8. Inverse Proportion Problems with step-by-step solutions and detailed explanations

 8. Inverse Proportion Problems with step-by-step solutions and detailed explanations

Below are 10 inverse proportion problems with step-by-step solutions and detailed explanations. Inverse proportion means that as one quantity increases, the other decreases such that their product remains constant, expressed as

y = \frac{k}{x}

, or equivalently,

x \cdot y = k

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

---

Problem 1

If 6 workers can complete a job in 12 days, how long will it take 8 workers to complete the same job?

Step-by-Step Solution:

1. Identify the proportion: The number of workers (( x )) and the time taken (( y )) are inversely proportional because the total work (worker-days) remains constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 6 workers take 12 days, substitute

   x = 6

   ,

   y = 12

   :
   k = 6 \cdot 12 = 72
   So, the total work is 72 worker-days.
3. Set up the equation for 8 workers: For

   x = 8

   , use

   8 \cdot y = 72

   :
   y = \frac{72}{8} = 9
4. Answer: It takes 9 days for 8 workers.

Explanation: As the number of workers increases, the time decreases because the same amount of work is distributed among more workers. The constant

k = 72

represents the total work in worker-days.

---

Problem 2

A car traveling at 60 km/h takes 2 hours to cover a certain distance. How long will it take if the speed is increased to 80 km/h?

Step-by-Step Solution:

1. Identify the proportion: Speed (( x )) and time (( y )) are inversely proportional because distance = speed × time remains constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given speed 60 km/h and time 2 hours, substitute

   x = 60

   ,

   y = 2

   :
   k = 60 \cdot 2 = 120
   So, the distance is 120 km.
3. Set up for speed 80 km/h: For

   x = 80

   , use

   80 \cdot y = 120

   :
   y = \frac{120}{80} = 1.5
4. Answer: It takes 1.5 hours at 80 km/h.

Explanation: Higher speed reduces the time to cover the same distance. The constant

k = 120

is the distance, and the inverse relationship ensures the product of speed and time is constant.

---

Problem 3

If 4 pumps can empty a tank in 10 hours, how long will it take 5 pumps to empty the same tank?

Step-by-Step Solution:

1. Identify the proportion: The number of pumps (( x )) and time (( y )) are inversely proportional because the total work (pump-hours) is constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 4 pumps take 10 hours, substitute

   x = 4

   ,

   y = 10

   :
   k = 4 \cdot 10 = 40
   So, the total work is 40 pump-hours.
3. Set up for 5 pumps: For

   x = 5

   , use

   5 \cdot y = 40

   :
   y = \frac{40}{5} = 8
4. Answer: It takes 8 hours for 5 pumps.

Explanation: More pumps mean less time to complete the same task. The constant

k = 40

represents the total work required to empty the tank.

---

Problem 4

If 15 machines produce 300 units in 5 days, how many machines are needed to produce 300 units in 3 days?

Step-by-Step Solution:

1. Identify the proportion: The number of machines (( x )) and days (( y )) are inversely proportional because the total work (machine-days) for 300 units is constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 15 machines take 5 days, substitute

   x = 15

   ,

   y = 5

   :
   k = 15 \cdot 5 = 75
   So, the total work is 75 machine-days.
3. Set up for 3 days: For

   y = 3

   , use

   x \cdot 3 = 75

   :
   x = \frac{75}{3} = 25
4. Answer: 25 machines are needed for 3 days.

Explanation: Fewer days require more machines to produce the same output. The constant

k = 75

is the total machine-days needed for 300 units.

---

Problem 5

If 8 pipes fill a tank in 6 hours, how long will it take 12 pipes to fill the same tank?

Step-by-Step Solution:

1. Identify the proportion: The number of pipes (( x )) and time (( y )) are inversely proportional because the total work (pipe-hours) is constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 8 pipes take 6 hours, substitute

   x = 8

   ,

   y = 6

   :
   k = 8 \cdot 6 = 48
   So, the total work is 48 pipe-hours.
3. Set up for 12 pipes: For

   x = 12

   , use

   12 \cdot y = 48

   :
   y = \frac{48}{12} = 4
4. Answer: It takes 4 hours for 12 pipes.

Explanation: More pipes reduce the time to fill the tank. The constant

k = 48

represents the total work in pipe-hours.

---

Problem 6

A group of 10 friends share a bill equally, and each pays $12. If only 8 friends share the same bill, how much does each pay?

