Let's break down and solve each problem step-by-step:

Problem 5: Three men can finish doing the interior designing of a house in 3 weeks. How many men are needed to finish the interior designing in a week?

Solution: This is an inverse proportion problem. As the number of men increases, the time taken decreases.

Let the number of men needed be $x$. We can set up the inverse proportion equation: $\text{men}_1 \times \text{time}_1 = \text{men}_2 \times \text{time}_2$ $3 \times 3 = x \times 1$ $9 = x$ Thus, 9 men are needed to finish the interior designing in 1 week.

Problem 6: The ratio of boys to girls in a badminton tournament is 4:3. Mariel counted that there are 12 more boys than girls. How many boys and girls are there in the tournament?

Solution: This is a ratio problem.

Let the number of boys be $4x$ and the number of girls be $3x$.

According to the problem, the number of boys is 12 more than the number of girls: $4x = 3x + 12$ Subtract $3x$ from both sides: $x = 12$

So, the number of boys is $4x = 4 \times 12 = 48$, and the number of girls is $3x = 3 \times 12 = 36$.

Therefore, there are 48 boys and 36 girls in the tournament.

Problem 7: It takes 3 salesmen 8 days to sell 5,000 boxes of soap. If 2 more salesmen are added, how long will it take them to sell the same number of boxes of soap?

Solution: This is another inverse proportion problem because as the number of salesmen increases, the time taken decreases.

Let the number of days needed be $x$. We can use the inverse proportion equation again: $\text{salesmen}_1 \times \text{time}_1 = \text{salesmen}_2 \times \text{time}_2$ $3 \times 8 = 5 \times x$ $24 = 5x$ Divide both sides by 5: $x = \frac{24}{5} = 4.8 \, \text{days}$ Thus, it will take the 5 salesmen approximately 4.8 days to sell the 5,000 boxes of soap.

Problem 8: Mr. Faustino allocates his monthly salary for bills, food, transportation, and other expenses at the ratio of 3:6:7:9, respectively. If he receives ₱ 28,450.00 each month, how much is his budget for food?

Solution: This is a ratio and proportion problem.

First, find the total ratio by adding the parts: $3 + 6 + 7 + 9 = 25$ The ratio for food is 6 parts out of 25 parts of the total salary. Therefore, the budget for food is: $\frac{6}{25} \times 28,450 = 6,828$ Thus, Mr. Faustino's budget for food is ₱6,828.

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Do you want more details or have any questions?

Here are five related questions to deepen your understanding:

1. What is the general formula for inverse proportion problems?
2. How can you solve ratio problems if there are no extra clues, such as a difference in quantity?
3. What happens to the time taken when more people are added in proportion-related work problems?
4. How can we adjust ratios when the total quantity changes?
5. How do you verify that you’ve set up a proportion equation correctly?

Tip: Always double-check whether a problem involves direct or inverse proportion. This distinction helps set up the correct equation.