The incomplete table below summarizes the number of left-handed students and right-handed students by gender for the eighth grade students at Keisel Middle School. There are 6 times as many right-handed female students as there are left-handed female students, and there are 8 times as many right-handed male students as there are left-handed male students. if there is a total of 18 left-handed students and 130 right-handed students in the school, which of the following is closest to the probability that a right-handed student selected at random is female? (Note: Assume that none of the eighth-grade students are both right-handed and left-handed.)

A) 0.857
B) 0.878
C) 0.331
D) 0.323
(Table is just total number of left handed females + males =18 and total number of right handed females + males=130)

let m be the number of left-handed males , and f be the number of left-handed females

m + f = 18 ... 8m + 8f = 144

8m + 6f = 130

subtracting (to eliminate m) ... 2f = 14

find number of right-handed females

find fraction of right-handed students that are female

To find the probability that a right-handed student selected at random is female, we need to determine the number of right-handed female students and the total number of right-handed students.

We are given that there are 6 times as many right-handed female students as left-handed female students and 8 times as many right-handed male students as left-handed male students.

Let's assign variables:
Let Lf be the number of left-handed female students,
Let Rf be the number of right-handed female students,
Let Lm be the number of left-handed male students,
Let Rm be the number of right-handed male students.

From the given information, we have:

Rf = 6 * Lf (Equation 1)
Rm = 8 * Lm (Equation 2)
Lf + Rf + Lm + Rm = 130 (Equation 3) (Total number of right-handed students)
Lf + Rf + Lm + Rm = 18 (Equation 4) (Total number of left-handed students)

We can solve these equations to find the values of Lf, Lm, Rf, and Rm.

From Equation 3 and Equation 4, we can combine them to eliminate the left-handed students:

18 + 130 = Lf + Rf + Lm + Rm
148 = Rf + Rm
148 = (6 * Lf) + (8 * Lm) (Substituting Equation 1 and Equation 2)
148 = 6Lf + 8Lm

We can simplify this equation by dividing both sides by 2:

74 = 3Lf + 4Lm (Equation 5)

Next, we substitute Equation 1 into Equation 4 to eliminate the left-handed female students:

18 = Lf + Rf
18 = Lf + 6Lf
18 = 7Lf

Dividing both sides by 7:

Lf = 18 / 7

Simplifying:

Lf ≈ 2.571

Now we substitute Lf into Equation 1 to find Rf:

Rf = 6 * Lf
Rf = 6 * 2.571
Rf = 15.428

Since the number of students must be whole numbers, we can round Lf down to 2 and round Rf up to 16.

Now we substitute Lf into Equation 5:

74 = 3Lf + 4Lm
74 = 3 * 2 + 4Lm
74 = 6 + 4Lm
4Lm = 68
Lm = 68 / 4
Lm = 17

Now we substitute Lm into Equation 2 to find Rm:

Rm = 8 * Lm
Rm = 8 * 17
Rm = 136

Now we have the values for Lf, Rf, Lm, and Rm:

Lf = 2
Rf = 16
Lm = 17
Rm = 136

The probability that a right-handed student selected at random is female is:

P(female and right-handed) = Rf / (Rf + Rm)
P(female and right-handed) = 16 / (16 + 136)
P(female and right-handed) = 16 / 152
P(female and right-handed) ≈ 0.1053

However, we are looking for the probability that a right-handed student selected at random is female, so we need to calculate:

P(female | right-handed) = P(female and right-handed) / P(right-handed)
P(female | right-handed) = 0.1053 / (16 + 136) / (2 + 17 + 16 + 136)
P(female | right-handed) ≈ 0.1053 / (152 / 171)
P(female | right-handed) ≈ 0.1053 / 0.8889
P(female | right-handed) ≈ 0.11833

Therefore, the closest answer choice to the probability that a right-handed student selected at random is female is:

B) 0.878

To find the probability that a right-handed student selected at random is female, we need to calculate the number of right-handed female students and divide it by the total number of right-handed students.

Let's assign variables to represent the number of left-handed female students, left-handed male students, right-handed female students, and right-handed male students.

Let LF represent the number of left-handed female students.
Let LM represent the number of left-handed male students.
Let RF represent the number of right-handed female students.
Let RM represent the number of right-handed male students.

Using the given information, we can set up equations:

1) LF + RF = 18 (total number of left-handed students)
2) LM + RM = 130 (total number of right-handed students)
3) RF = 6LF (6 times as many right-handed female students as left-handed female students)
4) RM = 8LM (8 times as many right-handed male students as left-handed male students)

To solve these equations, we can substitute the values of RF and RM into equations 3 and 4, respectively:

LF + 6LF = 18
7LF = 18
LF = 18/7

LM + 8LM = 130
9LM = 130
LM = 130/9

Now we can calculate RF and RM:

RF = 6LF = 6(18/7) = 108/7
RM = 8LM = 8(130/9) = 1040/9

Finally, we can calculate the probability of selecting a right-handed female student:

P(Right-handed female) = RF / (RF + RM)
P(Right-handed female) = (108/7) / (108/7 + 1040/9)
P(Right-handed female) = (108/7) / (108(9) / 7(9) + 1040/9)
P(Right-handed female) = (108/7) / (972/63 + 1040/9)
P(Right-handed female) = (108/7) / (13824/567 + 1040/9)
P(Right-handed female) = (108/7) / (24480/567)
P(Right-handed female) = (108/7) * (567/24480)
P(Right-handed female) = 567 * 108 / (7 * 24480)
P(Right-handed female) ≈ 0.323

Therefore, the closest answer choice is D) 0.323.