ACTIVITY: Solve the following problems and identify the kind of proportion involved. Write your answer in your notebook! 


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? 

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

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? 

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?

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**.

---

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.

ACTIVITY: Solve the following problems and identify the kind of proportion involved. Write your answer in your notebook! 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? 6. The ratio of boys to girls in a badminton tournament game is 4:3. Mariel counted that there are 12 more boys than girls. How many boys and girls are there in the tournament? 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? 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?

Explore how to solve real-world problems involving inverse proportion, ratios, and basic algebra. Learn how many men are needed to complete a task in less time, calculate the ratio of boys and girls in a tournament, determine the impact of adding more workers on task completion time, and allocate a budget based on given ratios. Suitable for Grades 7-9.

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