7. Direct Proportion Problems with step-by-step solutions and detailed explanations

 7. Direct Proportion Problems with step-by-step solutions and detailed explanations

Below are 10 direct proportion problems with step-by-step solutions and detailed explanations. Direct proportion means that as one quantity increases, the other increases by a constant ratio, expressed as

y = kx

, where ( k ) is the constant of proportionality. Each problem will include the problem statement, solution steps, and an explanation of the concept.

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Problem 1

If 5 pens cost $10, how much will 8 pens cost?

Step-by-Step Solution:

1. Identify the proportion: The cost of pens is directly proportional to the number of pens. Let ( x ) be the number of pens and ( y ) be the cost. Thus,

   y = kx

   .
2. Find the constant of proportionality (( k )): Given 5 pens cost $10, substitute

   x = 5

   ,

   y = 10

   :
   10 = k \cdot 5 \implies k = \frac{10}{5} = 2
   So, the cost per pen is $2.
3. Set up the equation for 8 pens: Using

   y = 2x

   , for

   x = 8

   :
   y = 2 \cdot 8 = 16
4. Answer: The cost of 8 pens is $16.

Explanation: The number of pens and their cost are directly proportional because the cost increases linearly with the number of pens. The constant

k = 2

represents the cost per pen. We used this to find the cost for any number of pens.

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Problem 2

A car travels 120 miles in 2 hours. How far will it travel in 5 hours at the same speed?

Step-by-Step Solution:

1. Identify the proportion: Distance is directly proportional to time when speed is constant. Let ( x ) be time (hours) and ( y ) be distance (miles). Thus,

   y = kx

   .
2. Find the constant (( k )): Given 120 miles in 2 hours, substitute

   x = 2

   ,

   y = 120

   :
   120 = k \cdot 2 \implies k = \frac{120}{2} = 60
   So, the speed is 60 miles per hour.
3. Calculate distance for 5 hours: Using

   y = 60x

   , for

   x = 5

   :
   y = 60 \cdot 5 = 300
4. Answer: The car travels 300 miles in 5 hours.

Explanation: Distance and time are directly proportional because the car travels at a constant speed. The constant

k = 60

is the speed, and multiplying it by time gives the distance.

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Problem 3

If 3 workers can complete a job in 12 days, how long will it take 5 workers to complete the same job?

Step-by-Step Solution:

1. Identify the proportion: The number of workers is inversely proportional to the time taken (more workers, less time), but the work rate per worker is directly proportional. Total work is constant, so we use the concept of work rate. Let ( x ) be the number of workers and ( y ) be the work rate. However, for direct proportion, consider the work done per day.
2. Find the work rate: 3 workers take 12 days, so the total work is

   3 \cdot 12 = 36

   worker-days. Work rate per worker =

   \frac{1}{12}

   job per day.
3. Set up for 5 workers: With 5 workers, the work rate is

   5 \cdot \frac{1}{12} = \frac{5}{12}

   job per day.
4. Find time for 5 workers: Time = Total work ÷ Work rate =

   1 \div \frac{5}{12} = \frac{12}{5} = 2.4

   days.
5. Answer: It takes 2.4 days for 5 workers.

Explanation: This is a twist on direct proportion. The work rate per worker is constant, and more workers increase the total work rate proportionally, reducing the time inversely. We calculated the time by considering the total work.

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Problem 4

If 4 kg of apples cost $12, how much will 7 kg of apples cost?

Step-by-Step Solution:

1. Identify the proportion: Cost is directly proportional to weight. Let ( x ) be weight (kg) and ( y ) be cost ($). Thus,

   y = kx

   .
2. Find the constant (( k )): Given 4 kg costs $12, substitute

   x = 4

   ,

   y = 12

   :
   12 = k \cdot 4 \implies k = \frac{12}{4} = 3
   So, the cost is $3 per kg.
3. Calculate cost for 7 kg: Using

   y = 3x

   , for

   x = 7

   :
   y = 3 \cdot 7 = 21
4. Answer: The cost of 7 kg of apples is $21.

Explanation: The cost per kg is constant, making cost directly proportional to weight. The constant

k = 3

is the price per kg, and we scale it for 7 kg.

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Problem 5

A recipe for 4 people requires 6 eggs. How many eggs are needed for 10 people?

Step-by-Step Solution:

1. Identify the proportion: The number of eggs is directly proportional to the number of people. Let ( x ) be people and ( y ) be eggs. Thus,

   y = kx

   .
2. Find the constant (( k )): Given 6 eggs for 4 people, substitute

   x = 4

   ,

   y = 6

   :
   6 = k \cdot 4 \implies k = \frac{6}{4} = 1.5
   So, 1.5 eggs per person.
3. Calculate eggs for 10 people: Using

   y = 1.5x

   , for

   x = 10

   :
   y = 1.5 \cdot 10 = 15
4. Answer: 15 eggs are needed for 10 people.

Explanation: The number of eggs scales directly with the number of people, as the recipe proportions remain constant. The constant

k = 1.5

represents eggs per person.

