80% of the students voted for John as class president. If he received 128 votes, how many students voted?
Let's use a proportion to solve for the total number of students who voted:
80% = 128/x
Here, x represents the total number of students who voted. To solve for x, we'll cross-multiply:
0.8x = 128
And now we'll divide both sides by 0.8:
x = 160
So there were 160 students who voted in the election.
Which equation represents the line that passes through the points (0, 3) and (4, 2)
We can use the point-slope form of a linear equation to find the equation of the line that passes through the points (0, 3) and (4, 2).
Point-slope form: y - y1 = m(x - x1)
We first need to find the slope of the line using the two points:
slope = (y2 - y1) / (x2 - x1)
slope = (2 - 3) / (4 - 0)
slope = -1/4
Now we can choose either point and substitute the values into the point-slope form:
y - 3 = (-1/4)(x - 0)
Simplifying:
y - 3 = (-1/4)x
y = (-1/4)x + 3
So the equation that represents the line passing through the points (0, 3) and (4, 2) is:
y = (-1/4)x + 3.
Which equation represents the line that passes through the points (0, 3) and (4, 2) ?
A. (y - 2) = - 1/4 * (x - 4)
B. (y + 2) = - 1/4 * (x + 4)
C. (y - 4) = - 1/4 * (x - 2)
D. (y + 4) = - 1/4 * (x + 2)
The correct equation is A. (y - 2) = - 1/4 * (x - 4).
Estimate: 20 is 22% of what number?
To solve the problem, we can use the formula:
part/whole = percentage/100
where "part" is the number we are trying to find, "whole" is the total number we're looking for (in this case), and "percentage" is 22.
So, we can write:
20/whole = 22/100
To solve for "whole", we can cross-multiply:
20 * 100 = 22 * whole
2000 = 22 * whole
To isolate "whole", we can divide both sides by 22:
whole = 2000/22
whole ≈ 90.91
Therefore, 20 is approximately 22% of 90.91.
Determine the vertex form and the maximum or minimum value of the function. f(x) = 2x ^ 2 + 8x + 3
We can convert the quadratic function f(x) = 2x^2 + 8x + 3 to vertex form by completing the square:
f(x) = 2(x^2 + 4x) + 3
f(x) = 2(x^2 + 4x + 4 - 4) + 3
f(x) = 2((x + 2)^2 - 4) + 3
f(x) = 2(x + 2)^2 - 5
Therefore, the vertex form of the function is f(x) = 2(x + 2)^2 - 5, and the vertex is (-2, -5).
Since the coefficient of the x^2 term is positive, the parabola opens upwards. Therefore, the vertex is the minimum point of the function, and the minimum value of the function is -5.
10/9 - 7/6 =
We need to find a common denominator to subtract these fractions. The denominator should be the least common multiple (LCM) of 9 and 6, which is 18.
10/9 - 7/6 can be written as:
(10/9) * (2/2) - (7/6) * (3/3)
= 20/18 - 21/18
Now we can subtract the numerators:
= (20 - 21)/18
= -1/18
Therefore, 10/9 - 7/6 = -1/18.
1.6 divided by 100
To divide 1.6 by 100, we can simply move the decimal point two places to the left:
1.6 ÷ 100 = 0.016
40 1/10 × 2 1/2 =
To multiply 40 1/10 by 2 1/2, we need to convert the mixed numbers to improper fractions:
40 1/10 = 401/10
2 1/2 = 5/2
Now we can multiply these fractions:
401/10 × 5/2
We can simplify the fractions before multiplying:
401/10 = 40 + 1/10 = 400/10 + 1/10 = 401/10
So we have:
401/10 × 5/2 = 401/10 × 5/2
= (401 × 5)/(10 × 2)
= 2005/20
We can simplify the fraction by dividing both the numerator and denominator by 5:
2005/20 = 401/4
Therefore, 40 1/10 × 2 1/2 = 401/4 or 100.25.
If the replacement set is the set of integers, find the solution set for the inequality: x - 9 < - 15
To solve for x in the inequality x - 9 < - 15, we'll isolate x by adding 9 to both sides:
x - 9 + 9 < - 15 + 9
x < - 6
Therefore, the solution set for the inequality when the replacement set is the set of integers is all integers less than -6.
In set-builder notation, we can write the solution set as:
{x | x < -6, x ∈ ℤ}.
If the replacement set is the set of integers, find the solution set for the inequality: x - 9 < - 15
A. { - 8, - 7, - 6 ,...}
B. { - 9, - 8, - 7 ,...}
C. {...,-8,-7,-6}
D. {...,-9,-8,-7}
The solution set for the inequality x - 9 < - 15, when the replacement set is the set of integers, is the set of integers less than -6.
The correct set is A. { - 8, - 7, - 6 ,...}.
