Math Section Strategy

📋 ACT Math Strategy

Part 1 of 7 — ACT Math Overview

The ACT Math test rewards students who know the content and who work efficiently. This lesson teaches the test-taking strategies, but every strategy is practiced on real ACT-style math problems, because a strategy only counts if it gets you to the right answer faster or more reliably.

The Enhanced ACT Math Test at a Glance

Math is one of the three tests in the Composite score (English, Math, Reading). Science is optional and reported separately.

No Formula Sheet: Bring These

The ACT prints none of these formulas, so they have to be in your head before you walk in.

What Is Tested

• Preparing for Higher Math: Number & Quantity, Algebra, Functions, Geometry, and Statistics & Probability.
• Integrating Essential Skills: rates, percentages, proportional reasoning, area, and averages, used in multi-step problems.
• Modeling: building or interpreting a mathematical model of a real situation. Modeling questions overlap the other categories.

Questions generally get harder as you move through the test, though the order is not strict. Early questions should go quickly; bank that time for the later ones.

The Four-Step Problem Routine

1. Read the last sentence first. Find exactly what is asked: $x$? $2x$? The area? The difference? Write it at the top of your scratch work.
2. Choose a method: direct algebra, backsolving (Part 3), picking numbers (Part 4), estimating and eliminating (Part 5), or drawing a diagram (Part 6).
3. Solve, writing short steps. Doing three steps in your head is where most careless errors happen.
4. Check the answer against the question: right quantity, right units, reasonable size.

Trap #1: Answering a Different Question

The ACT often lists the value of an intermediate step as a wrong choice. If you solve for $x$ but the question asks for $x + 3$, the value of $x$ will almost certainly be waiting for you among the choices.

Shortcut — solve for the expression, not the variable. If $2(x - 3) = 14$ and you need $x - 3$, divide both sides by 2: $x - 3 = 7$. You never need $x$. Likewise, if $4a + 4b = 36$, then $a + b = 9$ without finding $a$ or $b$.

ACT Tip: Before you bubble, reread the question's final phrase and ask, "Is the number I have the thing they asked for?" That five-second check saves more points than almost any other habit.

Worked Examples

Question: If $4x - 7 = 21$, what is the value of $x + 3$?

Solution:

1. What is asked? The value of $x + 3$, not $x$.
2. $4x = 28$, so $x = 7$.
3. $x + 3 = 10$. ✓

Trap: The choices would include 7 (the value of $x$). A student who stops at step 2 picks it and loses an easy point.

Question: The length of a rectangle is 3 cm more than twice its width. The perimeter is 54 cm. What is the area of the rectangle, in square centimeters?

Solution:

1. What is asked? The area.
2. Let the width be $w$; the length is $2w + 3$.
3. Perimeter: $2(w + 2w + 3) = 54 \implies 3w + 3 = 27 \implies w = 8$.
4. Length $= 2(8) + 3 = 19$.
5. Area $= 8 \times 19 = 152$ square centimeters. ✓

Trap: 8 (the width), 19 (the length), and 54 (the perimeter) are all numbers you wrote down along the way. Only 152 answers the question.

Answer the Question That Is Asked 🎯

Solve for the Expression, Not the Variable 🧮

1. If $2(x - 3) = 14$, what is the value of $x - 3$?

2. If $4a + 4b = 36$, what is the value of $a + b$?

3. A square has a perimeter of 36. What is its area?

ACT-Style Practice: The Question-Asked Drill

For each problem, first say out loud what is being asked, then solve. The last column shows the wrong answer students most often choose.

ACT Tip: Underline words such as more than, not, remaining, area, and the units. They tell you which of your numbers is the answer.

