1. Understanding Direct Proportion

In direct proportion, two quantities increase or decrease together. For example, if the amount of food rations for a family increases, the number of people that can be fed also increases. This relationship can be expressed with the formula y = kx, where k is the constant of proportionality.

Example Scenario

Consider a situation where 4 kg of rice can feed 6 people for a week. If we want to find out how much rice is needed to feed 9 people, we set up the proportion:

  • 4 kg / 6 people = x kg / 9 people

Solving for x gives:

x = (4 kg * 9 people) / 6 people = 6 kg

2. Exploring Inverse Proportion

Inverse proportion describes a relationship where one quantity increases as the other decreases. This can be seen in scenarios such as work days and completion time. For example, if more workers are assigned to complete a task, the time taken to finish it decreases.

Example Scenario

If 5 workers can complete a project in 10 days, how long will it take 10 workers to finish the same project?

Using the formula xy = k, where x is the number of workers and y is the number of days, we have:

  • 5 workers * 10 days = k

Thus, k = 50. Now, for 10 workers:

10 workers * y days = 50 → y = 5 days

3. Partitive Proportion Explained

Partitive proportion divides a whole into different parts based on a given ratio. For instance, if a budget of ₱100,000 is allocated to three departments in the ratio 2:3:5, we can find out how much each department receives.

Example Scenario

To find the total parts, add the ratio components: 2 + 3 + 5 = 10 parts. To find the amount for each department, we divide the total budget by the number of parts:

  • ₱100,000 / 10 = ₱10,000 per part

The distribution will be:

  • Department A: 2 parts → ₱20,000
  • Department B: 3 parts → ₱30,000
  • Department C: 5 parts → ₱50,000

4. Sample Practice Questions & Rationalizations

Q1: If 3 kg of sugar costs ₱150, how much will 7 kg cost?
A) ₱350
B) ₱350
C) ₱400
D) ₱450
Rationalization: Set up the proportion: 3 kg / ₱150 = 7 kg / x. Solving gives x = ₱350.
Q2: If 8 workers can complete a task in 12 days, how many days will it take 4 workers to complete it?
A) 24 days
B) 36 days
C) 48 days
D) 60 days
Rationalization: Using inverse proportion: 8 workers * 12 days = 96. For 4 workers: 4 workers * x days = 96 → x = 24 days.
Q3: A budget of ₱120,000 is divided among three departments in the ratio 3:2:5. How much does the second department receive?
A) ₱24,000
B) ₱30,000
C) ₱48,000
D) ₱60,000
Rationalization: Total parts = 10. Each part = ₱12,000. Second department gets 2 parts → ₱24,000.

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