Digital SAT Math Problems Free Practice 2025

This equation is a quadratic in sin(x). Solving the quadratic gives the solutions x = 7π/6, 11π/6, π/2.

Explanation: After one year: 300,000 * 1.10 = 330,000 After two years: 330,000 * 1.10 = 363,000

Explanation: Let 'x' be the common multiplier. Number of oranges = 3x = 15 Therefore, x = 5 Number of apples = 2x = 2 * 5 = 10

The y-intercept occurs when x = 0. Substituting x = 0 gives y = 3(0 + 2)² - 5 = 3(4) - 5 = 12 - 5 = 7. Therefore, the y-intercept is (0, 7).

Since the coefficient of x² is negative, the parabola opens downwards, meaning it has a maximum value. The x-coordinate of the vertex is x = -b/2a = -4/(2 * -1) = 2. Substituting x = 2 into the equation gives y = -(2)² + 4(2) - 3 = -4 + 8 - 3 = 1.

The sine function is negative in the third and fourth quadrants. sin x = -0.866 corresponds to x = 240° and x = 300°.

Let L be the length of the ladder. We can use the cosine function: cos(60°) = Adjacent/Hypotenuse = 7/L, 1/2 = 7/L, L = 14 feet.

The vertex of the parabola is (4, 1). Since the coefficient of the squared term is positive (1/2), the parabola opens upwards. Therefore, the range is all y-values greater than or equal to 1 (y ≥ 1).

Using the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So, x - 2 = 0 which means x = 2, or x + 3 = 0 which means x = -3.

The period of a function of the form cos(bx) is given by 2π/|b|. In this case, b = 2, so the period is 2π/2 = π.

The side opposite the 30° angle is half the hypotenuse, so 8 / 2 = 4.

Using the angle sum formula for sine: sin(75°) = (√6 + √2)/4.

Explanation: Average speed is calculated by dividing the total distance by the total time. 72 miles / 1.5 hours = 48 miles/hour

 Substitution or elimination gives x = 3, y = 1.

1. Subtract 7 from both sides: 5x = 20
2. Divide both sides by 5: x = 4

In a 45°-45°-90° triangle, the sides are in the ratio 1:1:√2. If the hypotenuse is 10√2, each leg is 10 because (10√2) / √2 = 10.

The longer leg is (hypotenuse/2) × √3. Here, (4√3 / 2) × √3 = 2 * 3 = 6.

Since sin(x) = 1/3, the adjacent side is negative. tan(x) = -√2/4.

 Elimination or substitution leads to x = 2, y = -1.

The perimeter of a rectangle is 2 × (length + width). 2 × (12 cm + 8 cm) = 40 cm.

1. Add 6 to both sides: 12x = 36
2. Divide both sides by 12: x = 3 1/2

Solving gives x = 2 and y = -1.

sin(60°) = √3/2, tan(30°) = √3/3, so (√3/2) * (√3/3) = 3/6 = 1/2. The previous answer was incorrect.

Explanation: After one year: 10,000 * 1.05 = 10,500 After two years: 10,500 * 1.05 = 11,025

The slope formula is m = (y2 - y1) / (x2 - x1) Substitute: m = (4 - (-2)) / (-1 - 3) = 6 / -4 = -3/2

The dot product u • v = (2)(-1) + (3)(4) = -2 + 12 = 10.

1. Subtract 10 from both sides: 7x ≥ 14
2. Divide both sides by 7: x ≥ 1

Explanation: First, find the speed: 18 miles / 2 hours = 9 miles/hour. Then, calculate the distance traveled in 5 hours: 9 miles/hour * 5 hours = 45 miles

The vertex form of a quadratic is a(x - h)² + k, where (h, k) is the vertex. In this case, h = 2 and k = 4, so the vertex is (2, 4).

The discriminant is b² − 4ac. Substituting a=5, b=−6, c=1, (−6)² − 4(5)(1)=36−20=16.

Explanation: Average speed is calculated by dividing the total distance by the total time. 120 miles / 3 hours = 40 miles/hour

Use slope-intercept form y = mx + b. With m = 2 and b = 3, the equation is y = 2x + 3.

