ACCUPLACER Arithmetic common mistakes
ACCUPLACER Arithmetic · Understand the traps and how to avoid them.
When subtracting, the candidate subtracts the smaller digit from the larger digit in each column regardless of position (e.g., computes 63 − 28 as 45 by doing 8−3 and 6−2).
The borrowing algorithm was memorized as a set of steps without understanding place value, so when the top digit is smaller the candidate silently flips the subtraction to avoid regrouping.
What to do instead
Re-anchor subtraction in place value: rewrite 63 − 28 as (50 + 13) − (20 + 8), regroup explicitly, and check every answer by adding it back (45 + 28 should equal 63).
When adding or subtracting fractions with different denominators, the candidate operates on numerators and denominators separately (e.g., 1/2 + 1/3 = 2/5).
The candidate generalizes the whole-number rule that columns must line up, treating numerator and denominator as independent quantities rather than recognizing the denominator names the size of the piece.
What to do instead
Have the candidate estimate first (1/2 + 1/3 must be near 1, so 2/5 is impossible), then convert to a common denominator using the least common multiple before combining numerators only.
The candidate cross-multiplies or flips fractions when multiplying and dividing (e.g., computes 2/3 × 3/4 by inverting the second fraction, or divides 1/2 ÷ 3/4 as 3/4 ÷ 1/2).
Multiply and divide rules get blended because both involve the two fractions, and the 'invert and multiply' step is recalled without the condition that it applies only to division.
What to do instead
Attach a verbal rule to each operation: multiply means 'straight across, then simplify'; divide means 'keep the first, change to multiply, flip the second.' Have the candidate state the rule aloud before computing.
The candidate misaligns decimal points when adding or subtracting decimals of different lengths (e.g., writes 4.5 + 0.37 with the 7 under the 5).
The candidate aligns the rightmost digits out of habit from whole-number work, ignoring that the decimal point marks the boundary between ones and tenths.
What to do instead
Pad with trailing zeros to equal length (4.50 + 0.37) or explicitly line up the decimal points before writing any digits; verify with a magnitude estimate.
When multiplying decimals, the candidate places the decimal point by aligning it under the factors rather than counting decimal places in the product (e.g., 0.3 × 0.4 = 0.12 is written as 1.2 or 0.012).
The addition rule (line up the points) is over-applied to multiplication, and the candidate has no check on the size of the answer.
What to do instead
Count total decimal places in both factors and move the point that many places left in the product; confirm with a whole-number estimate (0.3 × 0.4 ≈ 1/3 of 0.4, so about 0.1).
The candidate moves the decimal point the wrong direction or the wrong number of places when dividing by a decimal (e.g., 6 ÷ 0.2 computed as 0.3 or 3 instead of 30).
The 'move the point in both numbers' procedure is applied without tracking that the divisor must become a whole number, and the direction of the shift is not tied to any meaning.
What to do instead
Rewrite as a fraction (6 ÷ 0.2 = 6/0.2), multiply top and bottom by 10 to clear the decimal, then divide; sanity-check that dividing by a number less than 1 must give an answer larger than the dividend.
The candidate computes a percent of a number by moving the decimal the wrong way (e.g., finds 15% of 80 as 1.2 or 120 instead of 12).
The candidate treats 'percent' as a symbol to manipulate rather than as 'per hundred,' so the conversion 15% = 0.15 is not anchored to meaning.
What to do instead
Have the candidate translate every percent problem into an equation with the fraction form (15/100 × 80) and estimate first (10% of 80 is 8, so 15% is a bit more than 8).
The candidate confuses 'what percent of' with 'percent of' and solves for the wrong unknown (e.g., asked '12 is what percent of 48?' the candidate answers 4 or 576).
The part-whole-base relationship is not labeled, so the candidate plugs numbers into a remembered formula without identifying which quantity is the whole.
What to do instead
Force the setup 'part = percent × whole' and write which given number is which before solving; test the answer by checking that the percent is less than 100 when the part is smaller than the whole.
The candidate sets up a proportion with the ratios in inconsistent order (e.g., miles over hours on one side and hours over miles on the other), producing an inverted answer.
The candidate focuses on the numbers and cross-multiplication mechanics while ignoring that the units must correspond across the two ratios.
What to do instead
Require labeling of units in every proportion and check that matching units sit in the same position on both sides before cross-multiplying; verify by reasoning about whether the answer should be larger or smaller than the given value.
The candidate mishandles remainders in whole-number division, reporting the remainder as part of the quotient or dropping it when the context requires rounding up (e.g., 47 ÷ 5 answered as 9.4 when 9 full boxes and 2 left over is needed).
The candidate treats division as a pure calculation and does not re-read the problem to see whether the question asks for a quotient, a remainder, or a number of full groups.
What to do instead
After computing, have the candidate restate the question ('how many full boxes?' vs. 'how many left over?') and check with multiplication: 9 × 5 = 45, remainder 2, and 45 + 2 = 47.