Understanding Compound Proportion: Detailed Notes and Solved MCQs

Introduction:

Compound Proportion (also called Joint Proportion) deals with comparing two or more quantities that are linked by ratios. When multiple ratios are involved, we use the concept of compound proportion to find the unknown value, ensuring that the relationships between the different quantities are maintained. 

Types of Proportions:

1. Direct Proportion: Two quantities are in direct proportion if they increase or decrease together in such a way that their ratio remains constant.

   • Example: More hours worked → More pay earned.

2. Inverse Proportion: Two quantities are in inverse proportion if one quantity increases while the other decreases, such that their product remains constant.

   • Example: More workers → Less time to complete a job.

3. Compound Proportion: It is the combined effect of two or more direct or inverse proportions working simultaneously. The unknown quantity is calculated by using a combination of these proportions.

Steps to Solve Compound Proportion Problems:

1. Identify the type of proportion (direct or inverse) involved between the known quantities and the unknown quantity.
2. Set up the ratios for each proportion.
3. Formulate an equation involving all proportions and solve for the unknown quantity.

Formula for Compound Proportion:

Suppose two or more quantities $A_1, A_2, …, A_n$A1​,A2​,…,An​ are in proportion to other quantities $B_1, B_2, …, B_n$B1​,B2​,…,Bn​, respectively, and we need to find an unknown quantity $x$x. We use the relation:

$$x = left(frac{A_1}{B_1}right) times left(frac{A_2}{B_2}right) times … times left(frac{A_n}{B_n}right) times text{(constant)}$$x=(B1​A1​​)×(B2​A2​​)×…×(Bn​An​​)×(constant)

This equation allows us to solve for the unknown by considering all the factors involved.

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Solved MCQs / Problems:

Problem 1:

If 8 workers can finish a task in 12 days, how many days will it take for 6 workers to complete the same task, assuming the rate of work remains constant?

• a) 8 days
• b) 10 days
• c) 14 days
• d) 16 days

Solution:

Here, we have an inverse proportion because fewer workers will take more time to complete the task. We set up the inverse proportion as:

$$frac{8 text{ workers}}{6 text{ workers}} = frac{text{New time}}{12 text{ days}}$$6 workers8 workers​=12 daysNew time​

Cross-multiplying gives:

$$text{New time} = frac{8}{6} times 12 = 16 text{ days}$$New time=68​×12=16 days

Answer: d) 16 days

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Problem 2:

A car covers 120 km in 3 hours. How long will it take to cover 200 km at the same speed?

• a) 4 hours
• b) 5 hours
• c) 6 hours
• d) 7 hours

Solution:

This is a direct proportion because more distance requires more time. The ratio of distance to time is constant.

$$frac{120 text{ km}}{3 text{ hours}} = frac{200 text{ km}}{x text{ hours}}$$3 hours120 km​=x hours200 km​

Cross-multiplying gives:

$$x = frac{200 times 3}{120} = 5 text{ hours}$$x=120200×3​=5 hours

Answer: b) 5 hours

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Problem 3:

If 12 machines can produce 240 units in 8 hours, how many units can 15 machines produce in 10 hours, working at the same rate?

• a) 300 units
• b) 450 units
• c) 400 units
• d) 500 units

Solution:

This is a compound proportion because the number of machines and the number of hours both change. We use both direct proportions (machines and hours) together.

Let the number of units be $x$x.

$$frac{12 text{ machines}}{15 text{ machines}} times frac{8 text{ hours}}{10 text{ hours}} = frac{240 text{ units}}{x text{ units}}$$15 machines12 machines​×10 hours8 hours​=x units240 units​

Simplifying,

$$frac{12}{15} times frac{8}{10} = frac{240}{x}$$1512​×108​=x240​

Cross-multiply and solve for $x$x:

$$x = frac{240 times 15 times 10}{12 times 8} = 450 text{ units}$$x=12×8240×15×10​=450 units

Answer: b) 450 units

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Problem 4:

10 pumps can empty a tank in 6 hours. How many hours would it take 8 pumps to empty the tank?

• a) 5 hours
• b) 7.5 hours
• c) 8 hours
• d) 9 hours

Solution:

Since the number of pumps is inversely proportional to the time taken:

$$frac{10 text{ pumps}}{8 text{ pumps}} = frac{6 text{ hours}}{x text{ hours}}$$8 pumps10 pumps​=x hours6 hours​

Cross-multiplying gives:

$$x = frac{10}{8} times 6 = 7.5 text{ hours}$$x=810​×6=7.5 hours

Answer: b) 7.5 hours

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Problem 5:

A recipe requires 5 cups of flour to make 10 cakes. How many cakes can be made with 8 cups of flour?

• a) 12 cakes
• b) 14 cakes
• c) 16 cakes
• d) 18 cakes

Solution:

This is a direct proportion since more flour will make more cakes.

$$frac{5 text{ cups}}{8 text{ cups}} = frac{10 text{ cakes}}{x text{ cakes}}$$8 cups5 cups​=x cakes10 cakes​

Cross-multiplying gives:

$$x = frac{10 times 8}{5} = 16 text{ cakes}$$x=510×8​=16 cakes

Answer: c) 16 cakes

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Conclusion:

Compound proportion is an essential mathematical concept that applies in real-world situations involving multiple related variables. By analyzing the proportional relationships and solving the equations systematically, we can solve complex problems involving direct and inverse proportions.