Substituting x=2, f(2)=3(2)²−5(2)+2=4.

Explanation: Increase: 8% 8% of 150,000 = 0.08 * 150,000 = 12,000 New population: 150,000 + 12,000 = 162,000

The longer leg is (shortest side) × √3. Solving x√3 = 12, we get x = 12/√3 which simplifies to x= 4√3.

Explanation: Fuel needed is calculated by dividing the total distance by the miles per gallon. 360 miles / 20 miles/gallon = 18 gallons

The area of a parallelogram is base × height. 15 cm × 8 cm = 120 cm².

The sum of interior angles of a polygon with n sides is (n-2) × 180°. Setting this equal to 900° gives (n-2) × 180° = 900°. Solving for n gives n = 7.

Expanding (x−3)² = 9 gives x−3 = ±3, leading to solutions x=0 and x=6.

This is a difference of squares: x² - 9 = (x - 3)(x + 3) = 0. Setting each factor to zero gives x = 3 and x = -3.

1. Subtract 10 from both sides: 8x = 40
2. Divide both sides by 8: x = 5

Explanation: Increase: 15% 15% of 1,200 = 0.15 * 1,200 = 180 New rent: 1,200 + 180 = 1,320

After rotating 240°, the rider has moved past the top of the circle. The angle from the vertical is 240° - 180° = 60°. The vertical distance from the center of the circle is 50cos(60°) = 25 feet. Since the center of the circle is 50 feet above the ground, the rider's height is 50 + 25 = 75 feet.

Using the angle difference formula for cosine: cos(15°) = cos(45° - 30°) = (√6 + √2)/4.

 Rewrite in slope-intercept form: 5y = -2x + 15 y = (-2/5)x + 3. The slope is -2/5.

This is a downward-opening parabola with a vertex at (0, 2). Since it opens downwards, the y-values are less than or equal to the y-coordinate of the vertex, so y ≤ 2.

Using the Law of Cosines, the length of side a is √61.

Using the half-angle formula for sine: sin(θ/2) = √[(1 - cos(θ))/2] = √[(2/3)/2] = √(1/3) = √3/3.

Let the original price be x. After a 25% markup: 1.25x. After a 20% markdown: 1.25x * 0.80 = 1x. The final price is the same as the original, resulting in a 0% change.

The perimeter of a square is 4 × Side length = 4 × 8 = 32.

Using the quadratic formula with a=2, b=3, and c=-2 gives x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4. Therefore, x = 1/2 or x = -2.

The increase is 108,000 - 90,000 = 18,000. The percent increase is (18,000/90,000) * 100 = 20%.

Let x be the original population. After a 20% decrease: 0.80x = 8,000. Dividing both sides by 0.80: x = 10,000.

Substituting x=-1: 2(-1)² + 3(-1) - 5 = 2 - 3 - 5 = -6.

Length = Area / Width = 72 square units / 8 units = 9 units.

The range is the difference between the maximum and minimum values: 10 - 6 = 4.

12% of $80,000 is $9,600. Adding the increase: $80,000 + $9,600 = $89,600.

Average speed = Distance / Time = 750 miles / 2.5 hours = 300 mph.

Make the 'q' values the same. Multiply the first ratio by 3 and the second by 6: p:q = 15:18 and q:r = 18:24. Thus, p:r = 15:24, which simplifies to 5:8.

Using the quadratic formula with a=2, b=-1, and c=-1 gives x = (1 ± √(1 + 8)) / 4 = (1 ± 3) / 4. Therefore, x = 1 or x = -1/2.

Cost for 6 months = 6 * $25/month = $150. Total cost including joining fee = $150 + $100 = $250.

The rates are inversely proportional to the times. The ratio of times is 6:9, which simplifies to 2:3. Therefore, the ratio of rates is 3:2.

Factoring gives (x - 5)(x + 2) = 0. Therefore, x = 5 or x = -2.

Subtracting 2x from both sides gives 4x + 8 = −12. Subtracting 8 gives 4x = −20. Dividing by 4 gives x = −5.

