Imagine you scored ninety on your first ACT practice test, then eighty-five on your second. Simple math says your average is eighty-seven point five, right? But what if that second test counted twice as much? Suddenly, your average drops to eighty-six point seven, and those decimal points could mean the difference between your dream school and your safety school. Every ACT features three to five questions on averages and weighted averages, testing your ability to navigate between simple means and complex weighted calculations. These questions often hide in word problems about grades, speeds, or combined groups, requiring you to recognize when weights matter and when they do not. Master these concepts, and you will confidently tackle problems that leave other test takers averaging out their guesses.
Imagine you scored ninety on your first ACT practice test, then eighty-five on your second. Simple math says your average is eighty-seven point five, right? But what if that second test counted twice as much? Suddenly, your average drops to eighty-six point seven, and those decimal points could mean the difference between your dream school and your safety school. Every ACT features three to five questions on averages and weighted averages, testing your ability to navigate between simple means and complex weighted calculations. These questions often hide in word problems about grades, speeds, or combined groups, requiring you to recognize when weights matter and when they do not. Master these concepts, and you will confidently tackle problems that leave other test takers averaging out their guesses.
$Average = frac{Sum of all values}{Number of values}$
The three key relationships: - Sum = Average × Count - Count = Sum ÷ Average - Missing value = (Target average × Total count) - Current sum
For consecutive or evenly spaced numbers, the average equals the middle value (or the average of the first and last). The ACT rarely asks for a straightforward average — instead, they test your ability to manipulate these relationships.
$Weighted Average = frac{sum(Value × Weight)}{sum(Weights)}$
Use weighted averages when not all values count equally. Common scenarios include grade calculations with percentage weights, combining groups of different sizes, and average speed problems where total distance divided by total time gives the correct answer.
The formula is really just: total weighted contribution divided by total weight. Think of it as "how much does each piece contribute to the whole?"
Imagine losing easy points on the ACT because you could not quickly convert 0.375 to three-eighths. That is the reality for thousands of students every year! Converting between fractions, decimals, and percents appears in 4–6 questions per test, often hidden within word problems or multi-step calculations. But here is the secret—mastering these conversions is not about memorizing endless tables. It is about recognizing patterns and developing lightning-fast mental shortcuts. When you can instantly see that 0.625 = (5)/(8) = 62.5%, complex problems suddenly become simple. These skills create a ripple effect throughout the entire math section, making algebra, geometry, and statistics problems significantly easier.
Every conversion follows a logical path. Master these five moves and you can convert anything:
Fraction to Decimal: Divide numerator by denominator. (3)/(4) = 3 ÷ 4 = 0.75
Decimal to Percent: Multiply by 100 (move decimal 2 places right). 0.75 × 100 = 75%
Percent to Decimal: Divide by 100 (move decimal 2 places left). 75% ÷ 100 = 0.75
Decimal to Fraction: Count decimal places for the denominator, then simplify. 0.75 = (75)/(100) = (3)/(4)
Percent to Fraction: Remove the % sign, put over 100, simplify. 35% = (35)/(100) = (7)/(20)
Certain fraction families create beautiful patterns. Memorize these and you will save enormous time:
Eighths family: (1)/(8) = 0.125, (3)/(8) = 0.375, (5)/(8) = 0.625, (7)/(8) = 0.875. Notice: decimals always end in 125, 375, 625, or 875.
Thirds: (1)/(3) = 0.overline{3} approx 33.3%, (2)/(3) = 0.overline{6} approx 66.7%
Ninths: The numerator becomes the repeating digit! (1)/(9) = 0.overline{1}, (4)/(9) = 0.overline{4}, (7)/(9) = 0.overline{7}
Fifths: Always produce clean decimals ending in 0, 2, 4, 6, or 8. (1)/(5) = 0.2, (2)/(5) = 0.4, (3)/(5) = 0.6, (4)/(5) = 0.8
Sixths: (1)/(6) = 0.1overline{6} approx 16.7%, (5)/(6) = 0.8overline{3} approx 83.3%
Forty-two percent of ACT test takers miss geometry problems that combine multiple concepts. Yet these integrated questions — mixing area with volume or surface area with cross-sections — appear on every single ACT math test. You will encounter 3 to 5 of these multi-step geometry problems, often positioned as medium to hard difficulty questions. The secret to conquering them lies not in memorizing more formulas, but in recognizing how simpler shapes combine and understanding when to use each measurement type. These problems test whether you can decompose complex figures, apply the right formulas in sequence, and avoid common calculation traps. By mastering the integration strategies in this lesson, you will transform these intimidating multi-part problems into systematic, solvable puzzles.
Every complex shape on the ACT can be broken into basic components: rectangles, triangles, circles, semicircles for 2D; prisms, cylinders, pyramids, cones, and spheres for 3D.
The systematic approach: 1. Identify the basic shapes within the composite. 2. Sketch and label all dimensions. 3. Calculate each component separately. 4. Combine using addition or subtraction.
Two-dimensional shapes become 3D through extrusion (shape × height = volume), revolution (rotating around an axis), or stacking. Key connections:
Rectangle to rectangular prism Triangle to triangular prism or pyramid Circle to cylinder or sphere Trapezoid to trapezoidal prism
For surface area of joined shapes, remember that joined faces disappear — count only exposed surfaces.
