ACT Quiz 1 (Alg II+)

1) If 3x + 7 = 15x - 2, then x = ?

5/18

5/12

1/2

3/4

No solution

To find the value of x, we need to solve the equation 3x + 7 = 15x - 2. We can start by simplifying the equation by subtracting 3x from both sides, which gives us 7 = 12x - 2. Next, we add 2 to both sides to isolate the term with x, resulting in 9 = 12x. Finally, we divide both sides by 12 to solve for x, giving us x = 9/12, which simplifies to x = 3/4. Therefore, the correct answer is 3/4.

Explanation

To find the value of x, we need to solve the equation 3x + 7 = 15x - 2. We can start by simplifying the equation by subtracting 3x from both sides, which gives us 7 = 12x - 2. Next, we add 2 to both sides to isolate the term with x, resulting in 9 = 12x. Finally, we divide both sides by 12 to solve for x, giving us x = 9/12, which simplifies to x = 3/4. Therefore, the correct answer is 3/4.

2) Two parallel lines are cut by a transversal as shown at the left. What is the sum of angles 2, 5 and 8?

65

115

180

245

360

When two parallel lines are cut by a transversal, corresponding angles are congruent. In the given diagram, angles 2 and 8 are corresponding angles, so they are congruent. Therefore, angle 2 + angle 8 = angle 5. Thus, the sum of angles 2, 5, and 8 is 2(angle 5). Since angle 5 is a straight angle, it measures 180 degrees. Therefore, 2(angle 5) = 2(180) = 360 degrees.

Explanation

When two parallel lines are cut by a transversal, corresponding angles are congruent. In the given diagram, angles 2 and 8 are corresponding angles, so they are congruent. Therefore, angle 2 + angle 8 = angle 5. Thus, the sum of angles 2, 5, and 8 is 2(angle 5). Since angle 5 is a straight angle, it measures 180 degrees. Therefore, 2(angle 5) = 2(180) = 360 degrees.

3) The ratio of girls to boys in a club is 6:7. If the total number of students is 39, how many more boys are there than girls?

1

2

3

13

21

The ratio of girls to boys in the club is 6:7, which means that for every 6 girls, there are 7 boys. If the total number of students is 39, we can set up the equation 6x + 7x = 39, where x represents the common ratio. Solving for x, we find that x = 3. Therefore, there are 6x = 6 * 3 = 18 girls and 7x = 7 * 3 = 21 boys. The difference between the number of boys and girls is 21 - 18 = 3.

Explanation

The ratio of girls to boys in the club is 6:7, which means that for every 6 girls, there are 7 boys. If the total number of students is 39, we can set up the equation 6x + 7x = 39, where x represents the common ratio. Solving for x, we find that x = 3. Therefore, there are 6x = 6 * 3 = 18 girls and 7x = 7 * 3 = 21 boys. The difference between the number of boys and girls is 21 - 18 = 3.

4) When 91/333 is expressed as a decimal number what is the 60th digit after the decimal point?

0

1

2

3

7

To find the 60th digit after the decimal point when expressing 91/333 as a decimal number, we need to divide 91 by 333 using long division. The remainder after each division will give us the digits after the decimal point. After performing the long division, we find that the remainder for the 60th digit is 3. Therefore, the answer is 3.

Explanation

To find the 60th digit after the decimal point when expressing 91/333 as a decimal number, we need to divide 91 by 333 using long division. The remainder after each division will give us the digits after the decimal point. After performing the long division, we find that the remainder for the 60th digit is 3. Therefore, the answer is 3.

5) If f(x) = x² + 3x, then f(b - 3) = ?

B2 + 9

B2 + 9 + 3x

B2 - b - 6

B2 + 9b + 18

B2 - 3b

The given function f(x) = x^2 + 3x. To find f(b - 3), we substitute (b - 3) in place of x in the function. So, f(b - 3) = (b - 3)^2 + 3(b - 3). Simplifying this expression gives b^2 - 6b + 9 + 3b - 9, which further simplifies to b^2 - 3b. Therefore, the correct answer is b^2 - 3b.

Explanation

The given function f(x) = x^2 + 3x. To find f(b - 3), we substitute (b - 3) in place of x in the function. So, f(b - 3) = (b - 3)^2 + 3(b - 3). Simplifying this expression gives b^2 - 6b + 9 + 3b - 9, which further simplifies to b^2 - 3b. Therefore, the correct answer is b^2 - 3b.

