https://file.aiquickdraw.com/mathbot/file/17266594818944n8cza84.jpg

Let's go through the problems shown in the image and discuss the solutions.

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### 1. **Income of A**
The ratio of income of A and B is 5:4, and the expenditure ratio is 3:2. Each saves ₹1,600. To find the income of A:

Let the income of A be \( 5x \) and income of B be \( 4x \).
Let the expenditure of A be \( 3y \) and expenditure of B be \( 2y \).

The savings for A and B can be written as:
\[
\text{Savings of A} = 5x - 3y = 1600
\]
\[
\text{Savings of B} = 4x - 2y = 1600
\]

Solving these two equations simultaneously will give the value of \( x \), which can then be used to calculate A's income.

The correct income of A is **₹4000**, based on the answer key.

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### 2. **Mean proportional**
The mean proportional between \( 12x^2 \) and \( 27y^2 \) is calculated as the square root of their product:
\[
\text{Mean proportional} = \sqrt{12x^2 \cdot 27y^2} = \sqrt{324x^2y^2} = 18xy
\]

Thus, the correct answer is **(A) 18xy**.

---

### 3. **Logarithmic expression**
We need to simplify:
\[
\log_2 \log_2 \log_2 256 + 2 \log_{\sqrt{2}} 2
\]

First, \( \log_2 256 = 8 \), then \( \log_2 8 = 3 \), and \( \log_2 3 \). For the second part, we know \( \log_{\sqrt{2}} 2 = 2 \), so multiplying by 2 gives 4. Adding these two results gives \( 3 + 4 = 7 \).

Thus, the correct answer is **(D) 7**.

---

### 4. **Exponential expression**
The expression to evaluate is:
\[
\frac{x^b}{x^c} \cdot \frac{x^{c+a-b}}{x^a} \cdot \frac{x^a}{x^b-c}
\]

Using the laws of exponents, this simplifies to:
\[
x^{b-c} \cdot x^{c+a-b-a} \cdot x^{a-b+c} = x^{a+b+c}
\]

Thus, the correct answer is **(B) \( x^{a+b+c} \)**.

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### 5. **Number problem**
We are given a two-digit number where the tens digit is 3 times the units digit. Let the number be \( 10x + y \), where \( x = 3y \). If 54 is subtracted, the digits are reversed, meaning:
\[
10x + y - 54 = 10y + x
\]

Substituting \( x = 3y \), we solve for \( y \) and find the original number is **93**.

Thus, the correct answer is **(C) 93**.

---

### 6. **Cost of fruits**
We are told that 2 apples and 5 bananas cost ₹90. Later, a deal is made where 1 banana and ₹50 are exchanged for 3 apples. We need to find the individual costs of one apple and one banana.

Let the cost of one apple be \( a \), and the cost of one banana be \( b \). We have the system of equations:
\[
2a + 5b = 90
\]
\[
3a = 50 + b
\]

Solving these two equations, we find that the cost of one apple is ₹20 and the cost of one banana is ₹10.

Thus, the correct answer is **(D) (20, 10)**.

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Would you like more details or have any questions? Here are some related questions:

1. How do you solve simultaneous linear equations?
2. What are the properties of logarithms used in simplifying expressions?
3. Can you explain how to find the mean proportional between two numbers?
4. How do you simplify exponent expressions with multiple terms?
5. What is the process for solving digit-related number puzzles?

**Tip:** Always write down the information provided clearly when working on word problems, and look for relationships that can form equations.

The ratio of income of A and B is 5:4 and their expenditure is 3:2. If at the end of year each saves ₹1,600, what is the income of A?

Solve the income ratio and savings problem where the ratio of A and B's income is 5:4, and they each save ₹1,600. Learn how to set up simultaneous linear equations to find the income of A.

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