Study guide
The enhanced ACT math section presents 45 questions in 50 minutes — about 67 seconds each — and every question now has four answer choices instead of five. The content spans pre-algebra through basic trigonometry, with the bulk drawn from algebra and geometry fundamentals. This chapter covers the concepts that generate the most questions, plus the pacing habits that protect your score.
Pre-Algebra Workhorses: Percentages, Ratios, and Averages
Percent problems reduce to one relationship: part equals percent times whole. Percent change is always the change divided by the original value, so a price that rises from $80 to $92 changed by 12/80, or 15 percent. Successive percent changes multiply rather than add: a 20 percent increase followed by a 20 percent decrease leaves you at 1.20 times 0.80, which is 0.96 — a 4 percent net loss, not zero. Ratios describe relative size; the key move is introducing a multiplier. If a paint mix uses red and blue in a 3:5 ratio and contains 40 quarts total, write the parts as 3k and 5k, so 8k = 40, k = 5, giving 15 quarts of red. Distinguish part-to-part ratios (3:5) from part-to-whole (3:8). Averages should be handled through sums, because the sum is what the test manipulates. Average equals sum divided by count, so sum equals average times count. Suppose Priya's four quiz scores average 82: her total is 328. If she wants an 85 average across five quizzes, she needs a 425 total, so the fifth score must be 97. The classic changed-value problem works the same way: if a class of five students averages 84 (sum 420) and a recording error replaces a 70 with a 90, the sum rises by 20 to 440, and the new average is 88. Never re-average from scratch — adjust the sum.
Algebra: Linear Equations, Systems, Quadratics, and Exponents
Linear questions test whether you can isolate a variable and interpret slope-intercept form, y = mx + b, where m is the rate of change and b the starting value. A cell plan costing $25 monthly plus $0.10 per gigabyte is y = 0.10x + 25; the test loves asking what the 25 or the 0.10 represents in context. Systems of two equations yield to substitution or elimination; line up the equations and add or subtract to kill a variable. Conceptually, a system's solution is where the lines cross, so parallel lines (equal slopes, different intercepts) give no solution and identical lines give infinitely many. Quadratics center on factoring and the zero-product property: if (x - 3)(x + 5) = 0, then x is 3 or -5. Be ready to factor simple trinomials, recognize the difference of squares (x squared minus 49 equals (x-7)(x+7)), and fall back on the quadratic formula when factoring stalls. Graphically, the solutions are the x-intercepts of a parabola. Exponent rules are pure memorization: multiplying same bases adds exponents, dividing subtracts them, a power raised to a power multiplies them, a negative exponent means reciprocal, and a fractional exponent means a root — 8 to the 2/3 power is the cube root of 8, squared, which is 4. Function notation questions like f(x) = 3x - 4, find f(2a), are substitution exercises: replace every x with 2a and simplify to 6a - 4.
Coordinate Geometry: Slope, Distance, and Midpoint
Three formulas cover most coordinate questions. Slope is rise over run: the change in y divided by the change in x between two points. Between (2, 3) and (6, 11), slope is 8/4 = 2. Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals, so a line perpendicular to slope 2 has slope -1/2. Expect questions that hand you an equation in a disguised form like 3x + 4y = 12 and ask for the slope — solve for y to get y = -3/4 x + 3, so the slope is -3/4 and the y-intercept is (0, 3). The midpoint of a segment is simply the average of the endpoints' coordinates: between (2, 3) and (6, 11), the midpoint is (4, 7). The distance between two points comes from the Pythagorean theorem applied to the horizontal and vertical legs: for those same points, the legs are 4 and 8, so the distance is the square root of 16 + 64, which is the square root of 80. If you forget the distance formula, sketch the right triangle — that picture is the formula. Two related patterns appear regularly: finding where a line crosses an axis (set the other coordinate to zero) and the equation of a circle, (x - h) squared plus (y - k) squared equals r squared, with center (h, k) and radius r. A quick sketch on scratch paper prevents most sign errors in this territory.
