KBAT Problems
KBAT Problem: Setting Up a Variation
A KBAT-style problem where the hard part is translating a worded real situation into a correct variation relationship before any calculation.
Translate the words
A quantity depends on two others, directly on one, inversely on another. The KBAT challenge is writing the correct joint-variation equation from the sentence.
Once the relationship is right, finding the constant and answering is routine.
Understand the problem
The time t to fill a tank varies directly with the tank's volume V and inversely with the pump's rate r. When V = 200 litres and r = 25 litres per minute, t = 8 minutes.
Find t when V = 350 litres and r = 20 litres per minute. What it really asks: build the correct joint-variation equation first, the calculation only works if the relationship is right.
Plan and solve
- Directly with V and inversely with r means t = kV/r, where k is the constant of variation.
- Substitute the known set: 8 = k × 200 / 25 = k × 8, so k = 1.
- The model is therefore t = V/r.
- Substitute the new values: t = 350 / 20 = 17.5 minutes.
Check and a variant
Check the sense: a bigger tank with a slower pump should take longer, and 17.5 > 8 confirms it; the units litres ÷ (litres per minute) also cancel to minutes. A variant the examiner could add: instead of finding t, ask what pump rate fills 350 litres in exactly 10 minutes.
Rearranging t = V/r gives r = V/t = 350/10 = 35 litres per minute, the same model, solved for a different unknown.
Frequently asked questions
How do I tell direct variation from inverse variation in the words?
Ask what happens to one quantity when the other grows. If both rise together, it is direct, so that variable goes on top.
If one rises while the other falls, it is inverse, so that variable goes on the bottom. Test each pairing with a quick sentence before you write the equation.
Do I always have to find the constant k?
Yes, whenever you are given one complete set of values. Finding k turns the proportional relationship into a usable equation you can substitute into.
Skipping it leaves you with a shape but no numbers. Here k came out as exactly 1, but do not assume that; always substitute and check.
Why is setting up the relationship harder than the calculation?
The calculation is a fixed routine once the equation exists, but building the equation needs you to read the situation and choose direct or inverse correctly for each variable. That judgement is where marks are won or lost in KBAT.
Slow down on the sentence, then speed up on the arithmetic.