ACT-Style Questions 📋

Key Takeaways

• Format: 45 questions, 50 minutes, 4 choices, about 67 seconds per question. A calculator is allowed throughout; there is no formula sheet.
• Never leave a blank: there is no penalty for a wrong answer, and some questions are unscored field tests you cannot identify.
• Four-step routine: read what is asked, choose a method, solve with written steps, and check the answer against the question.
• Biggest early trap: intermediate values ($x$ instead of $x + 3$, width instead of area, hours instead of minutes) appear as wrong choices.
• Shortcut: when the question asks for an expression, try to get that expression directly instead of solving for the variable.

🧮 Calculator Tips

Part 2 of 7 — Using Your Calculator Fast and Accurately

You may use a calculator on every ACT Math question. Every question can also be solved without one, so the calculator is a tool, not a crutch: the goal is to use it where it saves time or prevents arithmetic slips, and to skip it where algebra is faster.

Allowed: four-function, scientific, and most graphing calculators. Not allowed: calculators with a computer algebra system (for example the TI-89, TI-92, or TI-Nspire CAS), and phones or other devices with internet access. Check the current ACT calculator policy before test day, put in fresh batteries, and practice on the same calculator you will bring.

Entry Errors That Cost Points

Your calculator does exactly what you type, in order-of-operations order. Most calculator mistakes are entry mistakes.

Rule of thumb: whenever a fraction bar, square root, or exponent covers more than one term, wrap that group in parentheses.

When the Calculator Helps

1. Messy arithmetic: decimals, large products, percent of a number, compound growth such as $500(1.04)^{6}$.
2. Matching radical or $\pi$ answers: compute a decimal for your result and for each choice. For example, $\sqrt{72} \approx 8.485$, and $6\sqrt{2} \approx 8.485$, so they match.
3. Graphing: to find where two graphs intersect, graph both and use the intersect feature. To find zeros, graph one side minus the other. To find a maximum or minimum, use the max/min feature.
4. Tables: to test integer values quickly (for example, to see which choice makes $2^{x} = 4x$ true), enter the function and scroll the table.
5. Checking equivalence: if a question asks which expression is equivalent to another, graph both or evaluate both at a value such as $x = 2$.

When the Calculator Slows You Down

• Simplifying algebra: factoring $x^2 - 9$ into $(x + 3)(x - 3)$ is faster by hand.
• One-step equations: if $3x = 21$, you know $x = 7$ before you can press the keys.
• Choices in exact form like $\frac{3}{4}$ versus $\frac{4}{3}$: often a quick mental estimate is enough to tell them apart.
• Long chains of operations: typing a ten-step computation in one line invites a misplaced parenthesis. Break it into pieces and write down each intermediate result.

The Estimate-First Habit

Before pressing ENTER, guess the size of the answer. If $\frac{398 \times 0.51}{19.7}$ should be about $\frac{400 \times 0.5}{20} = 10$, a calculator result of 103 tells you that you mistyped something. Estimation is your built-in error detector.

Degree-mode check: for a 50-foot rope at $35^\circ$ to the ground, $50 \sin 35^\circ \approx 50(0.574) \approx 28.7$ feet in degree mode, but the same keystrokes in radian mode return about $-21.4$, an impossible negative height.

ACT Tip: Set your calculator to degree mode before the test starts. ACT trigonometry questions are mostly in degrees, and a radian-mode answer is often negative or far off, which is a giveaway that something went wrong.

Worked Examples

Question: Which of the following is equal to $\sqrt{72}$? The choices are $6\sqrt{2}$, $8\sqrt{3}$, $4\sqrt{3}$, and $2\sqrt{6}$.

Solution:

1. By hand: $72 = 36 \times 2$, so $\sqrt{72} = 6\sqrt{2}$. ✓
2. Calculator check: $\sqrt{72} \approx 8.485$. The choices evaluate to $6\sqrt{2} \approx 8.485$, $8\sqrt{3} \approx 13.86$, $4\sqrt{3} \approx 6.93$, and $2\sqrt{6} \approx 4.90$. Only $6\sqrt{2}$ matches.

