MB Applied Math 40S common mistakes
MB Applied Math 40S · Understand the traps and how to avoid them.
When computing a loan or investment payment, entering the annual nominal rate directly into the formula instead of dividing it by the compounding frequency, so a 6% rate compounded monthly is used as 0.06 per period rather than 0.005.
The formula sheet shows i as the rate per compounding period, but students read 'annual rate of 6%' and substitute it unchanged. Textbooks often write the formula with an 'i' that looks like the quoted rate, and students have not internalized that i = annual rate / compounds per year.
What to do instead
Before touching the formula, write a small table with columns for annual rate, compounding frequency, rate per period, and number of periods, and fill it in first. Then substitute only those derived values. Practise by converting ten quoted rates (e.g., 4.5% compounded quarterly, 3.2% compounded semi-annually) into per-period rates until the division is automatic.
Mixing up the number of compounding periods n with the number of years t, for example using n = 5 instead of n = 60 for a five-year loan compounded monthly.
Students focus on the interest rate arithmetic and treat n as 'how many years' because that is the number given in the problem statement. The exponent is the least visible part of the formula, so the error goes unnoticed until the answer is wildly off.
What to do instead
Adopt a habit of writing 'n = years x compounds per year' as an explicit line of working before substitution, and label the exponent in the formula with that product. Sanity-check the final answer against an estimate: a five-year loan at 6% compounded monthly should have a total interest roughly a sixth of the principal, not triple it.
Choosing the wrong formula for a present-value versus future-value situation, such as using the compound-interest accumulation formula to find how much must be deposited today to reach a goal.
Both formulas share the same symbols, and students pattern-match on the word 'interest' rather than on whether the unknown is a starting amount or an ending amount. The distinction between 'what will this grow to' and 'what is needed now' is not tied to a decision rule.
What to do instead
Before solving, write one sentence naming the unknown: 'I am looking for the amount today, so this is a present value.' Then select the rearranged formula deliberately. Build a two-column reference sheet headed 'growing forward' and 'discounting back', each with a worked example, and cover the formula sheet while doing the first three practice problems.
Reporting a probability as a raw ratio of counts without checking that the denominator is the correct sample space, particularly in 'at least one' and conditional problems where the denominator should be restricted to a subset.
Students count favourable outcomes correctly but default to dividing by the total population, because early probability work always used the full set. Conditional probability requires redefining the sample space, and that conceptual step is easy to skip under time pressure.
What to do instead
For every probability question, write the denominator in words before calculating, e.g., 'out of the students who take French'. If the problem contains 'given that' or 'among those', draw a two-way table and shade only the relevant row or column. For 'at least one' problems, compute the complement and subtract from 1.
Treating mutually exclusive events and independent events as the same condition, so the addition rule and multiplication rule are applied interchangeably.
Both terms describe a relationship between two events, and both have short rules that look similar. Students memorize 'add for or, multiply for and' without connecting the rules to whether events can co-occur or whether one changes the other's probability.
What to do instead
Write the two definitions side by side with a concrete example each: drawing one card that is both red and black is impossible (mutually exclusive, so P(A and B) = 0), while rain today and rain tomorrow can co-occur and may be independent. Before computing, answer two yes/no questions in writing: can both happen at once, and does one change the other's probability?
Describing the spread of a data set using the mean and standard deviation when the distribution is strongly skewed or contains outliers, instead of the median and interquartile range.
Students learn mean and standard deviation first and treat them as the default summary. The rule that skewed data call for resistant measures is stated once in the text and rarely practised, so it is not triggered by the appearance of the data.
What to do instead
Make a habit of sketching the data or noting its shape before choosing a summary. Learn the trigger phrase: if the mean and median differ noticeably or a value sits far from the bulk, report median and IQR and say why. Practise with three data sets, one symmetric and two skewed, and write a one-sentence justification for the chosen measures each time.
Reading a transformation such as y = f(x - 3) + 2 as a shift three units left and two units up, reversing the horizontal direction.
The horizontal shift is counterintuitive because the sign inside the bracket is opposite to the direction of movement, while the vertical shift follows the sign outside. Students apply one consistent rule to both and get the horizontal direction backwards.
What to do instead
Test with a single point rather than reasoning abstractly: if f(0) = 5, then for y = f(x - 3) the input 3 now produces 5, so the point moved right. Do this point-mapping check on every transformation question, and remember the summary 'inside the bracket is backwards, outside is as written'.
Applying a vertical stretch to a function without accounting for the order of transformations, so y = 2f(x) + 3 is graphed by shifting up 3 and then doubling, giving y = 2f(x) + 6.
Students list transformations and apply them in reading order left to right. The standard form a·f(b(x - h)) + k encodes a specific order, and the interaction between the stretch and the vertical translation is not obvious without testing points.
What to do instead
Map three key points through the transformation algebraically instead of transforming the graph step by step. Take a known point (x, y), compute the new coordinates using the full expression, and plot those. Verify the vertical intercept and one other point against the equation before drawing the curve.
Solving a trigonometric equation over a stated interval but reporting only the principal solution from the inverse function, missing the second or further solutions in the given domain.
Calculators return a single value, and students accept it as the answer. The periodic nature of sine and cosine and the symmetry of reference angles require deliberate extension, which is a step students have not automated.
What to do instead
After obtaining the reference angle, always sketch the unit circle and mark which quadrants satisfy the sign condition, then generate every solution in the interval and list them. Finish by counting solutions against the expected number for the period and interval, and substitute each back into the original equation to confirm.
Using the sine law to find an angle in a triangle where the given information is side-side-angle, and reporting the acute angle without considering the ambiguous obtuse possibility.
The sine law computation yields one value, and the inverse sine function returns only the acute angle. Students do not check whether a second valid triangle exists, because the ambiguous case is a single topic in the course rather than a routine check.
What to do instead
Whenever the sine law is used to find an angle, immediately compute 180 degrees minus that angle and test whether the supplementary angle plus the known angle is still less than 180 degrees. If it is, both triangles are valid and both must be reported. Practise with a side-side-angle set where both cases exist until the check is reflexive.
Confusing surface area and volume formulas for composite three-dimensional objects, for example using the sphere surface area formula where the volume formula is required, or forgetting to omit the shared face when two solids are joined.
The formulas for spheres, cones, and cylinders are numerous and similar in appearance, and composite-object problems require deciding which faces are exposed. Students select a formula by matching a keyword rather than by reasoning about what is being measured.
What to do instead
Before calculating, state in words what the question asks for: 'area of the outside surface' or 'space enclosed'. For composite objects, sketch the assembled solid and shade the exposed surfaces, explicitly crossing out internal faces. Keep a formula card organized by solid and by whether the formula gives a squared or cubed result, and check units at the end.
Carrying out a measurement or scale conversion without tracking units, so a length given in metres is substituted into a formula expecting centimetres, or an area conversion is done with a single factor of 100 instead of 10 000.
Unit labels are dropped early in the working because they feel like clutter, and area and volume conversions involve squared and cubed factors that students treat as linear. The arithmetic then proceeds correctly on inconsistent numbers.
What to do instead
Write units beside every substituted number and cancel them as you go. When converting areas or volumes, write the linear conversion first and then square or cube the whole factor, e.g., 1 m = 100 cm so 1 m² = 10 000 cm². Convert all given quantities to a single unit before starting any calculation.