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Basic Arithmetic
25 + 37
Answer: 62
Solution: Add the ones place: 5 + 7 = 12. Carry the 1. Add tens: 2 + 3 + 1 = 6. Result: 62
156 - 89
Answer: 67
Solution: Subtract: 156 - 89. Borrow from tens: 16 - 9 = 7. Borrow from hundreds: 14 - 8 = 6. Result: 67
12 * 15
Answer: 180
Solution: 12 * 15 = 12 * (10 + 5) = 120 + 60 = 180
144 / 12
Answer: 12
Solution: 144 / 12 = 12 (since 12 * 12 = 144)
2^8
Answer: 256
Solution: 2^8 = 2*2*2*2*2*2*2*2 = 256
Fractions
1/2 + 1/3
Answer: 5/6
Solution: Find LCD = 6. Convert: 3/6 + 2/6 = 5/6
3/4 - 1/6
Answer: 7/12
Solution: Find LCD = 12. Convert: 9/12 - 2/12 = 7/12
2/3 * 3/4
Answer: 1/2
Solution: Multiply: (2*3)/(3*4) = 6/12 = 1/2
5/6 divided by 2/3
Answer: 5/4
Solution: Flip and multiply: 5/6 * 3/2 = 15/12 = 5/4
Simplify 24/36
Answer: 2/3
Solution: GCD(24,36) = 12. Divide both: 24/12 = 2, 36/12 = 3. Result: 2/3
Percentages
15% of 80
Answer: 12
Solution: 15% of 80 = 0.15 * 80 = 12
25% of 200
Answer: 50
Solution: 25% of 200 = 0.25 * 200 = 50
What is 45 as a percentage of 180?
Answer: 25%
Solution: (45/180) * 100 = 0.25 * 100 = 25%
Increase 80 by 20%
Answer: 96
Solution: 80 + (20% of 80) = 80 + 16 = 96
Decrease 150 by 30%
Answer: 105
Solution: 150 - (30% of 150) = 150 - 45 = 105
Square Roots
sqrt(144)
Answer: 12
Solution: sqrt(144) = 12 (since 12 * 12 = 144)
sqrt(225)
Answer: 15
Solution: sqrt(225) = 15 (since 15 * 15 = 225)
sqrt(81)
Answer: 9
Solution: sqrt(81) = 9 (since 9 * 9 = 81)
sqrt(196)
Answer: 14
Solution: sqrt(196) = 14 (since 14 * 14 = 196)
sqrt(625)
Answer: 25
Solution: sqrt(625) = 25 (since 25 * 25 = 625)
Algebra
Solve for x: 2x + 5 = 13
Answer: x = 4
Solution: 2x + 5 = 13. Subtract 5: 2x = 8. Divide by 2: x = 4
Solve for x: 3x - 7 = 14
Answer: x = 7
Solution: 3x - 7 = 14. Add 7: 3x = 21. Divide by 3: x = 7
Solve for x: 5x + 3 = 2x + 15
Answer: x = 4
Solution: 5x + 3 = 2x + 15. Subtract 2x: 3x + 3 = 15. Subtract 3: 3x = 12. x = 4
Solve for x: x^2 = 49
Answer: x = 7 or x = -7
Solution: x^2 = 49. Take square root: x = +/-sqrt(49) = +/-7
Solve for x: 2(x + 3) = 16
Answer: x = 5
Solution: 2(x + 3) = 16. Divide by 2: x + 3 = 8. Subtract 3: x = 5
Quadratics
Solve x^2 - 5x + 6 = 0
Answer: x = 2 or x = 3
Solution: Factor: (x-2)(x-3) = 0. Solutions: x = 2 or x = 3
Solve x^2 + 4x - 12 = 0
Answer: x = 2 or x = -6
Solution: Factor: (x-2)(x+6) = 0. Solutions: x = 2 or x = -6
Factor x^2 - 9
Answer: (x+3)(x-3)
Solution: Difference of squares: x^2 - 9 = (x+3)(x-3)
Factor x^2 + 6x + 9
Answer: (x+3)^2
Solution: Perfect square trinomial: x^2 + 6x + 9 = (x+3)^2
Find the vertex of y = x^2 - 4x + 3
Answer: (2, -1)
Solution: Vertex x = -b/2a = 4/2 = 2. y = (2)^2 - 4(2) + 3 = -1. Vertex: (2, -1)
Trigonometry
sin(30 degrees)
Answer: 0.5
Solution: sin(30) = 1/2 = 0.5 (standard angle)
cos(60 degrees)
Answer: 0.5
Solution: cos(60) = 1/2 = 0.5 (standard angle)
tan(45 degrees)
Answer: 1
Solution: tan(45) = sin(45)/cos(45) = 1
sin(90 degrees)
Answer: 1
Solution: sin(90) = 1 (standard angle)
cos(0 degrees)
Answer: 1
Solution: cos(0) = 1 (standard angle)
Calculus - Derivatives
Find the derivative of x^3 + 2x^2 - 5
Answer: 3x^2 + 4x
Solution: Power rule: d/dx(x^3) = 3x^2, d/dx(2x^2) = 4x, d/dx(-5) = 0. Result: 3x^2 + 4x
Find the derivative of 5x^4
Answer: 20x^3
Solution: Power rule: d/dx(5x^4) = 5 * 4x^3 = 20x^3
d/dx of sin(x)
Answer: cos(x)
Solution: Standard derivative: d/dx(sin(x)) = cos(x)
d/dx of e^x
Answer: e^x
Solution: Standard derivative: d/dx(e^x) = e^x
d/dx of ln(x)
Answer: 1/x
Solution: Standard derivative: d/dx(ln(x)) = 1/x
Calculus - Integrals
integrate x^2 dx
Answer: (x^3)/3 + C
Solution: Power rule: integral of x^2 = x^(2+1)/(2+1) + C = x^3/3 + C
integrate 3x^2 dx
Answer: x^3 + C
Solution: Power rule: integral of 3x^2 = 3 * x^3/3 + C = x^3 + C
integrate cos(x) dx
Answer: sin(x) + C
Solution: Standard integral: integral of cos(x) = sin(x) + C
integrate e^x dx
Answer: e^x + C
Solution: Standard integral: integral of e^x = e^x + C
integrate 1/x dx
Answer: ln|x| + C
Solution: Standard integral: integral of 1/x = ln|x| + C
Geometry
Area of circle with radius 5
Answer: 25pi or 78.54
Solution: Area = pi * r^2 = pi * 25 = 25pi (approx 78.54)
Circumference of circle with radius 7
Answer: 14pi or 43.98
Solution: Circumference = 2 * pi * r = 2 * pi * 7 = 14pi (approx 43.98)
Area of triangle with base 10 and height 6
Answer: 30
Solution: Area = (1/2) * base * height = (1/2) * 10 * 6 = 30
Pythagorean theorem: find c when a=3, b=4
Answer: 5
Solution: c^2 = a^2 + b^2 = 9 + 16 = 25. c = sqrt(25) = 5
Volume of sphere with radius 3
Answer: 36pi or 113.1
Solution: Volume = (4/3) * pi * r^3 = (4/3) * pi * 27 = 36pi (approx 113.1)