https://file.aiquickdraw.com/mathbot/file/1725678455487xjk42rpa.jpg

Let’s break down and address the questions from the image one by one.

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**ii.** Ms. Bhagya spent \( \frac{3}{16} \) of her monthly earnings on food, \( \frac{5}{18} \) on rent, and \( \frac{7}{12} \) on other expenses. Find the fractional part left of her earnings.

- **Step 1:** Total spent is the sum of these fractions.
\[
\text{Total spent} = \frac{3}{16} + \frac{5}{18} + \frac{7}{12}
\]
- **Step 2:** Subtract the total spent from 1 (whole earnings) to find the fraction left.

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**iii.** A solution of vinegar and water contains 2 liters of vinegar and 6 liters of water. How much pure vinegar must be added so that the resulting solution will be \( \frac{2}{5} \) vinegar?

- **Step 1:** Let the amount of pure vinegar to be added be \( x \) liters.
- **Step 2:** Write an equation where the new ratio of vinegar to the total solution equals \( \frac{2}{5} \).

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**iv.** Mr. Perera’s son Kumar has just completed 15 years of age. To provide Rs. 25,000 on Kumar's 25th birthday, Mr. Perera decides to invest a fixed sum. Calculate the amount to invest assuming 8% compounded semi-annually.

- **Step 1:** Use the compound interest formula:
\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]
Where \( A = 25,000 \), \( r = 0.08 \), \( n = 2 \), and \( t = 10 \) years.
- **Step 2:** Solve for \( P \), the principal amount.

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**v.** Solve the inequality:
\[
\frac{2x + 7}{6} \leq \frac{3x - 4}{9}
\]

- **Step 1:** Eliminate denominators by multiplying both sides by 18.
- **Step 2:** Solve for \( x \).

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**vi.** Amila, Babu, and Christy earn a total of Rs. 970 per week. Amila's take-home salary plus twice the sum of the others’ is Rs. 1640. Christy’s take-home salary plus twice each of the others' is Rs. 1600. Find the take-home salary of each.

- **Step 1:** Let the salaries of Amila, Babu, and Christy be \( A \), \( B \), and \( C \) respectively.
- **Step 2:** Form a system of equations:
  - \( A + B + C = 970 \)
  - \( A + 2(B + C) = 1640 \)
  - \( C + 2(A + B) = 1600 \)
- **Step 3:** Solve this system using matrices or substitution.

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Would you like me to solve one of these problems in detail, or provide additional steps for all? 

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Here are 5 relative questions that can help you expand on the topics:

1. How can you calculate the amount left of Ms. Bhagya’s earnings using decimal approximations instead of fractions?
2. How would the solution in the vinegar-water problem change if instead of pure vinegar, a mixture of vinegar and water were added?
3. What happens to the investment calculation in question iv if the interest were compounded quarterly instead of semi-annually?
4. Can the inequality in question v be solved graphically? If so, how would you approach it?
5. What alternative methods (besides matrices) could be used to solve the system of equations in question vi?

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**Tip:** Always double-check your solution by substituting the found values back into the original equations or inequalities to ensure accuracy!

ii. Ms. Bhagya spent 3/16 of her monthly earnings for food, 5/18 for rent, and 7/12 for other expenses. Find the fractional part left of her earnings.

iii. A solution of vinegar and water contains 2 litres of vinegar and 6 litres of water. How much pure vinegar must be added to this solution so that the resulting solution will be 2/5 vinegar?

iv. Mr. Perera’s son Kumar has just completed 15 years of age. In order to provide a sum of Rs. 25,000 on Kumar’s 25th Birthday, Mr. Perera has decided to invest a fixed sum of money. Calculate the amount of money that should be invested assuming the account earns interest at a rate of 8% compounded semi-annually.

v. Solve the following inequality: (2x + 7)/6 ≤ (3x - 4)/9

vi. Amila, Babu and Christy earn a total of Rs. 970 per week as the take-home salary. Amila’s take-home salary plus twice each of the other’s is Rs. 1640. Christy’s take-home salary plus twice each of the other’s is Rs. 1600. Using matrices, find the take-home salary of each.

Learn how to solve various math problems, including fractions, compound interest, inequalities, and systems of linear equations with matrices. Suitable for students in Grades 9-11. This detailed solution covers core mathematical concepts like fractions, ratios, and algebraic inequalities.

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