4 high-yield, syllabus-aligned questions on Number & numeration, covering 10 individually creditable mark points. Each one shows the answer that scores full marks under our guide, which phrase earns which mark, and a common incomplete answer — so you can see the difference rather than guess at it.
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Every question and solution is reviewed by examiners versed in WAEC and JAMB, each with more than thirty years of experience.
Mark allocations are TopMarks’ own, written against the published syllabus and Chief Examiners’ reports. They are a guide to how these answers are usually credited, not an official marking scheme.
Convert2 marks
Convert 11011 in base two to a number in base ten.
TopMarks model answer
11011 in base two = (1 x 16) + (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) = 16 + 8 + 0 + 2 + 1 = 27. Therefore 11011 in base two is 27 in base ten.
Suggested marking guide
1Expands the number in powers of two
2Correct answer
The working, line by line
1Write the place values from the right: 1, 2, 4, 8, 16 — In base two the place values double each time. Writing them above the digits stops every place-value error.
2(1 x 16) + (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) — Include the zero term. It costs one second and it proves you did not skip a place.
3= 16 + 8 + 0 + 2 + 1 = 27 — Sum the products.
Why it scores
Half the marks here are for the expansion, so writing “27” alone throws one away. The expansion is also your own check: if you have five digits, you must have five terms, and the largest place value must be 16.
A common incomplete answer
“11011 = 27”
What it costs: One mark of two. The answer is right, but with no expansion shown there is nothing for the method mark to attach to.
Key ideas to include: base two, place value, powers of two, expansion.
Simplify3 marks
Simplify 3√12 + 2√27 - √48, leaving your answer in surd form.
1√12 = √(4 x 3) = 2√3 — Split each surd so that one factor is a perfect square. That is the whole technique.
2√27 = √(9 x 3) = 3√3 and √48 = √(16 x 3) = 4√3 — Use the LARGEST perfect square factor. Using 4 instead of 16 for 48 leaves you with 2√12, which is not fully simplified.
36√3 + 6√3 - 4√3 = 8√3 — Surds with the same root add like terms. Never add the numbers under the root sign.
Why it scores
Each simplification is a separate mark, which means the working IS the answer here. Note also what “in surd form” forbids: converting to 13.86 loses the final mark even though the arithmetic is right, because the question asked for an exact answer.
A common incomplete answer
“3√12 + 2√27 - √48 = 8√3”
What it costs: Two marks of three. The answer is correct, but neither simplification is shown, so both method marks are lost.
Key ideas to include: surd, perfect square factor, like surds, exact form.
Evaluate3 marks
Given that log 2 = 0.3010 and log 3 = 0.4771, evaluate log 72 without using tables.
TopMarks model answer
First express 72 in terms of 2 and 3: 72 = 8 x 9 = 2³ x 3². Therefore log 72 = log(2³ x 3²) = 3 log 2 + 2 log 3 = 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572.
Suggested marking guide
1Expresses 72 in terms of 2 and 3
2Applies the laws of logarithms
3Correct value
The working, line by line
172 = 8 x 9 = 2³ x 3² — The whole question turns on this line. Break the number down until only the values you were GIVEN remain.
2log 72 = 3 log 2 + 2 log 3 — log(ab) = log a + log b, and log(aⁿ) = n log a. The index comes down in front.
3= 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572 — Substitute the given values and add.
Why it scores
The phrase “without using tables” is the instruction: you must build 72 out of the two logarithms you were handed. Every question of this type is the same puzzle — factorise the number into the bases you have been given — and the factorisation line is worth a full mark on its own.
A common incomplete answer
“72 = 8 x 9, so log 72 = log 8 + log 9.”
What it costs: Two marks of three. The factorisation is credited, but the logs are never converted into multiples of log 2 and log 3, so the given values cannot be used and no answer is reached.
Key ideas to include: laws of logarithms, log(ab) = log a + log b, index rule, factorise.
Express2 marks
Express 0.00045678 (a) in standard form (b) correct to three significant figures.
TopMarks model answer
In standard form, 0.00045678 = 4.5678 x 10⁻⁴. Correct to three significant figures, 0.00045678 = 0.000457.
Suggested marking guide
1Correct standard form
2Correct to three significant figures
Why it scores
Leading zeros are NOT significant figures. The first significant figure in this number is the 4, so three significant figures runs 4, 5, 6 — and the 7 that follows rounds the 6 up to 7. Counting the zeros is what turns 0.000457 into the wrong answer 0.000456 or 0.00045.
A common incomplete answer
“0.00045678 = 4.5678 x 10⁻⁴”
What it costs: One mark of two. The standard form is correct, but part (b) was not attempted at all.
Key ideas to include: standard form, significant figures, leading zeros, rounding.
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