KBAT Problems
KBAT Problem: When One Question Hides Two Topics
A KBAT problem that deliberately combines two chapters, the difficulty is recognising both are needed and switching between them.
Spot both topics
A single question might need trigonometry to find a length and then area to finish, or statistics to summarise before a probability. The KBAT skill is noticing the handover point and not getting stuck expecting one topic only.
Understand the problem
An original two-topic problem. A rectangular field measures 40 m by 30 m, and a straight path is cut along its diagonal.
The question asks how much shorter the diagonal is than walking along two sides, as a percentage. It quietly needs two chapters: Pythagoras' theorem to get the diagonal, then percentage change to compare, recognising both are required is the real difficulty.
Plan and solve
- The diagonal is the hypotenuse of a right triangle: d = √(40² + 30²) = √(1600 + 900) = √2500 = 50 m.
- Walking two sides = 40 + 30 = 70 m.
- Distance saved = 70 − 50 = 20 m.
- Percentage shorter = (20 ÷ 70) × 100% = 28.57% ≈ 28.6%.
Check and a variant
Check the triangle: 50² = 2500 = 40² + 30², so the diagonal is exact, and 20 out of 70 is under a third, matching 28.6%. A twist: the path costs RM15 per metre, then the diagonal costs 50 × 15 = RM750, pulling in a third idea.
The common error is dividing the saving by 50 instead of 70; the percentage must be measured against the original longer route.
Frequently asked questions
How do I spot that a question hides two topics?
Look for a value you cannot reach in one step. Here you are asked for a percentage, but the lengths you need aren't all given, the diagonal must be found first with Pythagoras.
Whenever the quantity asked for depends on something you must compute separately, expect a second topic to be doing that job.
Which number is the base for the percentage?
The original quantity you are comparing against, here the 70 m two-side route, because you are asking how much shorter the diagonal is than that. Dividing by 50 would answer a different question.
Percentage change is always (difference ÷ original) × 100%, so identifying the 'original' correctly is essential.
How should I lay out a multi-topic answer?
Finish one topic before starting the next and label each stage. Compute the diagonal with Pythagoras, write it down, then move to the percentage using that value.
Clear stages let a marker award the Pythagoras marks and the percentage marks separately, and they stop you carrying an unfinished piece into the wrong step.