Math Question Types
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SAT and ACT math sections test similar content but with different formats and emphases. Understanding the question types and domains helps you prepare strategically. This lesson covers the major math domains and provides practice questions in each category.
SAT Math Domains
- Algebra (~35%): Linear equations, systems, inequalities
- Advanced Math (~35%): Quadratics, polynomials, functions, radicals
- Problem-Solving & Data Analysis (~15%): Ratios, percentages, statistics
- Geometry & Trigonometry (~15%): Angles, triangles, circles, trig ratios
Key Algebra Skills
- Solving linear equations and inequalities
- Systems of equations (substitution and elimination)
- Interpreting linear functions (slope, y-intercept)
- Translating word problems into equations
- Working with absolute value
Key Advanced Math Skills
- Solving quadratic equations (factoring, quadratic formula)
- Working with exponents and radicals
- Polynomial operations and factoring
- Function notation and transformations
- Exponential growth and decay
Key Geometry & Trig Skills
- Properties of angles (complementary, supplementary, vertical)
- Triangle properties (area, Pythagorean theorem, similar triangles)
- Circle properties (area, circumference, arc length)
- Coordinate geometry (distance, midpoint)
- Basic trigonometry (SOH-CAH-TOA)
Calculator Strategy
The SAT has calculator-allowed and no-calculator sections. Even on calculator sections, many problems are faster to solve mentally or by hand. Use your calculator strategically for complex calculations, graphing, and checking answers.
Examples
Work through these examples to see different math question types and solution strategies.
Example 1: Linear Equation (Algebra)
If 3(x - 2) = 2x + 7, what is the value of x?
Solution:
Step 1: Distribute: 3x - 6 = 2x + 7
Step 2: Subtract 2x from both sides: x - 6 = 7
Step 3: Add 6 to both sides: x = 13
Answer: x = 13
Check: 3(13 - 2) = 3(11) = 33; 2(13) + 7 = 26 + 7 = 33. Correct.
Example 2: Systems of Equations (Algebra)
If 2x + y = 10 and x - y = 2, what is the value of x?
Solution (Elimination):
Add the equations to eliminate y:
(2x + y) + (x - y) = 10 + 2
3x = 12
x = 4
Answer: x = 4
To find y: 4 - y = 2, so y = 2. Check: 2(4) + 2 = 10. Correct.
Example 3: Quadratic Equation (Advanced Math)
What are the solutions to x^2 - 5x + 6 = 0?
Solution (Factoring):
Factor the quadratic: (x - 2)(x - 3) = 0
Set each factor equal to zero:
x - 2 = 0 or x - 3 = 0
x = 2 or x = 3
Answer: x = 2 or x = 3
Note: You could also use the quadratic formula: x = (5 plus or minus sqrt(25-24))/2 = (5 plus or minus 1)/2
Example 4: Geometry - Triangle
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
Solution (Pythagorean Theorem):
a^2 + b^2 = c^2
5^2 + 12^2 = c^2
25 + 144 = c^2
169 = c^2
c = 13
Answer: 13
Note: 5-12-13 is a Pythagorean triple (like 3-4-5 and 8-15-17). Memorizing these saves time.
Example 5: Word Problem (Problem-Solving)
A store sells notebooks for $4 each and pens for $2 each. If Maria spent $26 on 8 items total, how many notebooks did she buy?
Solution:
Let n = number of notebooks, p = number of pens
Set up equations:
n + p = 8 (total items)
4n + 2p = 26 (total cost)
From equation 1: p = 8 - n
Substitute into equation 2: 4n + 2(8 - n) = 26
4n + 16 - 2n = 26
2n = 10
n = 5
Answer: 5 notebooks
Check: 5 notebooks + 3 pens = 8 items. 5($4) + 3($2) = $20 + $6 = $26. Correct.
Practice Problems
Work through these SAT/ACT-style math questions covering all major domains.
1. If 4x - 7 = 2x + 9, what is the value of x?
Show Answer
C) 8 - Subtract 2x from both sides: 2x - 7 = 9. Add 7: 2x = 16. Divide by 2: x = 8.
2. What is the slope of the line that passes through the points (2, 5) and (6, 13)?
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B) 2 - Slope = (y2 - y1)/(x2 - x1) = (13 - 5)/(6 - 2) = 8/4 = 2.
3. Which of the following is equivalent to (x^3)(x^4)?
Show Answer
A) x^7 - When multiplying with the same base, add exponents: x^(3+4) = x^7.
4. If f(x) = 2x^2 - 3x + 1, what is f(3)?
Show Answer
B) 10 - f(3) = 2(3)^2 - 3(3) + 1 = 2(9) - 9 + 1 = 18 - 9 + 1 = 10.
5. What are the solutions to x^2 - 9 = 0?
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C) x = 3 or x = -3 - Factor as difference of squares: (x + 3)(x - 3) = 0. Solutions: x = -3 or x = 3.
6. A circle has a radius of 5. What is its area?
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B) 25pi - Area of circle = pi*r^2 = pi*(5)^2 = 25pi.
7. In a right triangle, if one leg is 8 and the hypotenuse is 10, what is the length of the other leg?
Show Answer
B) 6 - Using Pythagorean theorem: 8^2 + b^2 = 10^2. 64 + b^2 = 100. b^2 = 36. b = 6. (This is a 6-8-10 triangle, a multiple of 3-4-5.)
8. If the product of two consecutive positive integers is 72, what is the larger integer?
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B) 9 - Let the integers be n and n+1. n(n+1) = 72. Testing: 8(9) = 72. The larger integer is 9.
9. What is the value of sqrt(48) in simplest form?
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A) 4sqrt(3) - sqrt(48) = sqrt(16 * 3) = sqrt(16) * sqrt(3) = 4sqrt(3).
10. A car travels 180 miles in 3 hours. At this rate, how far will it travel in 5 hours?
Show Answer
A) 300 miles - Rate = 180/3 = 60 mph. Distance in 5 hours = 60 * 5 = 300 miles.
Check Your Understanding
Review these key concepts and strategies.
1. When solving a system of two linear equations, which method is usually fastest when one equation is already solved for a variable?
Show Answer
B) Substitution - When one equation is already solved for a variable (like y = 2x + 3), substitution is most efficient.
2. Which Pythagorean triple includes the numbers 3 and 4?
Show Answer
A) 3-4-5 - The most common Pythagorean triple. 3^2 + 4^2 = 9 + 16 = 25 = 5^2.
3. To factor x^2 - 16, you should recognize this as:
Show Answer
B) A difference of squares - x^2 - 16 = (x + 4)(x - 4). The pattern is a^2 - b^2 = (a + b)(a - b).
4. If a line has a slope of 2 and passes through (0, 3), what is its equation?
Show Answer
B) y = 2x + 3 - In slope-intercept form (y = mx + b), m = 2 (slope) and b = 3 (y-intercept, since the line passes through (0, 3)).
Key Math Formulas
- Slope: m = (y2 - y1)/(x2 - x1)
- Slope-intercept form: y = mx + b
- Quadratic formula: x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a
- Pythagorean theorem: a^2 + b^2 = c^2
- Area of circle: A = pi*r^2
- Circumference: C = 2*pi*r
- Distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2)
Next Steps
- Review any question types you found challenging
- Memorize key formulas and Pythagorean triples
- Practice translating word problems into equations
- Continue to the Timed Modules for test-like practice conditions