = \frac{1800}{14700} = \frac{12}{98} = \frac{6}{49} \approx 0.1224 = 12.24\%
$$

So monthly interest rate = 12.24%

Annual = 12.24 × 12 = 146.9%

But this is not feasible.

Alternatively, perhaps the interest is earned on the full ₹15,000, and the 2% charge is a separate fee, so deposit = ₹15,000, interest = 1800, so r = 12% per month.

Still high.

But since the problem asks to use matrix method, let’s try to represent it.

---

#### Matrix Representation

Let’s define a vector for the flows.

Let’s say:

- Input: Donation = ₹30,000
- Output:
- Care: ₹15,000
- Admin fee: ₹300
- Deposit: ₹14,700

So, the deposit amount is:

$$
P = 30,000 - 15,000 - 300 = 14,700
$$

Now, let’s define a matrix to represent the allocation.

Let’s create a coefficient matrix.

Let:
- $ x $ = donation amount
- We want to express the deposit as a function of x.

From above:
- Spent on care: $ 0.5x $
- Charge: $ 0.02 \times 0.5x = 0.01x $
- Deposit: $ x - 0.5x - 0.01x = 0.49x $

So, deposit = $ 0.49x $

With $ x = 30,000 $, deposit = $ 0.49 \times 30,000 = 14,700 $

Now, interest = $ P \times r = 0.49x \times r $

Set equal to 1800:

$$
0.49 \times 30,000 \times r = 1800 \Rightarrow 14,700 \times r = 1800 \Rightarrow r = \frac{1800}{14700} = 12.24\%
$$

Now, to use matrix method, we can write:

Let’s define:

$$
\vec{v} = \begin{bmatrix} \text{Donation} \\ \text{Care} \\ \text{Charge} \\ \text{Deposit} \end{bmatrix}
$$

But we can write:

$$
\begin{bmatrix} \text{Care} \\ \text{Charge} \\ \text{Deposit} \end{bmatrix} =
\begin{bmatrix}
0.5 & 0 \\
0.01 & 0 \\
0.49 & 0 \\
\end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
$$

But perhaps simpler:

Let’s define a transformation matrix for the allocation.

Let’s say the donation is $ x $, and we want to compute the deposit.

The deposit is:

$$
\text{Deposit} = x - 0.5x - 0.01x = 0.49x
$$

So, in matrix form:

$$
\text{Deposit} = [0.49] \times x
$$

Then, interest = deposit × r = 0.49x × r

Set equal to 1800:

$$
0.49 \times 30,000 \times r = 1800 \Rightarrow r = \frac{1800}{14700} = 12.24\%
$$

So, monthly interest rate = 12.24%

But this is not realistic.

However, since the problem asks for it, we'll go with it.

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#### ✔ Answer to Problem 6:

- Deposit amount = ₹14,700
- Required interest = ₹1800/month
- Rate of interest required = $ \frac{1800}{14700} \times 100 = 12.24\% $ per month

- Annual rate = 12.24 × 12 = 146.9% — extremely high

This suggests that either:
- The numbers are incorrect (e.g., ₹180 instead of ₹1800)
- Or the assumption about the deposit is wrong

But based on given data, this is the result.

---

#### Should people donate to such trusts?

Yes, if the trust is transparent, accountable, and uses funds effectively. Even if the interest rate is high, it might be due to short-term goals or assumptions.

But 12.24% monthly interest is unrealistic — banks do not offer such rates.

So, likely the numbers are misstated.

But assuming the math is correct, the answer is:

> The trust needs 12.24% per month interest rate, which is not feasible in reality.

Therefore, people should donate only if the trust is trustworthy and the financial model is sound.

---

✔ Final Answers:

#### Problem 5:
- Daily expenses:
- Family A: ₹1050
- Family B: ₹1150
- Family C: ₹2300
- More children increase dependency, raising expenses and societal burden, unless supported by social welfare.

#### Problem 6:
- Deposit = ₹14,700
- Required interest = ₹1800/month
- Rate of interest = 12.24% per month (≈146.9% per annum)
- Not feasible in real banking.
- People should donate only if the trust is transparent and financially sound.

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Let me know if you'd like to revise the numbers or explore a different interpretation.