A single percent symbol can mean the difference between profit and loss, between getting into your dream college or settling for second best. On the ACT Math section, you will face 2–4 questions that directly test percents, markups, discounts, and percent change, with these skills also appearing inside other multi-step problems. These are not just abstract concepts — they are the mathematics of real-world commerce, the calculations that drive every business decision from your local coffee shop to Wall Street. What makes these problems particularly powerful is their ability to combine multiple concepts in a single question. You might calculate a markup, apply a discount, add tax, and find the percent change all in one problem. The students who excel do not just memorize formulas — they understand the underlying relationships.
Instead of calculating percent amounts and then adding or subtracting, use multipliers to transform multi-step problems into single calculations.
20% increase to multiply by 1.20 20% decrease to multiply by 0.80 15% markup to multiply by 1.15 25% discount to multiply by 0.75 8% tax to multiply by 1.08
The universal formula: Final Value = Initial Value × Multiplier
For increase: Multiplier = 1 + frac{percent}{100} For decrease: Multiplier = 1 - frac{percent}{100}
A discount reduces the original price: Sale Price = Original × (1 - discount rate).
Critical rule: successive discounts do NOT add. A 20% discount followed by a 15% discount is NOT 35% off. Each discount applies to the already-reduced price.
20% off, then 15% off: multiply by 0.80 × 0.85 = 0.68. That is 32% off, not 35%.
Imagine shrinking the entire world to fit on your desk, then calculating the real distance between cities. That is the power of scale drawings, and on the ACT, these questions blend proportions with real-world applications in ways that can boost or bust your math score. You will face 2 to 4 questions combining proportions and scale drawings, making up about 5 to 8 percent of the Math section. Whether it is reading maps, interpreting blueprints, or solving shadow problems, success comes from understanding that scale is just a special type of proportion. The beauty of these problems? Once you grasp the core concept, every variation follows the same logical pattern. Let's build your toolkit so you can handle anything the ACT throws at you.
Every proportion problem boils down to (a)/(b) = (c)/(d). To solve it, cross-multiply: multiply diagonally to get ad = bc, then isolate the unknown.
Here is cross-multiplication in action:
(3)/(8) = (x)/(20)
Cross-multiply: 3 × 20 = 8 × x, so 60 = 8x, giving x = 7.5.
Always verify that units match on both sides of the equation. A common mistake is mixing feet with inches or hours with minutes, which throws your answer off by a factor of 12 or 60.
The unit rate method is often faster: find what one unit equals, then scale up. If 4 pencils cost 2.60 dollars, one pencil costs 0.65 dollars, so 10 pencils cost 6.50 dollars. When you see trigger phrases like 'for every,' 'if... then,' or 'at this rate,' your proportion radar should activate.
The golden formula: Actual Size = Drawing Size × Scale Factor.
Scale appears in three formats: - Ratio form: 1:100 means 1 unit on the drawing = 100 units in reality. - Fractional form: (1)/(100) means the drawing is (1)/(100) of actual size. - Verbal form: '1 inch = 50 feet' states the conversion explicitly.
When scales mix units, convert BEFORE multiplying. A (1)/(4)-inch = 1 foot scale means each quarter-inch on the blueprint corresponds to 1 real foot. So 1 blueprint inch = 4 feet (because 4 quarter-inches fit in 1 inch). If you need everything in inches, 1 blueprint inch = 48 real inches, because 4 feet × 12 inches/foot = 48 inches.
Every ACT test contains a hidden superpower that can save you precious minutes and boost your accuracy — it is called dimensional analysis. Picture this: you are racing through the math section when suddenly you face a problem asking you to convert 88 feet per second to miles per hour. Without the right technique, you would waste valuable time with multiple calculations and risk making errors. But with dimensional analysis, you can solve it in one elegant equation. On the ACT, you will encounter 3 to 5 questions that require unit conversions or rate calculations, often disguised within word problems, science contexts, or real-world scenarios. Once you master these techniques, what seems complex becomes mechanical, turning potential time-drains into quick wins.
A unit rate compares a quantity to exactly one unit of another quantity. It answers 'how much per one?'
$Unit Rate = frac{Total Quantity}{Total Units}$
Common types: speed (miles per hour), price (dollars per pound), efficiency (miles per gallon), density (grams per cubic centimeter), and work rate (problems per hour).
Look for keywords: 'per,' 'for each,' 'every,' or rate notation with '/' (km/h, dollars/lb). These signal that a unit rate is lurking in the problem.
Dimensional analysis converts between units by multiplying by fractions equal to 1. The key: arrange conversion factors so unwanted units cancel and desired units remain.
Why does this work? Because frac{5{,}280 feet}{1 mile} = 1 — both represent the same distance. Multiplying by 1 does not change the value, only the units.
Essential conversion factors to memorize: - 1 mile = 5,280 feet - 1 hour = 60 minutes = 3,600 seconds - 1 kilogram = 2.205 pounds - 1 mile = 1.609 kilometers - 1 gallon = 3.785 liters - 1 meter = 3.281 feet