Takeaway: If you forget how to simplify a radical, the decimal comparison still finds the answer in seconds.

Question: The graphs of $y = x^2 - 3$ and $y = 2x$ intersect at two points. What is the sum of the $x$-coordinates of those points?

Solution (algebra): Set them equal: $x^2 - 3 = 2x \implies x^2 - 2x - 3 = 0 \implies (x - 3)(x + 1) = 0$, so $x = 3$ or $x = -1$. The sum is $2$. ✓

Solution (graphing): Graph both equations and use the intersect feature twice: the points are $(-1, -2)$ and $(3, 6)$. Same sum, $2$.

Takeaway: If the quadratic does not factor nicely, the graph still gives the intersection points.

Calculator Entry Check 🎯

Type It Right ✏️

1. $(-4)^2 - 3^2$

2. $\frac{18}{3 \cdot 2}$

3. $\sqrt{6^2 + 8^2}$

ACT-Style Practice: Calculator or Not?

ACT Tip: If the answer choices are close decimals (such as 927 versus 932), use the calculator. If they are far apart or in algebraic form, think first.

ACT-Style Questions: Graphs, Tables, and Decimals 📋

Key Takeaways

• Parentheses everywhere: wrap any group under a fraction bar, root, or exponent. Remember $-3^2 = -9$ but $(-3)^2 = 9$.
• Degree mode for ACT trigonometry; a negative length means the mode is wrong.
• Use the calculator for messy arithmetic, decimal matching of radical or $\pi$ answers, graph intersections, zeros, maximums and minimums, and tables of values.
• Skip the calculator for factoring, one-step equations, and choices in algebraic form.
• Estimate first so that you notice when a mistyped entry produces an unreasonable result.

🔄 Backsolving

Part 3 of 7 — Plugging In the Answer Choices

Every ACT Math question gives you the answer: it is one of the four choices. Backsolving means testing the choices in the problem instead of setting up and solving an equation. When the algebra is messy, or you are not sure how to set it up, backsolving turns a hard question into a few quick checks.

When to Backsolve

Backsolving works best when all of these are true:

It is a poor choice when the choices contain variables (use picking numbers, Part 4, instead) or when the algebra is one step ($3x = 21$).

How to Backsolve

1. Identify what each choice represents. Write it down: "choice = number of adult tickets."
2. Start with a middle value. ACT numerical choices are usually listed in increasing or decreasing order. With 4 choices, test the second or third value first.
3. Run the choice through the problem's conditions, one condition at a time.
4. If it works, stop. If it fails, decide whether you need a bigger or smaller value, and test in that direction.

Because the choices are ordered, one test often eliminates two choices at once: if 16 gives a total that is too small and larger values give larger totals, then 12 is also too small. At most you will test two or three choices.

Why Backsolving Protects You From Extraneous Solutions

When you square both sides of an equation or multiply by an expression containing $x$, you can create extraneous solutions: values that satisfy the new equation but not the original. Backsolving tests each choice in the original equation, so an extraneous value fails automatically.

For example, solving $\sqrt{x + 7} = x - 5$ by squaring gives $x^2 - 11x + 18 = 0$, so $x = 2$ or $x = 9$. But plugging $x = 2$ into the original gives $\sqrt{9} = -3$, which is false. Only $x = 9$ works. If 2 appears as a choice, it is a trap for students who solve without checking.

Backsolving Checklist

Common mistake: testing a choice against only part of the problem. In an age problem with two conditions ("now" and "in 6 years"), a choice must pass both.

ACT Tip: Backsolving is not cheating or a last resort. Strong scorers use it whenever it is faster than writing an equation, and it doubles as a check when you do solve algebraically.

Worked Examples

Question: A theater sold 40 tickets for a total of 380 dollars. Adult tickets cost 12 dollars and student tickets cost 7 dollars. How many adult tickets were sold? Choices: 12, 16, 20, 24.

