KBAT Problems
KBAT Problem: A Number-Base Puzzle
A KBAT problem where a real counting situation is set in an unfamiliar base, testing whether you understand place value, not just conversion steps.
Read the place values
A device stores counts in base five, and you are told a reading. The KBAT step is realising each column is worth five times the one to its right, so 24 in base five is 2×5 + 4 = 14 in base ten.
Convert and check
Convert to base ten to reason about the real quantity, answer the question, then convert back if the answer must stay in base five. Checking by converting the final answer the other way catches slips.
Understand the problem
A shelf counter stores stock in base five. It shows 143₅ jars in store, and 24₅ jars are sold.
How many jars remain, given in base five? What it really asks: you must reason about a real subtraction while the numbers are written in an unfamiliar base, so place value, not a memorised conversion trick, is the key.
Plan and solve
- Read the place values in base five: columns are worth 25, 5 and 1 from left to right.
- Convert to base ten: 143₅ = 1×25 + 4×5 + 3 = 48, and 24₅ = 2×5 + 4 = 14.
- Subtract in base ten: 48 − 14 = 34 jars remain.
- Convert 34 back to base five: 34 = 1×25 + 1×5 + 4, so the answer is 114₅ jars.
Check and a variant
Check by converting the answer the other way: 114₅ = 1×25 + 1×5 + 4 = 34, and adding back the 14 sold gives 48 = 143₅, so it is consistent. A variant the examiner could add: subtract directly in base five without converting.
Take 143₅ − 24₅; the units 3 − 4 need a borrow worth five, giving 8 − 4 = 4, then 3 − 2 = 1 and 1 stays, so 114₅, the same answer, testing column borrowing in an unusual base.
Frequently asked questions
Should I convert to base ten first, or work in the given base?
Either is accepted if the working is clear. Converting to base ten is safer for most students because the arithmetic feels familiar; working directly in the base is faster once you trust the borrowing.
Whichever you pick, show the place-value reasoning so the marker can follow your method to the answer.
Why does a base-five puzzle count as higher-order thinking?
Because the real challenge is understanding place value, not repeating a conversion recipe. Setting the problem in an unfamiliar base stops you running on autopilot and forces you to reason about what each column is worth.
That transfer of a familiar idea to a new setting is exactly what KBAT questions test.
Is the little subscript five important, or just decoration?
It is essential. The subscript tells you which base a number is written in, so 24 in base five and 24 in base ten are different values.
Always write the base as a subscript on every number, including your final answer, or the marker cannot tell what you mean and marks are lost.