Grade 7 Direct and Inverse Proportion

Concept: Two quantities are in **direct proportion** if they increase or decrease at the same rate. As one quantity increases, the other quantity increases proportionally, and as one decreases, the other decreases proportionally.

Relationship: If quantities X and Y are in direct proportion, their ratio X/Y is constant. We can write this as X is proportional to Y, or X = kY, where k is the constant of proportionality.

Example: The cost of apples and the number of apples. If one apple costs 10 rupees, then 2 apples cost 20 rupees, 3 apples cost 30 rupees, and so on.

Explanation: As the number of apples increases, the total cost increases proportionally. The ratio Cost / Number of Apples is constant (10/1 = 20/2 = 30/3 = 10).

Step 1: Identify the quantities and the type of proportion. The quantities are the number of pens and their cost. As the number of pens increases, the cost increases, so this is a **direct proportion**.

Step 2: Set up a proportion using the given information. Let the cost of 12 pens be x rupees.

Number of Pens : Cost

5 : 60 :: 12 : x

In direct proportion, the ratio is constant: 5/60 = 12/x.

Step 3: Solve the proportion for x. Cross-multiply:

5 multiplied by x = 60 multiplied by 12

5x = 720

Step 4: Isolate x by dividing both sides by 5.

x = 720 / 5

x = 144

Result: The cost of 12 pens is 144 rupees.

Concept: Two quantities are in **inverse proportion** if an increase in one quantity causes a proportional decrease in the other quantity, and vice versa. Their product is constant.

Relationship: If quantities X and Y are in inverse proportion, their product X multiplied by Y is constant. We can write this as X is inversely proportional to Y, or X = k/Y, or XY = k, where k is the constant of proportionality.

Example: The time taken to complete a job and the number of workers doing the job. If 2 workers take 6 days to paint a room, how long will 4 workers take?

Explanation: As the number of workers increases, the time taken to complete the job decreases. The product of the number of workers and the time taken is constant (2 workers multiplied by 6 days = 12 worker-days).

Step 1: Identify the quantities and the type of proportion. The quantities are the number of workers and the time taken in days. As the number of workers decreases, the time taken increases, so this is an **inverse proportion**.

Step 2: In inverse proportion, the product of the quantities is constant. Set up the equation:

Number of Workers multiplied by Time Taken = Constant

Let the time taken by 4 workers be y days.

Using the initial information: 6 workers multiplied by 8 days = Constant

Using the new information: 4 workers multiplied by y days = Constant

So, 6 multiplied by 8 = 4 multiplied by y

Step 3: Solve the equation for y.

48 = 4y

Step 4: Isolate y by dividing both sides by 4.

y = 48 / 4

y = 12

Result: It will take 4 workers 12 days to complete the task.