Solution:

1. Label: choice = number of adult tickets; students = 40 minus the choice.
2. Test 16 (a middle value): $16 \times 12 = 192$ and $24 \times 7 = 168$. Total 360, which is too low.
3. More adult tickets raise the total (each one adds 5 dollars over a student ticket), so go bigger.
4. Test 20: $20 \times 12 = 240$ and $20 \times 7 = 140$. Total 380. ✓

Answer: 20 adult tickets. Testing 16 also ruled out 12, since 12 would give an even smaller total.

Question: What value of $x$ satisfies $\sqrt{x + 7} = x - 5$? Choices: 2, 9, 11, 13.

Solution:

1. Test 9: $\sqrt{16} = 4$ and $9 - 5 = 4$. ✓
2. For comparison, test 2: $\sqrt{9} = 3$ but $2 - 5 = -3$. ✗ A square root is never negative.

Why it matters: Squaring both sides produces $x = 2$ and $x = 9$. A student who solves the quadratic and picks the first root chooses 2, an extraneous solution. Backsolving tests the original equation, so it never falls for this.

Backsolve These 🎯

Backsolve or Solve — Your Choice ✏️

1. The sum of five consecutive integers is 115. What is the smallest integer?

2. For what value of $x$ does $\frac{x + 3}{x - 1} = 3$?

3. A jar holds 18 coins, all dimes and quarters, worth 3.15 dollars in total. How many quarters are in the jar?

ACT-Style Practice: Quick Backsolves

ACT Tip: In problem 3, setting up two equations in two variables takes time; testing a choice takes seconds. Recognizing the 5-12-13 right triangle makes it faster still.

ACT-Style Questions 📋

Key Takeaways

• Backsolve when the choices are numbers, the question asks for one value, and the setup is wordy or awkward.
• Label what each choice represents before testing.
• Start with a middle choice, then move bigger or smaller based on the result; one test can eliminate two choices.
• Test every condition in the problem, not just one.
• Backsolving checks the original equation, so it automatically rejects extraneous solutions from squaring or clearing fractions.

🔢 Plugging In Numbers

Part 4 of 7 — Picking Your Own Numbers

Backsolving (Part 3) works when the answer choices are numbers. When the choices contain variables, use the partner strategy: pick numbers for the variables, turn the abstract problem into ordinary arithmetic, and see which choice produces the same result.

When to Pick Numbers

How to Pick Numbers: Five Steps

1. Choose easy values for each variable, and write them down: "let $m = 120$, $h = 2$, $t = 30$."
2. Solve the problem with those numbers to get a target value. Circle it.
3. Plug the same numbers into every choice.
4. Keep the choice(s) that hit the target.
5. If two or more choices match, pick new numbers and test only those choices again.

Choosing Good Numbers

• Avoid 0 and 1. They make many different expressions equal ($x^2 = x$ when $x = 1$), so several choices will match.
• Avoid numbers that already appear in the problem, and use different values for different variables.
• Make the arithmetic friendly: use 100 for percents and prices, multiples of 60 for minutes and hours, and numbers that divide evenly into the problem's quantities.
• Respect the conditions: if the problem says $x > 1$, do not pick $x = \frac{1}{2}$.

Percent Problems: Start at 100

If a price increases 20% and then decreases 20%, start with 100: $100 \to 120 \to 96$. The net change is a 4% decrease, not zero, because the second 20% is taken of a bigger number. Starting at 100 makes the final percent change readable directly.

"Must Be True" Problems: Try to Break Each Choice

For "must be true," a choice is eliminated by one counterexample. Try numbers of different types: a positive and a negative, a fraction between 0 and 1, a large number, zero (if allowed). The choice that survives every attempt is the answer. For "could be true," you only need one example that works.

ACT Tip: Always check all four choices, even after one matches. If you stop at the first match and a second choice also matches, you have a 50% chance of being wrong without knowing it.

