ACT Math study guide · Chapter 47
The Error Ledger
The archetype index in the previous chapter is organized by what a question looks like. This ledger is organized the other way, by the specific mistake an answer choice is built to catch. The same small number of mistakes reappears across many different topics, and naming the mistake in advance makes it far easier to catch in the moment.
Stopping at an intermediate value. Solving 3x+5=20 for x gives x=5. If the question asks for 2x+1 instead of x itself, reporting 5 is the answer to a question that was not asked; the actual answer is 2(5)+1=11. This mistake appears whenever a problem is solved correctly but one step early.
Percent symmetry. A price that increases 20% and then decreases 20% does not return to its original value. Starting at 100: 100×1.20=120, then 120×0.80=96, a net decrease of 4%, because the second change applies to the new value, not the original one.
The wrong conditioned denominator. Of 50 people, 30 own a dog, and of those dog owners, 12 also own a cat. The probability that a dog owner also owns a cat is 12/30=40%, using only the dog owners as the denominator. Dividing by the full 50 instead gives 24%, which answers a different question than the one asked.
A function's inside sign, reversed. If f(x)=(x−3)², then f(5)=(5−3)²=4. Shifting in the wrong direction and computing (5+3)²=64 instead is one of the most common transformation errors, since the sign inside a function's argument is easy to reverse without noticing.
Matrix multiplication done entry by entry. Multiplying [[1,2],[3,4]] by [[5,6],[7,8]] gives a top-left entry of 1(5)+2(7)=19, a row times a column. Multiplying entry by entry instead, as if the matrices were ordinary numbers, gives 1(5)=5, which is not how matrix multiplication is defined.
Trusting a diagram's scale. A triangle drawn to look isosceles, with only two leg lengths of 6 and 8 actually stated, is not automatically isosceles; without a right-angle mark or a stated third length, nothing supports assuming the third side matches one of the given legs. If a right angle is given between the two legs, the third side is a hypotenuse of 10, found from the two stated lengths, not from how the figure looks.
Combining quantities in different units. A rectangle measuring 2 feet by 18 inches has an area of 3 square feet, found by converting 2 feet to 24 inches first: 24×18=432 square inches, equivalent to 3 square feet. Multiplying 2×18=36 directly, without converting either quantity, mixes two different units into a meaningless result.
Applying right-triangle trigonometry to a triangle with no right angle. A triangle with sides 7 and 10 and an included angle of 50° has a third side of about 7.68, found with the Law of Cosines. SOHCAHTOA only relates the sides and angles of a right triangle, so it cannot be applied here no matter how the triangle is labeled.
Solving further than the question requires. A full symbolic solution is not always the fastest route when four answer choices are given. Full treatment of when testing a choice or plugging in a value is faster than solving directly is in Backsolving, Plugging In, and Estimating.
Assuming every adjacent question is harder than the one before it. ACT's current design framework specifies an overall increase in difficulty across the Math section, but that is a form-level pattern, not a guarantee that each question is harder than the last one. Full treatment of pacing without relying on strict item-by-item escalation is in Pacing the ACT Math Section.