BCS Written Mathematics common mistakes
BCS Written Mathematics · Understand the traps and how to avoid them.
When a problem involves a percentage increase followed by a percentage decrease (or successive changes), the candidate adds or subtracts the percentages and applies the net figure once to the original amount, e.g. treating +20% then −20% as no change.
Successive percentage changes are mentally compressed into a single operation because the candidate has only practised one-step percentage problems; the second percentage is silently applied to the original base instead of the new, changed base.
What to do instead
Always write the multiplier chain explicitly: a 20% rise is ×1.20, a 20% fall is ×0.80, so the combined factor is 0.96, a 4% net fall. Before computing, state in one line what the base of each percentage is. Drill five mixed increase/decrease chains and verify each by computing the actual rupee or taka amounts at every stage.
In ratio, proportion and work-rate problems, the candidate inverts a ratio when setting up the equation (e.g. writes time ratio as 3:5 when the speed ratio is 3:5), or applies a unitary-method step to the wrong quantity, producing an answer that is plausible but wrong by a reciprocal factor.
Ratios are manipulated as bare number pairs without attaching units or labels, so the candidate loses track of which quantity is in the numerator; inverse proportionality (more workers, less time) is confused with direct proportionality.
What to do instead
Annotate every ratio term with its unit and quantity name before forming any equation, and write 'direct' or 'inverse' above each pair. For work-rate problems, convert everything to a rate per unit time first (work per day), then add rates rather than times. Check the final answer for reasonableness: doubling the workers must halve the time.
When solving a linear or quadratic equation, the candidate divides both sides by an expression containing the unknown (e.g. dividing by x or by (x−2)) and thereby loses one or more roots, or drops a root by taking only the positive square root.
Dividing by a variable expression feels like legitimate simplification and is faster than factorising; the candidate has not internalised that division by a quantity that may be zero is invalid and destroys solutions.
What to do instead
Adopt the rule: never divide by an expression containing the unknown - factorise and use the zero-product property instead. For x² = k, always write x = ±√k. After solving, count the roots against the degree of the equation and state both roots explicitly, even when one is rejected on contextual grounds.
In inequalities and modulus problems, the candidate multiplies or divides an inequality by a negative quantity without reversing the sign, or solves |x − a| < b as a single inequality instead of a double inequality.
Inequalities are treated with the same mechanical steps as equations; sign reversal is a rule with no counterpart in equation-solving, so it is omitted under time pressure, and the modulus is expanded only in its positive branch.
What to do instead
Before dividing, write the sign of the divisor in the margin and, if negative, immediately flip the inequality arrow. For modulus, memorise the two templates: |X| < b gives −b < X < b, and |X| > b gives X < −b or X > b. Verify each final interval by testing one interior and one exterior numerical value in the original inequality.
In geometry, the candidate writes the correct theorem in words but assigns the wrong correspondence when naming congruent or similar triangles (e.g. writes ΔABC ≅ ΔDEF when the actual correspondence is ΔABC ≅ ΔEDF), leading to wrong equal sides and a failed proof.
The candidate treats the vertex order in the congruence statement as a formality rather than as the encoding of which vertices, angles and sides correspond; diagrams are read by visual position rather than by marked equal parts.
What to do instead
Before writing any congruence or similarity statement, list the three matched pairs (A↔E, B↔D, C↔F) from the given equal angles and sides, then write the statement in that exact order. In similarity problems, immediately write the side proportion using the matched order and check that each ratio pairs corresponding sides.
When computing areas and volumes, the candidate mixes units (radius in centimetres with height in metres) or uses the diameter in place of the radius in formulas such as πr², (4/3)πr³ and (1/3)πr²h.
Formulas are recalled as symbol strings without the definition of each symbol, and unit conversion is skipped because all numbers 'look metric'; the factor-of-four error from using diameter is not caught by a plausibility check.
What to do instead
At the start of every mensuration problem, write r = ... and h = ... in a single common unit before substituting. Keep a one-line formula card that defines each symbol. Sanity-check the result: a sphere of radius 3 cm has volume about 113 cm³, so an answer of 450 cm³ signals a diameter-for-radius slip.
In trigonometric problems the candidate applies the sine rule or cosine rule without checking whether the given triangle data correspond to the ambiguous (two-solution) case, and reports only the acute angle when an obtuse angle is also valid.
The sine rule is memorised as a single formula a/sin A = b/sin B with no attached condition; the candidate does not connect sin θ = sin(180° − θ) to the geometry, and textbooks' single-answer examples reinforce the habit.
What to do instead
When using the sine rule to find an angle, always compute both candidates θ and 180° − θ, then test each against the angle-sum constraint (the two known angles plus the candidate must be under 180°). Prefer the cosine rule when three sides are known, since it returns a unique angle. State explicitly which candidate is rejected and why.
In differentiation and integration, the candidate omits the chain rule factor (writes d/dx sin 3x = cos 3x) or, in integration by substitution, forgets to convert dx into du and to change the limits for a definite integral.
The outer-function derivative is applied and the inner derivative is skipped because simple examples with coefficient 1 dominate practice; in substitution, dx is carried along mechanically as a symbol rather than as part of the differential.
What to do instead
Force the habit of naming the inner function u and writing du/dx explicitly before differentiating or substituting. For definite integrals, write the new limits in u above the integral sign before integrating. Differentiate the result back (or differentiate the antiderivative) as a routine final check.
In probability, the candidate adds probabilities for events that are not mutually exclusive without subtracting the intersection, or multiplies P(A)P(B) for events that are not independent (e.g. drawing two cards without replacement).
The addition and multiplication rules are recalled in their simplest forms, and the qualifiers 'mutually exclusive' and 'independent' are not checked against the described experiment; sampling without replacement is not recognised as changing the sample space.
What to do instead
Before computing, write one line classifying the events: mutually exclusive or not, independent or not. Use P(A∪B) = P(A) + P(B) − P(A∩B) whenever overlap is possible, and for sampling without replacement adjust the denominator and numerator after each draw. Verify that the total probability over all outcomes sums to 1.
In matrix work, the candidate computes a determinant or inverse without first checking that the matrix is square and non-singular, or multiplies matrices in the wrong order (writing AB when the problem requires BA), producing an answer that fails verification.
Matrix multiplication is treated as commutative by analogy with ordinary arithmetic, and the existence conditions for a determinant and inverse are not checked because the candidate works procedurally from the formula.
What to do instead
Before any determinant or inverse, state the order of the matrix and confirm it is square; compute the determinant first and, if it is zero, declare the inverse undefined. For products, write the orders side by side (e.g. 2×3 times 3×2) to confirm compatibility, and remember AB ≠ BA in general. Verify an inverse by checking that A A⁻¹ gives the identity.