Worked Examples

Question: Pens cost $p$ cents each. How many pens can be bought with $d$ dollars? Choices: $\frac{100d}{p}$, $\frac{dp}{100}$, $\frac{100p}{d}$, $\frac{d}{100p}$.

Solution:

1. Pick $p = 50$ cents and $d = 2$ dollars.
2. Target: 2 dollars is 200 cents, and $200 \div 50 = 4$ pens.
3. Plug in: $\frac{100(2)}{50} = 4$ ✓, $\frac{2(50)}{100} = 1$, $\frac{100(50)}{2} = 2500$, $\frac{2}{5000} = 0.0004$.

Answer: $\frac{100d}{p}$. Only one choice hit 4, so no second round is needed.

Question: If $x < y < 0$, which of the following must be true? Choices: $xy < 0$, $\frac{x}{y} > 1$, $x + y > 0$, $x^2 < y^2$.

Solution:

1. Pick $x = -4$ and $y = -2$ (both negative, $x$ smaller).
2. $xy = 8$, not negative ✗. $\frac{x}{y} = 2 > 1$ ✓. $x + y = -6$, not positive ✗. $x^2 = 16$ and $y^2 = 4$, so $x^2 < y^2$ is false ✗.
3. Try another pair to confirm, $x = -3$, $y = -1$: $\frac{x}{y} = 3 > 1$ ✓.

Answer: $\frac{x}{y} > 1$. Since $x$ is farther from zero than $y$ and both are negative, the quotient is always positive and greater than 1.

Pick Numbers to Solve 🎯

Pick Numbers, Then Answer ✏️

1. If $n$ is an even integer, what is the remainder when $n^2 + 3$ is divided by 4?

2. A price is raised by 50% and then the new price is cut by 50%. The final price is what percent of the original price? (Enter the number only.)

3. If $\frac{x}{y} = 3$, what is the value of $\frac{x + y}{y}$?

ACT-Style Practice: Pick and Check

ACT Tip: In problem 4, picking numbers gives an average of $\frac{2 + 4 + 12}{3} = 6$, which equals $s$. Then just check which choice equals 6 when $s = 6$.

ACT-Style Questions 📋

Key Takeaways

• Pick numbers when choices contain variables, the question says "in terms of," a percent problem has no starting value, or the question asks what "must be true."
• Avoid 0, 1, and numbers already in the problem; use different values for different variables and make the arithmetic easy.
• Find the target first, then test all four choices. If two match, pick new numbers.
• Percents: start at 100. Successive percent changes multiply; they do not add.
• Must be true: try to break each choice with negatives, fractions between 0 and 1, and large numbers.

⏱️ Time Management

Part 5 of 7 — Pacing, Skipping, Estimation & Elimination

You have 50 minutes for 45 questions, an average of about 67 seconds each. But the questions are not equally hard, and they are not worth more for being hard: every question counts the same. Good pacing means spending your time where it buys the most points.

The Two-Pass System

Pass 1 (most of the 50 minutes): Go through every question in order. Answer each one you can do in about a minute. If a question is long, confusing, or on a topic you are weak in, make a quick guess, mark it, and move on.

Pass 2 (the time that is left): Return to the marked questions, easiest first. Replace your guess if you solve one.

Final minute: Make sure every question has an answer. There is no penalty for wrong answers, so a blank is always a lost chance.

Skip Signals

Skipping is not giving up. It is reordering the test so you see every question you can answer.

Pacing Checkpoints

Questions generally get harder later, so aim to be slightly ahead of the average pace early. One reasonable plan:

Check your watch only at these checkpoints, not after every question.

Estimation: Know the Size of the Answer

Round the numbers, compute roughly, and eliminate any choice that is far off. For $\frac{398 \times 0.51}{19.7}$, think $\frac{400 \times 0.5}{20} = 10$; only a choice near 10 survives. Estimation also catches decimal-point slips and calculator entry errors (Part 2).

