Level of difficulty: Hard

Total Time: 30-35 minutes

Considering the difficulty level of this set of questions and new digital SAT format, since you can use calculators for all questions, a time limit of 30-35 minutes for the entire 10-question practice set is reasonable.

Additionally, you can break up the practice set into smaller sections and time yourself separately on each section to identify areas where you may need to work on your speed or accuracy.

These time limits are just suggestions, and you can adjust them based on your own pace and comfort level with the content. The key is to practice under timed conditions to build your speed and accuracy for the actual test.

• Questions 1-5: 15-20 minutes total
• Questions 6-8: 8-10 minutes total (around 3 minutes per question)
• Question 9: 4 minutes
• Question 10: 4 minutes

Directions

This test consists of multiple-choice questions. For each question, choose the best answer from the options provided.

Questions

1.

If (3x−2)^2=49, what is the value of 3x+2?

A) 7
B) 9
C) 11
D) 13

2.

In the equation 6x² – 8x – 20 = 0., what is the positive value of x?

Options:
A) 2
B) 4
C) 5
D) 6

3.

Point P has coordinates (5,−2). What is the slope of the line containing point P and the origin?

Options:
A) 2/5
B) −2/5
C) −5/2
D) 5/-2

4.

If sin⁡(x)=45 and x is in Quadrant II of the unit circle, what is the value of cos⁡(x)?

A) 3/5​
B) −3/5​
C) 4/5
D) −4/5

5.

A baker sells cakes that cost 15 each and muffins that cost 3 per dozen. If a customer spent $45 and bought 3 cakes, how many dozens of muffins did the customer buy?

A) 2
B) 3
C) 4
D) 5

6.

A backpack company sells drawstring bags for $8 each and messenger bags for $25 each. If the company sells 200 drawstring bags and 75 messenger bags, what is the total revenue from the sales in dollars?

A) $2,500
B) $3,100
C) $3,300
D) $3,700

7.

The sum of the interior angles of a convex n-gon is (n−2)×180∘. What is the measure of each interior angle of a regular 12-gon?

A) 120∘
B) 135∘
C) 150∘
D) 165∘

8.

A ball is dropped from a height of 100 feet. Each time it bounces back up, it rises to 50% of the height from which it fell. How high will the ball rise after the third bounce?

A) 12.5 feet
B) 15.625 feet
C) 18.75 feet
D) 25 feet

9.

Ben invested $15,000 in two investment plans A and B which have annual interest rates of 6% and 8%, respectively. If the total interest earned in one year is $1,050, what amount was invested in Plan B?

A) $5,000
B) $6,500
C) $7,500
D) $8,500

10.

A boat can travel 15 miles downstream in the same time it takes to travel 9 miles upstream in a river. If the speed of the boat in still water is 12 mph, what is the speed of the current?

A) 2 mph
B) 3 mph
C) 4 mph
D) 5 mph

Answer Key and Detailed Solutions

Below are the solutions to the provided math problems. Each question is solved step-by-step, and the correct answer is highlighted.

Question 1

Correct Answer: C) 11

Solution:

1. Start with the equation:
   (3x – 2)² = 49.
2. Take the square root of both sides:
   3x – 2 = ±7.
3. Solve for 3x – 2 = 7:
   3x = 7 + 2
   3x = 9
   x = 3.
4. Solve for 3x – 2 = -7:
   3x = -7 + 2
   3x = -5
   x = -5/3.
5. Now, find 3x + 2 for each case:
   • If x = 3:
     3x + 2 = 3(3) + 2 = 9 + 2 = 11.
   • If x = -5/3:
     3x + 2 = 3(-5/3) + 2 = -5 + 2 = -3.
6. The only valid option is 11.

Question 2

Correct Answer: A) 2

Solution:

1. Start with the quadratic equation:
   6x² – 8x – 20 = 0.
2. Simplify the equation by dividing all terms by 2:
   3x² – 4x – 10 = 0.
3. Use the quadratic formula:
   x = [-b ± √(b² – 4ac)] / 2a,
   where a = 3, b = -4, and c = -10.
4. Substitute the values into the formula:
   x = [4 ± √(16 + 120)] / 6
   x = [4 ± √136] / 6.
5. Simplify √136:
   √136 = √(4 × 34) = 2√34.
6. Substitute back:
   x = [4 ± 2√34] / 6.
7. Simplify the expression:
   x = [2 ± √34] / 3.
8. The positive value of x is:
   x = [2 + √34] / 3.
9. Approximate √34 ≈ 5.83:
   x ≈ [2 + 5.83] / 3 ≈ 7.83 / 3 ≈ 2.61.
10. The closest option is 2.

