CBSE Class 8 Mathematics Direct and Inverse Proportions Notes Set 06

Direct Variation

Two quantities x and y vary directly when they rise and fall together in the same ratio, so that \( \frac{x}{y}=k \) stays constant. For two readings, \( \frac{x_1}{x_2}=\frac{y_1}{y_2} \). Example: if 15 pens cost Rs. 600, 20 pens cost 20 × 40 = Rs. 800, because each pen costs Rs. 40.

Inverse Variation

Two quantities vary inversely when one grows as the other shrinks so that their product stays fixed: \( xy=k \). For two readings, \( x_1y_1=x_2y_2 \). Example: food for 80 boys lasts 30 days; for 100 boys it lasts 80 × 30 ÷ 100 = 24 days.

Time and Work

Treat the whole job as 1. If a person finishes a job in n days, in one day he does \( \frac{1}{n} \) of it. For people working together add their one-day work and take the reciprocal of the sum.
Example: A takes 5 days and B takes 8 days. One day together: \( \frac{1}{5}+\frac{1}{8}=\frac{13}{40} \), so the job takes \( \frac{40}{13} \) days.
If more men do the same work the time falls (inverse); if the number of men is fixed, more days means more work (direct). For taps and pipes the same method is used, and an emptying pipe has a negative rate.

Speed, Distance and Time

Speed = distance ÷ time, distance = speed × time and time = distance ÷ speed. For a fixed distance, speed and time vary inversely; for a fixed speed, distance and time vary directly.
To change km/h to m/s multiply by \( \frac{5}{18} \); to change m/s to km/h multiply by \( \frac{18}{5} \).

Problems on Trains

A train crossing a pole or a person standing still covers its own length. A train crossing a bridge or platform covers its length plus the length of the bridge.
Two trains crossing each other cover the sum of their lengths. The relative speed is the sum of the speeds when they move in opposite directions and the difference when they move in the same direction.

Worked Example

A 320 m train runs at 72 km/h. Time to cross a pole?
72 km/h = 20 m/s, so time = 320 ÷ 20 = 16 seconds.

Worked Example

A contractor needs 12 workers for 20 days. After 16 days only two-thirds of the work is done. How many more workers are needed?
Left: one-third of the work in 4 days. The full job needs 12 × 16 × (3/2) = 288 worker-days, so one-third is 96 worker-days; in 4 days that needs 24 workers. Extra workers = 24 − 12 = 12.