Grade 8 Variation

Concept: Two quantities, x and y, are in **direct variation** if their ratio is constant (x/y = k). This means as one increases, the other increases proportionally.

Equation: x = ky, where k is the constant of variation.

Problem: If the cost of apples (C) varies directly with their weight (W), and 5 kg of apples cost ₹250, find the cost of 8 kg of apples.

Step 1: Write the variation equation: C = kW.

Step 2: Use the given values (C = 250, W = 5) to find k.

250 = k * 5

k = 250 / 5 = 50

The constant of variation is 50.

Step 3: The equation is C = 50W.

Step 4: Find C when W = 8.

C = 50 * 8

C = 400

Result: The cost of 8 kg of apples is ₹400.

Concept: Two quantities, x and y, are in **inverse variation** if their product is constant (x * y = k). This means as one increases, the other decreases proportionally.

Equation: xy = k or x = k/y, where k is the constant of variation.

Problem: The time (T) taken to travel a fixed distance varies inversely with the speed (S). If a car travels at 60 km/h and takes 4 hours, how long will it take to travel the same distance at 80 km/h?

Step 1: Write the variation equation: T * S = k.

Step 2: Use the given values (T = 4, S = 60) to find k.

4 * 60 = k

k = 240

The constant of variation is 240.

Step 3: The equation is TS = 240.

Step 4: Find T when S = 80.

T * 80 = 240

T = 240 / 80

T = 3

Result: It will take 3 hours to travel the same distance at 80 km/h.

Step 1: Check for direct variation by calculating the ratio x/y for each pair of values.

For (2, 10): 2 / 10 = 0.2

For (4, 20): 4 / 20 = 0.2

For (6, 30): 6 / 30 = 0.2

Step 2: Since the ratio x/y is constant (0.2) for all pairs, the quantities x and y are in direct variation.

Note: If the ratio was not constant, we would check for inverse variation by calculating the product x*y for each pair. If the product was constant, it would be inverse variation.

Result: The table shows direct variation.

Step 1: Check for direct variation by calculating the ratio p/q.

For (2, 12): 2 / 12 ≈ 0.167

For (3, 8): 3 / 8 = 0.375

The ratio is not constant, so it is not direct variation.

Step 2: Check for inverse variation by calculating the product p*q for each pair of values.

For (2, 12): 2 * 12 = 24

For (3, 8): 3 * 8 = 24

For (4, 6): 4 * 6 = 24

Step 3: Since the product p*q is constant (24) for all pairs, the quantities p and q are in inverse variation.

Result: The table shows inverse variation.