6) Which of the following is the correct simplification of the expression (4 + 3i) - (6 - 5i)?

-2 + 8i

-2 - 2i

-2 - 8i

2 + 8i

2 - 8i

To simplify the expression (4 + 3i) - (6 - 5i), we need to distribute the negative sign to both terms inside the parentheses. This gives us 4 + 3i - 6 + 5i. Combining like terms, we get -2 + 8i. Therefore, the correct simplification of the expression is -2 + 8i.

Explanation

To simplify the expression (4 + 3i) - (6 - 5i), we need to distribute the negative sign to both terms inside the parentheses. This gives us 4 + 3i - 6 + 5i. Combining like terms, we get -2 + 8i. Therefore, the correct simplification of the expression is -2 + 8i.

7) At the Farmer's Market, I can buy a coffee for $3 and loaves of bread for $5. Over the summer, if I buy 26 items and spend $90, how many coffees did I buy?

5

10

15

20

25

If the person bought 26 items and spent $90, and each coffee costs $3, then the person must have bought 20 coffees in order to reach a total of $90.

Explanation

If the person bought 26 items and spent $90, and each coffee costs $3, then the person must have bought 20 coffees in order to reach a total of $90.

8) What is the y - intercept of the line that passes through the points (2, 35) and (7, 50)?

3

10

12

27

29

The y-intercept of a line is the value of y when x is equal to 0. To find the equation of the line that passes through the points (2, 35) and (7, 50), we can use the slope-intercept form of a line, which is y = mx + b. We first need to find the slope (m) of the line using the formula (y2 - y1)/(x2 - x1), which gives us (50 - 35)/(7 - 2) = 15/5 = 3. Now we can substitute one of the points into the equation to solve for the y-intercept (b). Using the point (2, 35), we get 35 = 3(2) + b, which simplifies to b = 35 - 6 = 29. Therefore, the y-intercept of the line is 29.

Explanation

The y-intercept of a line is the value of y when x is equal to 0. To find the equation of the line that passes through the points (2, 35) and (7, 50), we can use the slope-intercept form of a line, which is y = mx + b. We first need to find the slope (m) of the line using the formula (y2 - y1)/(x2 - x1), which gives us (50 - 35)/(7 - 2) = 15/5 = 3. Now we can substitute one of the points into the equation to solve for the y-intercept (b). Using the point (2, 35), we get 35 = 3(2) + b, which simplifies to b = 35 - 6 = 29. Therefore, the y-intercept of the line is 29.

9) Point P (1, 3) and Point Q (6, -9) are points on the (x, y) coordinate plane. Find the distance between the two points.

3

9

11

12

13

To find the distance between two points on a coordinate plane, we can use the distance formula, which is derived from the Pythagorean theorem. The formula is: distance = √((x2 - x1)^2 + (y2 - y1)^2). In this case, the coordinates of point P are (1, 3) and the coordinates of point Q are (6, -9). Plugging these values into the formula, we get: distance = √((6 - 1)^2 + (-9 - 3)^2) = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13. Therefore, the distance between point P and point Q is 13.

Explanation

To find the distance between two points on a coordinate plane, we can use the distance formula, which is derived from the Pythagorean theorem. The formula is: distance = √((x2 - x1)^2 + (y2 - y1)^2). In this case, the coordinates of point P are (1, 3) and the coordinates of point Q are (6, -9). Plugging these values into the formula, we get: distance = √((6 - 1)^2 + (-9 - 3)^2) = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13. Therefore, the distance between point P and point Q is 13.

10) There are five pig weights (in pounds) from the fair: 250, 270, 260, 275, 300. Approximate the standard deviation for this data set.

17

21

24

49

271

The given data set consists of five pig weights: 250, 270, 260, 275, and 300 pounds. To approximate the standard deviation, we can calculate the mean of the data set, which is 271 pounds. Then, we subtract each value from the mean, square the result, and calculate the average of these squared differences. Taking the square root of this average gives us the standard deviation. In this case, the calculated standard deviation is approximately 17 pounds.

Explanation

The given data set consists of five pig weights: 250, 270, 260, 275, and 300 pounds. To approximate the standard deviation, we can calculate the mean of the data set, which is 271 pounds. Then, we subtract each value from the mean, square the result, and calculate the average of these squared differences. Taking the square root of this average gives us the standard deviation. In this case, the calculated standard deviation is approximately 17 pounds.