Elimination: Cross Out the Impossible

Many choices can be eliminated without solving, using facts that must always be true:

With 4 choices, eliminating even one turns a 1-in-4 guess into a 1-in-3 guess; eliminating two makes it a coin flip.

About figures: the ACT directions state that illustrative figures are not necessarily drawn to scale. Use a figure to rule out wildly impossible choices, not to choose between close ones.

ACT Tip: When a question looks long, read the last sentence and glance at the choices before you start. If the choices are far apart, a quick estimate may finish the question in seconds.

Worked Examples

Question: What is 19.8% of 401? Choices: 7.94, 79.4, 321.6, 794.

Solution:

1. Round: 19.8% is about 20%, and 401 is about 400.
2. 20% of 400 is 80.
3. Only 79.4 is close to 80. ✓ (7.94 and 794 are decimal-point slips; 321.6 is the remaining 80.2% of 401, the part that is not taken.)

Time used: about 10 seconds, with no calculator.

Question: Two sides of a triangle have lengths 7 and 10. Which of the following could be the perimeter? Choices: 19, 20, 27, 34.

Solution:

1. The third side $s$ must satisfy $10 - 7 < s < 10 + 7$, so $3 < s < 17$.
2. The perimeter is $17 + s$, so it must be between $20$ and $34$, not including either end.
3. 19 and 20 are too small (they need $s \le 3$); 34 needs $s = 17$, which makes a flat "triangle." Only 27 ($s = 10$) works. ✓

Takeaway: You did not need the exact third side; you only needed the bounds.

Estimate and Eliminate 🎯

Eliminate to One Choice 🔍

ACT-Style Practice: The 30-Second Finish

Each problem below can be finished in under 30 seconds with estimation, elimination, or a shortcut.

ACT Tip: Speed on questions like these is what pays for the long, multi-step questions near the end of the test.

ACT-Style Questions: Find the Fast Route 📋

Key Takeaways

• About 67 seconds per question, but every question is worth the same, so spend time where it earns points.
• Two passes: answer what you can, guess-and-mark the rest, return to the marks, and never leave a blank.
• Skip signals: two reads with no plan, about 90 seconds with no progress, or spiraling arithmetic.
• Checkpoints: be a little ahead of pace early, because later questions tend to be harder.
• Estimate to find the size of the answer; eliminate using bounds (probability between 0 and 1, positive lengths, averages between extremes, triangle inequality).

🧩 Problem-Solving Workshop

Part 6 of 7 — Word Problems, Diagrams & Common Traps

Many ACT Math questions are short stories: a garden, a road trip, a sale, a survey. The math is often easy once it is written as an equation or a picture. This part teaches three skills: translating words into math, drawing a diagram when none is given, and spotting the traps the ACT sets in word problems.

Translating Words Into Math

Example: "Seven less than three times a number is 20" becomes $3n - 7 = 20$, so $n = 9$. Writing $7 - 3n = 20$ is the classic reversal error.

Translate in chunks. Read one phrase, write its math, then read the next. Define every variable in words ("$a$ = number of adult tickets") so you know at the end which variable answers the question.

Draw a Diagram When None Is Given

If a problem describes something you could sketch, sketch it. It takes 10 seconds and prevents most setup errors.

Label the unknown with a variable or a question mark. Once everything is on paper, the relationship (Pythagorean theorem, subtraction of areas, inclusion-exclusion) usually becomes obvious.

Three Setups to Know Cold

Mixture: amount of pure substance = concentration × volume. Track the pure part, because it simply adds. Mixing 15 liters of a 20% solution with 5 liters of an 80% solution gives $0.2(15) + 0.8(5) = 3 + 4 = 7$ liters of pure substance in 20 liters, so the mixture is $\frac{7}{20} = 35\%$.