Subtracting 3x from both sides gives 2x − 3 = 5. Adding 3 gives 2x = 8. Dividing by 2 gives x = 4.

Total pencils = 50 boxes * 20 pencils/box = 1000 pencils. Cost per pencil = $200 / 1000 pencils = $0.20/pencil.

Q1 is the median of the lower half (2, 3, 4, 4, 5), which is 4. Q3 is the median of the upper half (6, 7, 7, 7, 8), which is 7. IQR = 7 - 4 = 3.

From the second equation, y=x-3. Substituting this into the first equation gives 4x+(x-3)=11 which simplifies to 5x=14 so x=2. Substituting x=2 in y=x-3 gives y=-1.

Each term increases by 5. 10 + 5 = 15.

The volume of a rectangular prism is Length × Width × Height = 4 × 6 × 8 = 192.

30% of $3,000 = (30/100) * $3,000 = $900.

Expanding gives 6x² - 3x² - 12x + 3 = 3x² - 12x + 3.

5/6 = 10/12, 7/12 = 7/12, 2/3 = 8/12, 10/12 - 7/12 + 8/12 = 11/12

The lateral surface area of a cylinder is 2πrh: 2 * 3.14 * 4 * 10 = 251.2 units².

Subtracting 3x from both sides gives 3x + 9 = −3. Subtracting 9 gives 3x = −12. Dividing by 3 gives x = −4.

Cost for 3 days = 3 * $30 = $90. Cost for 200 miles = 200 * $0.50 = $100. Total cost = $90 + $100 = $190.

Substituting a=-2: 3(-2)² - 5(-2) + 7 = 12 + 10 + 7 = 29.

The volume of a cylinder is πr²h = 3.14 × 4² × 10 = 3.14 × 16 × 10 = 502.4.

The sequence consists of consecutive cubes: 1^3, 2^3, 3^3, 4^3, 5^3. The next term is 6^3 = 216.

Factoring gives (x - 2)(x - 4) = 0. Therefore, x = 2 or x = 4.

The volume of a cone is (1/3)πr²h: (1/3) * 3.14 * (3)² * 8 = (1/3) * 3.14 * 9 * 8 = 75.36 units³.

Substituting n = 9 into the formula gives a_9 = 2(9) + 5 = 18 + 5 = 23.

Subtract: 6.75 - 2.45 = 4.3

Expanding gives 4x - 8 + 6x + 3 = 10x - 5.

2.4 - 1.2 = 1.2, 3.8 × 1.2 = 4.56, 4.56 + 4.2 = 7.8

Subtracting 3x from both sides gives 4x + 4 = −8. Subtracting 4 gives 4x = −12. Dividing by 4 gives x = −3.

Expanding gives x² + 6x + 9 - (x² - 4) = x² + 6x + 9 - x² + 4 = 6x + 13.

Assuming we take 10 units from each jar (LCM of 5 and 9). Jar 1 has 6 sugar and 4 water. Jar 2 has approximately 4.4 sugar and 5.6 water. Adding gives 10.4 sugar and 9.6 water. Simplifying the ratio gives approximately 31:33.

Substituting n = 4 into the formula gives a_4 = 3(4^2) + 4 – 2 = 48 + 4 - 2 = 50.

Factoring gives (x + 1)(x + 7) = 0. Therefore, x = -1 or x = -7.

The volume of a sphere is (4/3)πr³ = (4/3) × 3.14 × 6³ = (4/3) × 3.14 × 216 = 904.32.

The mode is the value or range with the highest frequency, which is 30–39 as it has 15 participants.

A positive correlation means that as one variable increases, the other also tends to increase.

Substituting n = 6 into the formula gives a_6 = 6^2 – 2(6) + 3 = 36 – 12 + 3 = 27.

The ratio is 3:4 (sugar:flour). If you have 48 cups of flour, then (3/4) * 48 = 36 cups of sugar are needed.

The circumference of a circle is 2πr = 2 × 3.14 × 3 = 18.84.

2.4 - 1.2 = 1.2, 3.8 × 1.2 = 4.56, 4.56 + 4.2 = 7.8