ACT G01 Math Solutions: Step-by-Step Explanations

Hey guys! Let's break down the ACT G01 Math section. This guide will walk you through each question, offering clear, step-by-step explanations to help you understand the solutions. Whether you're stuck on a specific problem or just want to review, this is the place to be. So, grab your pencil and let’s get started!

Question 1

Original Question: If x + y = 23 and x - y = 7, then x = ?

Improved Question: Given two equations, x + y = 23 and x - y = 7, what is the value of x?

Explanation:

Alright, let's solve this system of equations. This is a classic setup where we can use either substitution or elimination. Elimination is going to be the easier route here. We have:

• x + y = 23
• x - y = 7

Notice that the y terms have opposite signs. If we add the two equations together, the y terms will cancel out:

(x + y) + (x - y) = 23 + 7

This simplifies to:

2x = 30

Now, just divide both sides by 2 to solve for x:

x = 15

So, the value of x is 15. That wasn't too bad, was it? Remember, spotting those easy elimination setups can save you loads of time!

Question 2

Original Question: What is the least common multiple of 30, 20, and 70?

Improved Question: Determine the least common multiple (LCM) of the numbers 30, 20, and 70.

Explanation:

Okay, let's tackle finding the least common multiple (LCM). First, we'll break down each number into its prime factors:

• 30 = 2 * 3 * 5
• 20 = 2 * 2 * 5 = 2² * 5
• 70 = 2 * 5 * 7

Now, to find the LCM, we need to take the highest power of each prime factor that appears in any of the factorizations. Let's list the prime factors we have: 2, 3, 5, and 7.

• The highest power of 2 is 2² (from the factorization of 20).
• The highest power of 3 is 3¹ (from the factorization of 30).
• The highest power of 5 is 5¹ (it appears in all factorizations, but only to the first power).
• The highest power of 7 is 7¹ (from the factorization of 70).

Multiply these together to get the LCM:

LCM = 2² * 3 * 5 * 7 = 4 * 3 * 5 * 7 = 12 * 35 = 420

So, the least common multiple of 30, 20, and 70 is 420. Got it? Great!

Question 3

Original Question: The expression (x - 3)² is equivalent to:

Improved Question: Expand the expression (x - 3)² and determine its equivalent form.

Explanation:

Alright, let's expand this binomial squared. Remember the formula (a - b)² = a² - 2ab + b². In our case, a = x and b = 3.

So, (x - 3)² = x² - 2(x)(3) + 3²

Simplify this:

x² - 6x + 9

Therefore, the expression (x - 3)² is equivalent to x² - 6x + 9. Easy peasy!

Question 4

Original Question: If 3x + 5 = 23, then x = ?

Improved Question: Solve the equation 3x + 5 = 23 for the value of x.

Explanation:

Let's isolate x in this equation. First, subtract 5 from both sides:

3x + 5 - 5 = 23 - 5

This simplifies to:

3x = 18

Now, divide both sides by 3 to solve for x:

x = 18 / 3

x = 6

So, the value of x is 6. Nailed it!

Question 5

Original Question: In the standard (x, y) coordinate plane, what is the slope of the line containing the points (3, 4) and (5, 8)?

Improved Question: Determine the slope of the line that passes through the points (3, 4) and (5, 8) in the standard (x, y) coordinate plane.

Explanation:

To find the slope, we'll use the formula: slope = (y₂ - y₁) / (x₂ - x₁). We have two points: (3, 4) and (5, 8).

Let's label them:

• (x₁, y₁) = (3, 4)
• (x₂, y₂) = (5, 8)

Now, plug these values into the slope formula:

slope = (8 - 4) / (5 - 3) = 4 / 2 = 2

So, the slope of the line is 2. Keep up the great work!

Question 6

Original Question: 20% of 80 is equal to what percent of 50?

Improved Question: Twenty percent of 80 is equivalent to what percentage of 50?

Explanation:

First, let's find 20% of 80:

20% of 80 = 0.20 * 80 = 16

Now, we want to know what percent of 50 is equal to 16. Let's set up an equation:

p% of 50 = 16

Convert the percentage to a decimal:

(p/100) * 50 = 16

Simplify:

0.5 * p = 16

Now, divide both sides by 0.5:

p = 16 / 0.5 = 32

So, 20% of 80 is equal to 32% of 50. You're on a roll!

Question 7

Original Question: The area of a circle with a diameter of 10 inches is:

Improved Question: Calculate the area of a circle that has a diameter of 10 inches.

Explanation:

First, remember that the radius (r) is half of the diameter (d). So, if the diameter is 10 inches, the radius is 5 inches.

r = d / 2 = 10 / 2 = 5 inches

Now, use the formula for the area of a circle, which is A = πr²:

A = π * (5)² = π * 25 = 25π

So, the area of the circle is 25π square inches. Excellent!

Question 8

Original Question: If f(x) = 2x² - 3x + 4, then f(-2) = ?

Improved Question: Given the function f(x) = 2x² - 3x + 4, evaluate f(-2).

Explanation:

We need to substitute -2 for x in the function:

f(-2) = 2*(-2)² - 3*(-2) + 4

Now, simplify:

f(-2) = 2*(4) + 6 + 4

f(-2) = 8 + 6 + 4

f(-2) = 18

So, f(-2) = 18. Fantastic job!

Question 9

Original Question: Which of the following is a solution to the equation x² + 5x + 6 = 0?

Improved Question: Find a solution to the quadratic equation x² + 5x + 6 = 0.

Explanation:

To solve this quadratic equation, we can factor it. We're looking for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.

So, we can factor the equation as:

(x + 2)(x + 3) = 0

Now, set each factor equal to zero and solve for x:

x + 2 = 0 => x = -2

x + 3 = 0 => x = -3

So, the solutions are x = -2 and x = -3. You're doing great!

Question 10

Original Question: What is the value of |5 - 9|?

Improved Question: Determine the absolute value of the expression |5 - 9|.

Explanation:

First, evaluate the expression inside the absolute value:

5 - 9 = -4

Now, take the absolute value of -4. Remember, the absolute value of a number is its distance from zero, so it's always non-negative.

|-4| = 4

So, the value of |5 - 9| is 4. Awesome!

Conclusion

Alright, we've walked through several ACT G01 Math questions step by step. I hope these explanations were helpful! Remember to practice regularly and review these concepts. You got this! Keep up the hard work, and you'll be well-prepared for the ACT math section. Good luck, and happy studying!