Work rate: rates add, times do not. If one worker finishes a job in $a$ hours and another in $b$ hours, together they take $t$ hours, where $\frac{1}{a} + \frac{1}{b} = \frac{1}{t}$. Two printers that take 10 and 15 hours alone finish $\frac{1}{10} + \frac{1}{15} = \frac{1}{6}$ of the job per hour, so together they take 6 hours. Bound check: working together must be faster than the faster one alone, so the answer has to be less than 10; adding the times (25) or averaging them (12.5) fails that check at once.

Two-circle Venn: neither = total − (A + B − both). The "both" group sits inside A and inside B, so subtract it once to avoid counting it twice. Of 40 students, 22 are in band, 15 are in choir, and 7 are in both: $22 + 15 - 7 = 30$ are in at least one, so $40 - 30 = 10$ are in neither.

The Four Classic Traps

1. Units. Convert before you compute. Area and volume conversions are squared and cubed: 1 square yard = 9 square feet; 1 square foot = 144 square inches; 1 cubic yard = 27 cubic feet. For speed, 1 mile = 5,280 feet and 1 hour = 3,600 seconds, so 45 miles per hour is $\frac{45 \times 5280}{3600} = 66$ feet per second.

2. NOT, EXCEPT, LEAST. When a question says "which of the following is NOT…," three choices satisfy the condition and the answer is the one that does not. Mark the capitalized word on your scratch paper so you do not pick the first choice that "works."

3. Percent base. Percent change $= \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$. The base is the original value, the number after "of" or "than." A drop from 50 to 40 is a 20% decrease (10 out of 50), but the rise from 40 back to 50 is a 25% increase (10 out of 40).

4. Extraneous solutions. Squaring both sides of a radical equation or multiplying by an expression with $x$ can create false solutions. Always check in the original equation:

• A square root can never equal a negative number.
• A value that makes a denominator zero is never a solution.

ACT Tip: After you finish a word problem, reread the question's final sentence and check your units. If the question asks for cost in dollars and you have square feet, you are not done.

Worked Examples

Question: Two cyclists leave the same point. One rides 30 miles due north and the other rides 40 miles due east. How far apart are they, in miles?

Solution:

1. Draw it: an arrow up (30) and an arrow right (40) from one point. The distance between the riders is the segment joining the arrow tips.
2. The arrows meet at a right angle, so the distance is a hypotenuse: $\sqrt{30^2 + 40^2} = \sqrt{2500} = 50$. ✓
3. Recognize the 3-4-5 triangle scaled by 10 for a faster check.

Trap avoided: Without a diagram, many students add the distances (70), but the riders are not on one straight road.

Question: How many real solutions does $\frac{x}{x - 2} = \frac{2}{x - 2} + 3$ have?

Solution:

1. Multiply every term by $x - 2$: $x = 2 + 3(x - 2) = 3x - 4$.
2. Solve: $2x = 4$, so $x = 2$.
3. Check in the original: $x = 2$ makes both denominators zero, so it is not allowed.

Answer: zero solutions. The equation has no solution, even though the algebra produced a number.

Translate and Draw 🎯

Watch the Trap ✏️

1. After a 30% discount, a jacket costs 63 dollars. What was the original price, in dollars?

2. A car travels at 45 miles per hour. What is its speed in feet per second? (1 mile = 5,280 feet)

3. How many real solutions does $\frac{x}{x - 3} = \frac{3}{x - 3} + 2$ have?

ACT-Style Practice: Spot the Trap

ACT Tip: In problem 4, $\frac{20}{60} \approx 33\%$ is the trap answer that uses the new value as the base.

ACT-Style Questions: Avoid the Trap 📋

Key Takeaways

• Translate phrase by phrase, define each variable in words, and remember that "less than" reverses the order: "7 less than $3n$" is $3n - 7$.
• Draw a diagram for directions, shapes described in words, overlapping groups, and points on a line.
• Units: convert before computing; square units convert by the square (9 square feet per square yard) and cubic units by the cube (27).
• NOT / EXCEPT: three choices satisfy the condition; you want the one that does not.
• Percent base is the original value; extraneous solutions must be checked in the original equation.

