LCA Mathematical Applications common mistakes
LCA Mathematical Applications · Understand the traps and how to avoid them.
When converting between units in a multi-step word problem (for example, scaling a recipe or a construction estimate), the candidate multiplies where division is required, or applies the conversion factor in the wrong direction, then reports the reciprocal or an off-by-a-factor-of-10 answer.
Under time pressure candidates pattern-match on the surface numbers instead of tracking the dimensional relationship; they treat 'per' and 'for each' as multiplication cues without checking whether the result should grow or shrink. Mixed metric prefixes and imperial units compound this because the factor is not always a clean power of ten.
What to do instead
Before computing, write the conversion as a fraction with units shown and cancel units explicitly (e.g., 3.5 m x 100 cm / 1 m). Do a sanity check on direction: if the target unit is smaller, the number must get larger. Practice 10 mixed-direction conversion chains and verify each with an estimate.
The candidate computes a percentage change by dividing the difference by the new value instead of the original value, or adds/subtracts percentage points as if they were percentages (e.g., treating a rise from 20% to 25% as a 5% increase rather than a 25% increase).
The base of a percentage is implicit in the wording, and candidates anchor on whichever number appears last or largest. Percentage points versus percent is a vocabulary distinction that is easy to skip over when reading quickly.
What to do instead
Always write the formula as (new - old) / old and label the base before substituting. When a problem involves two percentages, underline the base each one refers to. Drill a set of problems that deliberately mix percentage-point and percent-change language and state the base aloud for each.
When simplifying or solving an expression with exponents and radicals, the candidate distributes an exponent across a sum (writing (a+b)^2 = a^2 + b^2) or adds exponents when multiplying terms with different bases.
The distributive property is over-learned from linear expressions and gets misapplied to nonlinear operations. Exponent rules are memorized as strings without the underlying repeated-multiplication meaning, so similar-looking rules blur together.
What to do instead
Expand small cases by hand to test any exponent rule before using it: write (a+b)^2 as (a+b)(a+b) and multiply out. Keep a one-page rule sheet that pairs each exponent law with a numeric example, and re-derive rather than recall when unsure.
The candidate solves a linear or quadratic equation correctly but then answers the wrong question, for example giving the x-coordinate of a vertex when the maximum value was requested, or reporting the break-even quantity when the profit amount was asked.
Candidates treat 'solve the equation' as the goal and stop at the first clean number, skipping the final read-back. Word problems with business or modeling contexts bury the actual target in the last sentence.
What to do instead
Before solving, restate the question in your own words as 'Find ___'. After computing, substitute the answer back into the original context and check units and plausibility. Circle the requested quantity in the prompt and write the final answer with its unit.
In composite figures, the candidate adds areas of overlapping regions twice or uses the slant height of a cone or pyramid in place of the perpendicular height when computing volume.
Diagrams invite adding every visible piece, and overlapping regions are visually merged. Slant height and height are both labeled on many figures and the formulas look interchangeable at a glance.
What to do instead
Decompose the figure into non-overlapping pieces and record each piece's area separately before summing; subtract any region counted twice. For cones and pyramids, write the volume formula with h explicitly and compute h from the slant height using the Pythagorean theorem when only the slant is given.
The candidate applies a scale factor to area or volume as if it were linear, for example tripling the side of a square and expecting the area to triple, or doubling the radius of a sphere and expecting the volume to double.
Scale factors are introduced with lengths, and candidates carry the linear intuition into two and three dimensions without tracking how many dimensions the measured quantity has.
What to do instead
Attach a dimension tag to every quantity (length, area, volume) and apply the scale factor raised to that power. Verify with a concrete example: a 1x1 square scaled by 3 becomes 3x3, so area grows by 9. Memorize the pattern k, k^2, k^3 for length, area, volume.
The candidate reads a two-way table or conditional-probability problem and divides by the grand total when the question specifies a subpopulation ('given that the person is female...'), producing the joint probability instead of the conditional one.
The grand total is the most visually prominent number in the table, and the phrase 'given that' is a small linguistic cue that is easy to miss. Candidates conflate P(A and B) with P(A|B).
What to do instead
When you see 'given', 'of those', or 'among', immediately restrict the denominator to that row or column total and cross out the grand total. Write the conditional formula P(A|B) = P(A and B) / P(B) and identify which event is the condition before computing.
The candidate computes a mean, median, or standard deviation from grouped or summarized data by averaging the category values without weighting by frequency, or by treating class midpoints as if every observation fell exactly there.
Summary tables hide the raw data, and candidates default to a simple average of the listed numbers. The distinction between the midpoint approximation and the true mean is rarely made explicit in practice.
What to do instead
For grouped data, build a frequency-weighted sum (value x frequency) and divide by the total frequency; state that the result is an estimate based on midpoints. For median and quartiles, locate the position using cumulative frequencies rather than averaging the category labels.
The candidate sets up a modeling problem with the variables swapped, for example writing cost as a function of price when price is the decision variable, or misassigning the slope and intercept in a linear cost model (putting fixed cost as the slope).
Word problems describe relationships in prose, and candidates grab the first two numbers they see without deciding which is independent and which is dependent. Fixed versus variable cost is a business concept that maps poorly onto memorized y = mx + b.
What to do instead
Before writing an equation, define each variable in a sentence ('let x = number of units, let C = total cost') and identify the fixed quantity as the intercept and the per-unit rate as the slope. Test the model at x = 0 to confirm the intercept equals the fixed cost.
The candidate accepts a mathematically correct solution that violates a real-world constraint in the modeling context, such as a negative number of items, a fractional person, or a time beyond the stated range, and reports it as the answer.
Algebraic solving produces extraneous or out-of-domain roots, and candidates stop at the algebra without re-reading the constraints. Modeling problems deliberately include domain restrictions that pure equation-solving ignores.
What to do instead
After solving, list the natural domain (nonnegative, integer, within the stated interval) and test each candidate solution against it, discarding those that fail. State the constraint explicitly in the final answer, e.g., 'x = -3 is rejected because quantity cannot be negative.'