Step-by-Step Solution:

1. Identify the proportion: The number of friends (( x )) and the amount each pays (( y )) are inversely proportional because the total bill is constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 10 friends pay $12 each, substitute

   x = 10

   ,

   y = 12

   :
   k = 10 \cdot 12 = 120
   So, the total bill is $120.
3. Set up for 8 friends: For

   x = 8

   , use

   8 \cdot y = 120

   :
   y = \frac{120}{8} = 15
4. Answer: Each of the 8 friends pays $15.

Explanation: Fewer friends mean each pays more to cover the same bill. The constant

k = 120

is the total bill amount.

---

Problem 7

If 5 typists can type a document in 8 hours, how many typists are needed to type it in 4 hours?

Step-by-Step Solution:

1. Identify the proportion: The number of typists (( x )) and time (( y )) are inversely proportional because the total work (typist-hours) is constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 5 typists take 8 hours, substitute

   x = 5

   ,

   y = 8

   :
   k = 5 \cdot 8 = 40
   So, the total work is 40 typist-hours.
3. Set up for 4 hours: For

   y = 4

   , use

   x \cdot 4 = 40

   :
   x = \frac{40}{4} = 10
4. Answer: 10 typists are needed for 4 hours.

Explanation: Less time requires more typists to complete the same document. The constant

k = 40

represents the total work.

---

Problem 8

A light bulb’s brightness is inversely proportional to the square of the distance from it. If the brightness is 16 units at 2 meters, what is the brightness at 4 meters?

Step-by-Step Solution:

1. Identify the proportion: Brightness (( y )) is inversely proportional to the square of the distance (( x )), so

   y = \frac{k}{x^2}

   , or

   x^2 \cdot y = k

   .
2. Find the constant (( k )): Given brightness 16 units at 2 meters, substitute

   x = 2

   ,

   y = 16

   :
   k = 2^2 \cdot 16 = 4 \cdot 16 = 64
3. Set up for 4 meters: For

   x = 4

   , use

   4^2 \cdot y = 64

   :
   16 \cdot y = 64 \implies y = \frac{64}{16} = 4
4. Answer: The brightness at 4 meters is 4 units.

Explanation: Brightness follows an inverse square law, common in physics (e.g., light or gravity). Doubling the distance reduces brightness to a quarter, as

(4/2)^2 = 4

.

---

Problem 9

If 20 cows graze a field in 6 days, how many cows are needed to graze the same field in 4 days?

Step-by-Step Solution:

1. Identify the proportion: The number of cows (( x )) and days (( y )) are inversely proportional because the total grazing work (cow-days) is constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 20 cows take 6 days, substitute

   x = 20

   ,

   y = 6

   :
   k = 20 \cdot 6 = 120
   So, the total work is 120 cow-days.
3. Set up for 4 days: For

   y = 4

   , use

   x \cdot 4 = 120

   :
   x = \frac{120}{4} = 30
4. Answer: 30 cows are needed for 4 days.

Explanation: Fewer days require more cows to graze the same field. The constant

k = 120

is the total grazing work.

---

Problem 10

If 12 students can clean a room in 9 hours, how long will it take 18 students to clean the same room?

Step-by-Step Solution:

1. Identify the proportion: The number of students (( x )) and time (( y )) are inversely proportional because the total work (student-hours) is constant. Thus,

   x \cdot y = k

   .
2. Find the constant (( k )): Given 12 students take 9 hours, substitute

   x = 12

   ,

   y = 9

   :
   k = 12 \cdot 9 = 108
   So, the total work is 108 student-hours.
3. Set up for 18 students: For

   x = 18

   , use

   18 \cdot y = 108

   :
   y = \frac{108}{18} = 6
4. Answer: It takes 6 hours for 18 students.

Explanation: More students reduce the time to clean the room. The constant

k = 108

represents the total work.

---

General Notes on Inverse Proportion

• Formula:

  x \cdot y = k

  , or

  y = \frac{k}{x}

  , where ( k ) is the constant of proportionality.
• Key Steps:
  1. Identify the two quantities that are inversely proportional.
  2. Use given values to find ( k ).
  3. Use ( k ) to find the unknown quantity.
• Alternative Method: You can use ratios for inverse proportion. For example, in Problem 1:
  6 \cdot 12 = 8 \cdot y \implies y = \frac{6 \cdot 12}{8} = 9
  This method is equivalent and often intuitive.
• Key Insight: Inverse proportion often involves a fixed total (e.g., work, distance, or cost), and increasing one variable decreases the other to maintain the constant product.

These problems illustrate inverse proportion in practical scenarios, such as work rates, speed, and resource sharing. If you need more problems or further clarification, let me know!