Direct and Inverse Variations
ICSE Class 8 · Mathematics · Chapter 10
What this chapter covers
- Quantities that change together
- Direct variation
- Inverse variation
- Telling direct and inverse variation apart
- Word problems on variation
- Time and work problems
Key definitions
Direct variation is the relationship between two quantities in which the ratio of one quantity to the other always stays the same, so that when one increases or decreases, the other increases or decreases in the same proportion.
Inverse variation is the relationship between two quantities in which the product of the two quantities always stays the same, so that when one increases the other decreases in the same proportion.
One day's work is the fraction of the whole job that a person finishes in one day, found by dividing 1 by the number of days that person needs to finish the job alone.
Key points
- Two quantities vary directly when their ratio stays constant: (y)/(x) = k, or y = kx.
- For direct variation, (x1)/(y1) = (x2)/(y2): if one quantity doubles, the other doubles.
- Two quantities vary inversely when their product stays constant: xy = k, or y = (k)/(x).
- For inverse variation, x1 y1 = x2 y2: if one quantity doubles, the other halves.
- Both increasing, or one increasing while the other decreases, is not enough; the ratio or the product must stay fixed.
- Decide the type of variation from the situation, then use the unitary method or the ratio or product rule.
- In time and work problems the whole job is 1, and a person who needs n days does (1)/(n) of it each day.
- People working together add their one day's works; a pipe that empties a tank is subtracted.
- Joint wages are shared in the ratio of the work each person does.
Questions answered
State whether each pair varies directly, inversely, or neither: (a) the number of tickets bought and the total cost; (b) the speed of a train and the time to cover 300 km; (c) a child's age and weight.
(a) Direct. (b) Inverse. (c) Neither: both increase, but the ratio does not stay the same.
If y varies directly as x, and y = 24 when x = 6, find y when x = 9.
k = (24)/(6) = 4, so y = 4 × 9 = 36.
If y varies inversely as x, and y = 12 when x = 5, find y when x = 15.
k = 12 × 5 = 60, so y = (60)/(15) = 4.
Explain why "the more you give away, the less you keep" is not an example of inverse variation.
The sum of the two amounts stays the same, not their product. Doubling the amount given away does not halve the amount kept.
Five kilograms of mangoes cost ₹450. Find the cost of 8 kg.
1 kg costs 450 ÷ 5 = ₹90, so 8 kg cost 8 × 90 = ₹720.
Twelve pumps can empty a pond in 10 hours. How long will 8 pumps take, working at the same rate?
12 × 10 = 8 × t, so t = 120 ÷ 8 = 15 hours.
A bus covers 150 km in 3 hours. At the same speed, how far will it go in 5 hours?
Distance varies directly as time. In 1 hour it covers 50 km, so in 5 hours it covers 250 km.
A can finish a job in 20 days and B in 30 days. How long will they take together?
One day's work together = (1)/(20) + (1)/(30) = (3)/(60) + (2)/(60) = (1)/(12), so they take 12 days.