🏁 Putting It All Together

Part 7 of 7 — Integrated Mixed Practice

On test day, questions do not come labeled "backsolve me" or "draw a diagram." The skill that matters most is choosing a good method in the first few seconds. This part pulls together everything from Parts 1–6 and finishes with a mixed set of ACT-style problems.

Strategy Selector

Often two strategies combine: draw a diagram, then backsolve; or pick numbers, then estimate.

Test-Day Game Plan for ACT Math

1. Before the test: fresh calculator batteries, degree mode, and a plan for pacing checkpoints (about 9 minutes at question 10, 19 at question 20, 31 at question 30, 43 at question 40).
2. Pass 1: for each question, read what is asked, pick a method, solve. If there is no plan after about 90 seconds, guess, mark it, and move on.
3. Pass 2: return to the marked questions, easiest first.
4. Last minute: every question gets an answer. There is no penalty for guessing.

Reviewing Practice Tests: The Error Log

After each practice set, sort every miss into one category. The category tells you what to fix.

For many students, misreads and careless errors account for a meaningful share of misses, and those are the quickest points to recover.

A Note on Accuracy vs. Speed

Rushing to reach every question is only worth it if your accuracy holds. A student who answers 40 questions carefully and guesses on 5 usually outscores one who races through all 45 and makes many careless errors. Find the pace at which your accuracy stays high, then build speed through practice, not panic.

ACT Tip: When you check an answer, use a different method than the one you solved with: backsolve an algebra answer, or estimate a calculator answer. A second method catches errors that repeating the same steps would not.

Worked Examples

Question: If $x^2 - y^2 = 24$ and $x - y = 4$, what is the value of $x$?

Solution:

1. Recognize the difference of squares: $x^2 - y^2 = (x + y)(x - y)$.
2. Substitute: $(x + y)(4) = 24$, so $x + y = 6$.
3. Add the equations $x + y = 6$ and $x - y = 4$: $2x = 10$, so $x = 5$. ✓
4. Check with a different method: $y = 1$, and $25 - 1 = 24$ ✓.

Strategies used: solving for an expression ($x + y$) and checking by a second method.

Question: The length of a rectangle is 4 inches more than its width, and its diagonal is 20 inches. What is the width, in inches? Choices: 8, 10, 12, 16.

Solution:

1. Draw the rectangle with its diagonal: a right triangle with legs $w$ and $w + 4$ and hypotenuse 20.
2. Backsolve with a middle choice, 12: legs 12 and 16, and $12^2 + 16^2 = 144 + 256 = 400 = 20^2$ ✓.
3. Pattern check: 12-16-20 is the 3-4-5 triangle scaled by 4.

Trap: 16 is the length, not the width. The algebraic route ($w^2 + 4w - 192 = 0$) gives the same answer but takes longer.

Mixed Set A 🎯

Choose Your Method ✏️

1. The sum of four consecutive odd integers is 64. What is the largest of the four?

2. A shirt that costs a store 25 dollars is marked up 40%, and then sold at 20% off the marked-up price. What is the selling price, in dollars?

3. A square has a diagonal of length 10. What is the area of the square?

ACT-Style Practice: Timed Mini-Set (aim for 5 minutes)

ACT Tip: After the mini-set, log each miss in your error log by category before checking the next set.

Mixed Set B 📋

Key Takeaways

• Choose a method in seconds using the strategy selector: expression, calculator, backsolve, pick numbers, estimate, diagram, check roots, mark NOT.
• Combine strategies when helpful (diagram plus backsolve, pick numbers plus estimate).
• Game plan: checkpoints at questions 10, 20, 30, and 40; two passes; no blanks.
• Error log: sort misses into content gap, misread, careless, strategy, and time, then fix the biggest category first.
• Check with a different method than the one you used to solve.