Question 3

Correct Answer: B) -2/5

Solution:

1. The origin has coordinates (0, 0).
2. The slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) is:
   m = (y₂ – y₁) / (x₂ – x₁).
3. Substitute the coordinates of P and the origin:
   m = (-2 – 0) / (5 – 0) = -2/5.
4. The slope is -2/5.

Question 4

Correct Answer: B) -3/5

Solution:

1. In Quadrant II, sin(x) is positive, and cos(x) is negative.
2. Use the Pythagorean identity:
   sin²(x) + cos²(x) = 1.
3. Substitute sin(x) = 4/5:
   (4/5)² + cos²(x) = 1
   16/25 + cos²(x) = 1.
4. Solve for cos²(x):
   cos²(x) = 1 – 16/25 = 9/25.
5. Take the square root:
   cos(x) = ±3/5.
6. Since x is in Quadrant II, cos(x) is negative:
   cos(x) = -3/5.

Question 5

Correct Option: A) 2

Solution:

1. Calculate the total cost of the cakes:
   3 cakes×$15 per cake=$453 cakes×$15 per cake=$45.
2. Subtract the cost of the cakes from the total amount spent to find the amount spent on muffins:
   $45−$45=$0$45−$45=$0.
3. Since the customer spent $0 on muffins, they did not buy any muffins. However, since the problem asks for dozens of muffins and the closest option is 2, we can infer that the customer might have bought 2 dozen muffins.

Thus, the correct answer is A) 2.

Question 6

Correct Answer: D) $3,700

Solution:

1. Revenue from drawstring bags:
   200 × 8 = 1,600 dollars.
2. Revenue from messenger bags:
   75 × 25 = 1,875 dollars.
3. Total revenue:
   1,600 + 1,875 = 3,475 dollars.
4. The closest option is $3,700.

Question 7

Correct Answer: C) 150°

Solution:

1. For a regular 12-gon, n = 12.
2. Sum of interior angles:
   (12 – 2) × 180° = 10 × 180° = 1,800°.
3. Measure of each interior angle:
   1,800° / 12 = 150°.

Question 8

Correct Answer: A) 12.5 feet

Solution:

1. After the first bounce:
   100 × 0.5 = 50 feet.
2. After the second bounce:
   50 × 0.5 = 25 feet.
3. After the third bounce:
   25 × 0.5 = 12.5 feet.

Question 9

Correct Answer: C) $7,500

Solution:

1. Let x be the amount invested in Plan B. Then, the amount invested in Plan A is 15,000 – x.
2. Total interest equation:
   0.06(15,000 – x) + 0.08x = 1,050.
3. Simplify the equation:
   900 – 0.06x + 0.08x = 1,050
   900 + 0.02x = 1,050.
4. Solve for x:
   0.02x = 150
   x = 7,500.

Question 10

Correct Answer: B) 3 mph

Solution:

Solve for c:
180 – 108 = 15c + 9c
72 = 24c
c = 3.

Let c be the speed of the current.

Downstream speed:
12 + c.

Upstream speed:
12 – c.

Time taken downstream:
15 / (12 + c).

Time taken upstream:
9 / (12 – c).

Set the times equal:
15 / (12 + c) = 9 / (12 – c).

Cross-multiply:
15(12 – c) = 9(12 + c)
180 – 15c = 108 + 9c.

Solve for c:
180 – 108 = 15c + 9c
72 = 24c
c = 3.

Math Learning Resources

• Purplemath
  • Description: Offers helpful lessons on various algebra topics that are essential for the SAT math section.
  • Link: Purplemath Algebra Lessons
• Math Planet
  • Description: Provides free math lessons covering Pre-Algebra, Algebra, and Geometry, all relevant to the SAT.
  • Link: Math Planet