Q1.
Can we represent this problem with the following statement of proportionality — 30 : 60 :: 3 : x ? Will the travel time increase or decrease as the speed of the motorcycle increases?
Answer
No, that statement is wrong here. As the speed increases, the travel time decreases.
If we wrongly wrote 30 : 60 :: 3 : x, the rule of three would give
x = (60 × 3) ÷ 30 = 6 hours — longer than the motorcycle took
But a car at 60 km/h cannot take more time than a motorcycle at 30 km/h!
x = (60 × 3) ÷ 30 = 6 hours — longer than the motorcycle took
But a car at 60 km/h cannot take more time than a motorcycle at 30 km/h!
The correct statement keeps the product fixed:
speed × time = distance
30 × 3 = 60 × x
x = 90 ÷ 60 = 1.5 hours
30 × 3 = 60 × x
x = 90 ÷ 60 = 1.5 hours
Why it happens: Writing a : b :: c : d assumes the two quantities rise and fall together. Speed and time do not: the distance from Lucknow to Kanpur is fixed at 90 km, so every extra km/h eats into the time. Doubling the speed from 30 to 60 halves the time from 3 hours to 1.5 hours. The two quantities still change by the same factor — 2 and ½ — but in opposite directions, which is what makes this an inverse proportion.
Tip: Before setting up any proportion, ask yourself a single question: if this quantity gets bigger, does the other get bigger or smaller? Bigger together → divide (keep the quotient fixed). One bigger, one smaller → multiply (keep the product fixed).