Convert 23 L / m * in to milliliters per second (mL / s)
We can use dimensional analysis to convert 23 L/m*in to mL/s:
23 L/m*in * (1000 mL/L)/(60 s/min)
First, we multiplied by the conversion factor (1000 mL/L) to convert liters to milliliters.
Then, we multiplied by the conversion factor (1 min/60 s) to convert minutes to seconds.
Simplifying:
= 383.333... mL/s
Therefore, 23 L/m*in is equivalent to 383.333... mL/s (rounded to three decimal places).
A bag contains 4 green marbles, 3 red marbles, and 7 blue marbles. One marble is taken from the bag and put back after checking its color. A second marble is then taken out. What is the probability that the first is blue and the second red?
Since a marble is put back in the bag after the first draw, the number of marbles in the bag remains the same for the second draw. Therefore, the probability of drawing a blue marble on the first draw is:
P(blue on first draw) = 7/14 = 1/2
Since the marble is put back in the bag, the probability of drawing a red marble on the second draw is:
P(red on second draw) = 3/14
Now we can multiply the probabilities of the individual events to get the combined probability of both events happening:
P(blue on first draw and red on second draw) = P(blue on first draw) × P(red on second draw)
P(blue on first draw and red on second draw) = (1/2) × (3/14)
P(blue on first draw and red on second draw) = 3/28
Therefore, the probability that the first marble drawn is blue and the second marble drawn is red is 3/28.
What is the area of a circular region whose diameter is 12 centimeters?
A. 36pi square cm
B. 36pi cm
C. 12pi cm
D. 24pi square cm
The formula for the area (A) of a circle is:
A = πr^2
where r is the radius of the circle. We can use the diameter (d) to find the radius (r) using the formula:
d = 2r
or
r = d/2
In this case, the diameter is 12 centimeters, so the radius is:
r = 12/2 = 6 centimeters
Now we can use the formula for the area of a circle:
A = πr^2
A = π(6^2)
A = 36π
Therefore, the area of the circular region with a diameter of 12 centimeters is 36π square centimeters, which is answer choice A.
Find the x-intercept of the line: 8x - 4y = - 24
A. 3
B. -3
C. 6
D. -6
To find the x-intercept of the line 8x - 4y = -24, we substitute 0 for y and solve for x:
8x - 4(0) = -24
8x = -24
x = -24/8
x = -3
Therefore, the x-intercept of the line is -3, which is answer choice B.
Find the slope of the line through the points (- 1, - 4) and (2, - 2)
The formula for the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is:
m = (y2 - y1) / (x2 - x1)
We can use this formula to find the slope of the line passing through the points (-1, -4) and (2, -2):
m = (-2 - (-4)) / (2 - (-1))
m = 2 / 3
Therefore, the slope of the line passing through the points (-1, -4) and (2, -2) is 2/3.
Evaluate: 5 - 3|3(- 6)|
A. -49
B. 49
C. -59
D. 59
First, we need to simplify the absolute value expression inside the parentheses:
|3(-6)| = |-18| = 18
Now we can substitute the absolute value result into the original expression:
5 - 3|3(-6)| = 5 - 3(18)
= 5 - 54
= -49
Therefore, the value of the expression 5 - 3|3(-6)| is -49, which is answer choice A.
(8.1 × 10 ^ 3)/(2.7 × 10 ^ 6) =
A. 3 × 10 ^ 11
B. 3 × 10 ^ - 5
C. 3 × 10 ^ 6
D. 3 × 10 ^ - 11
To divide two numbers in scientific notation, we divide their coefficients and subtract their exponents.
So for (8.1 × 10^3) / (2.7 × 10^6), we have:
(8.1 / 2.7) × 10^(3-6)
3 × 10^(-3)
Therefore, (8.1 × 10^3) / (2.7 × 10^6) = 3 × 10^(-3), which is answer choice B.
25 7/12 A tree is feet high. Richard cuts off the 8 7/8 top feet. How tall is the tree now?
A. 17 17/24 feet
B. 16 17/24 feet
C. 17 7/24 feet
D.16 7/24 feet
To find the height of the tree after Richard cuts off the top 8 7/8 feet, we need to subtract 8 7/8 from 25 7/12:
25 7/12 - 8 7/8
We can convert both mixed numbers to improper fractions to make the subtraction easier:
25 7/12 = (25 × 12 + 7) / 12 = 307 / 12
8 7/8 = (8 × 8 + 7) / 8 = 71 / 8
Now we can subtract the fractions:
307/12 - 71/8
We need to find a common denominator for 12 and 8. The LCM of 12 and 8 is 24, so we can convert both fractions to have a denominator of 24:
(307/12) × (2/2) = 614/24
(71/8) × (3/3) = 213/24
Now we can subtract the fractions:
614/24 - 213/24 = 401/24
We can convert the improper fraction back to a mixed number to get the answer in the required format:
401/24 = 16 17/24
Therefore, the tree is now 16 17/24 feet tall, which is answer choice B.