The product stays constant
In an inverse proportion, x × y is always the same number.
Speed times time
From Lucknow to Kanpur, walking at 5 km/h takes 18 hours. A car at 60 km/h takes 1.5 hours.
Compare speed × time for the car (60 × 1.5) with walking (5 × 18). What do you expect?
What this lesson covers
The idea
Quantities x and y vary in inverse proportion if xy = k for a constant k, so corresponding values satisfy x1y1 = x2y2 = k and x1/x2 = y2/y1.
Speed times time
From Lucknow to Kanpur, walking at 5 km/h takes 18 hours. A car at 60 km/h takes 1.5 hours.
Compare speed × time for the car (60 × 1.5) with walking (5 × 18). What do you expect?
- More than walking
- The same as walking
- Less than walking
Same area, different shapes
A rectangle is x wide (the speed) and y high (the time). Its area x × y is the distance, 90 km. Pick a faster speed and change the time until the area is 90 again.
x × y = k
Quantities x and y vary in inverse proportion if xy = k for a constant k, so corresponding values satisfy x₁y₁ = x₂y₂ = k and x₁/x₂ = y₂/y₁.
Speed and time for the 90 km trip: 5 × 18 = 90, 15 × 6 = 90, 60 × 1.5 = 90. The product is always the same: k = 90 km, the distance.
Take x₁ = 5, x₂ = 60 and y₁ = 18, y₂ = 1.5: 5 ÷ 60 = 0.0833… 1.5 ÷ 18 = 0.0833… So x₁/x₂ = y₂/y₁. The fractions are equal.
This gives a quick way to solve problems. Two pumps fill a tank in 18 hours. With 4 pumps: 2 × 18 = 4 × x x = 2 × 18 ÷ 4 = 9 hours.
Notes
Quantities x and y vary in inverse proportion if xy = k for a constant k, so corresponding values satisfy x₁y₁ = x₂y₂ = k and x₁/x₂ = y₂/y₁.
Check yourself
6 pipes fill a tank in 10 hours. How many hours will 12 pipes (of the same kind) take?
Answer: 5 hours
6 × 10 = 12 × x, so x = 60 ÷ 12 = 5 hours. Twice the pipes, half the time.
A school has food for 80 students for 15 days. If 100 students are there, for how many days will the food last?
Answer: 12 days
80 × 15 = 1200, and 1200 ÷ 100 = 12 days. More students, fewer days.
Two quantities x and y are in inverse proportion. x₁ = 4, y₁ = 15 and x₂ = 6. Which equation gives y₂?
40 men can finish a job in 12 days. How many men are needed to finish it in 8 days?
Answer: 60 men
40 × 12 = 480 and x × 8 = 480, so x = 480 ÷ 8 = 60 men. Fewer days, more men.
- 4 × y₂ = 6 × 15. That is the cross-multiplication of a direct proportion, 4 : 6 :: 15 : y₂. In an inverse proportion x₁ × y₁ = x₂ × y₂.
- 4 ÷ 15 = 6 ÷ y₂. That keeps the quotient x ÷ y the same. In an inverse proportion the product x × y stays the same.
- 4 × 15 = 6 × y₂ — correct. Yes! x₁ × y₁ = x₂ × y₂, so 4 × 15 = 6 × y₂ and y₂ = 60 ÷ 6 = 10.
- 4 + 15 = 6 + y₂. Adding does not keep the proportion. The products are equal: x₁ × y₁ = x₂ × y₂.