Plane Geometry: Angles, Triangles, Circles, Area and Volume
Angle questions chain a few facts: angles on a straight line sum to 180 degrees, vertical angles are equal, and when parallel lines are cut by a transversal, alternate interior angles are equal while same-side interior angles sum to 180. Triangle facts do heavy lifting. The interior angles of any triangle sum to 180 degrees; an isosceles triangle has equal base angles opposite its equal sides. Similar triangles have equal angles and proportional sides, so if a 6-foot pole casts a 4-foot shadow while a tree casts a 30-foot shadow, the tree is 45 feet tall by the proportion 6/4 = h/30. The Pythagorean theorem, a squared plus b squared equals c squared, applies only to right triangles; memorize the common triples 3-4-5 and 5-12-13 and their multiples to skip the arithmetic. Core area formulas: triangle, one half base times height; rectangle, length times width; trapezoid, one half the sum of the parallel sides times the height. For circles with radius r, circumference is 2 pi r and area is pi r squared; an arc or sector is the fraction of the whole given by its central angle over 360, so a 90-degree sector is a quarter of the circle's area. Volume questions usually involve prisms and cylinders, where volume equals the area of the base times the height — a cylinder of radius 3 and height 10 holds 90 pi cubic units. Draw and label a figure whenever one is not provided.
Right-Triangle Trigonometry and Pacing the Section
ACT trig questions are mostly SOH-CAH-TOA: in a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. If a 13-foot ramp reaches a loading dock 5 feet high, the sine of the ramp's angle is 5/13, and the horizontal run is 12 by the 5-12-13 triple. Many questions require no angle computation at all — just building the correct ratio from labeled sides, or using one given ratio to find another. Remember the identity that sine squared plus cosine squared equals 1, and that tangent equals sine over cosine. Now, pacing. Forty-five questions in fifty minutes averages 67 seconds each, but you should not spend evenly: the section runs roughly from easier to harder, so early questions should take well under a minute, banking time for the final stretch. Use a two-pass strategy — solve everything quick on the first pass, circle anything that stalls you for more than about 90 seconds, and return after reaching the end. Because there are only four answer choices and no penalty for wrong answers, answer every question, even blind guesses in the final minute. Backsolving is powerful: plug answer choices into the problem, starting with a middle value so one test tells you whether to go higher or lower. Plugging in a concrete number for a variable turns abstract choices into arithmetic. A permitted calculator helps, but set up the problem on paper first — the calculator cannot fix a wrong equation.
Key terms
- Percent change
- — The change in a quantity divided by its original value, expressed as a percent.
- Ratio multiplier
- — The common factor k used to convert ratio parts like 3:5 into actual amounts 3k and 5k.
- Mean (average)
- — The sum of values divided by the number of values; equivalently, sum equals mean times count.
- Slope-intercept form
- — The line equation y = mx + b, where m is the slope and b the y-intercept.
- System of equations
- — Two or more equations solved together; the solution is the point where their graphs intersect.
- Zero-product property
- — If a product of factors equals zero, at least one factor must equal zero — the key to solving factored quadratics.
- Midpoint formula
- — The midpoint of a segment is the average of the endpoints' x-coordinates and the average of their y-coordinates.
- Pythagorean theorem
- — In a right triangle, the legs satisfy a squared plus b squared equals c squared, where c is the hypotenuse.
- Similar triangles
- — Triangles with equal corresponding angles and proportional corresponding sides.
- Sector
- — A pie-slice region of a circle whose area is the circle's area times the central angle over 360 degrees.
- SOH-CAH-TOA
- — The memory aid for sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.
- Backsolving
- — Testing answer choices in the problem instead of solving algebraically, usually starting from a middle value.
Exam tips
- The enhanced ACT math section has four answer choices and no wrong-answer penalty, so never leave a blank — a guess beats a blank every time.
- Convert every average problem to sums immediately (sum = average times count); changed-value questions become simple addition and subtraction.
- Reread what the question actually asks before answering — a classic trap offers the value of x as a choice when the question asked for 2x or for y.
- Use two passes: bank the early, easier questions quickly and circle anything eating more than about 90 seconds for a return trip.
- When algebra stalls, backsolve from a middle answer choice or plug in a small concrete number like 2 or 10 for the variable.