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Problem 6

A machine produces 200 units in 5 hours. How many units will it produce in 8 hours?

Step-by-Step Solution:

1. Identify the proportion: Units produced are directly proportional to time. Let ( x ) be time (hours) and ( y ) be units. Thus,

   y = kx

   .
2. Find the constant (( k )): Given 200 units in 5 hours, substitute

   x = 5

   ,

   y = 200

   :
   200 = k \cdot 5 \implies k = \frac{200}{5} = 40
   So, the production rate is 40 units per hour.
3. Calculate units for 8 hours: Using

   y = 40x

   , for

   x = 8

   :
   y = 40 \cdot 8 = 320
4. Answer: The machine produces 320 units in 8 hours.

Explanation: The production rate is constant, so units produced increase directly with time. The constant

k = 40

is the rate of production.

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Problem 7

If 2 liters of paint cover 10 square meters, how many liters are needed to cover 25 square meters?

Step-by-Step Solution:

1. Identify the proportion: Paint volume is directly proportional to area. Let ( x ) be area (sq m) and ( y ) be paint (liters). Thus,

   y = kx

   .
2. Find the constant (( k )): Given 2 liters for 10 sq m, substitute

   x = 10

   ,

   y = 2

   :
   2 = k \cdot 10 \implies k = \frac{2}{10} = 0.2
   So, 0.2 liters per sq m.
3. Calculate paint for 25 sq m: Using

   y = 0.2x

   , for

   x = 25

   :
   y = 0.2 \cdot 25 = 5
4. Answer: 5 liters of paint are needed.

Explanation: The amount of paint scales directly with the area, as the coverage rate is constant. The constant

k = 0.2

is the paint needed per square meter.

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Problem 8

A pump fills a tank with 300 liters of water in 15 minutes. How long will it take to fill 500 liters?

Step-by-Step Solution:

1. Identify the proportion: Time is directly proportional to volume. Let ( x ) be volume (liters) and ( y ) be time (minutes). Thus,

   y = kx

   .
2. Find the constant (( k )): Given 300 liters in 15 minutes, substitute

   x = 300

   ,

   y = 15

   :
   15 = k \cdot 300 \implies k = \frac{15}{300} = 0.05
   So, 0.05 minutes per liter.
3. Calculate time for 500 liters: Using

   y = 0.05x

   , for

   x = 500

   :
   y = 0.05 \cdot 500 = 25
4. Answer: It takes 25 minutes to fill 500 liters.

Explanation: The time to fill the tank scales directly with the volume, as the pump’s flow rate is constant. The constant

k = 0.05

is the time per liter.

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Problem 9

If 6 books weigh 9 kg, what is the weight of 10 books?

Step-by-Step Solution:

1. Identify the proportion: Weight is directly proportional to the number of books. Let ( x ) be books and ( y ) be weight (kg). Thus,

   y = kx

   .
2. Find the constant (( k )): Given 6 books weigh 9 kg, substitute

   x = 6

   ,

   y = 9

   :
   9 = k \cdot 6 \implies k = \frac{9}{6} = 1.5
   So, 1.5 kg per book.
3. Calculate weight for 10 books: Using

   y = 1.5x

   , for

   x = 10

   :
   y = 1.5 \cdot 10 = 15
4. Answer: The weight of 10 books is 15 kg.

Explanation: The weight per book is constant, so the total weight scales directly with the number of books. The constant

k = 1.5

is the weight per book.

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Problem 10

A factory produces 150 toys in 3 days. How many toys will it produce in 7 days?

Step-by-Step Solution:

1. Identify the proportion: Toys produced are directly proportional to time. Let ( x ) be days and ( y ) be toys. Thus,

   y = kx

   .
2. Find the constant (( k )): Given 150 toys in 3 days, substitute

   x = 3

   ,

   y = 150

   :
   150 = k \cdot 3 \implies k = \frac{150}{3} = 50
   So, 50 toys per day.
3. Calculate toys for 7 days: Using

   y = 50x

   , for

   x = 7

   :
   y = 50 \cdot 7 = 350
4. Answer: The factory produces 350 toys in 7 days.

Explanation: The production rate is constant, so the number of toys scales directly with time. The constant

k = 50

is the daily production rate.

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General Notes on Direct Proportion

• Formula:

  y = kx

  , where ( k ) is the constant of proportionality.
• Key Steps:
  1. Identify the two quantities that are directly proportional.
  2. Use given values to find ( k ).
  3. Use ( k ) to find the unknown quantity.
• Alternative Method: You can also use ratios for direct proportion. For example, in Problem 1:
  \frac{5 \text{ pens}}{$10} = \frac{8 \text{ pens}}{y} \implies y = \frac{8 \cdot 10}{5} = 16
  This method is equivalent and often quicker for simple problems.

Each problem above demonstrates a real-world application of direct proportion, from costs to production rates, making the concept practical and relatable. If you need further clarification or additional problems, let me know!