The incomplete table above summarizes the number of left-handed students and right-handed students by gender for the eighth grade students at Keisel Middle School. There are 5 times as many right-handed female students as there are left-handed female students, and there are 9 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 122 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) 0.410
B) 0.357
C) 0.333
D) 0.250
Answers
ᴀɴsᴡᴇʀ
ʟᴇᴛ x ʙᴇ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ʟᴇғᴛ ʜᴀɴᴅᴇᴅ ғᴇᴍᴀʟᴇ sᴛᴜᴅᴇɴᴛs ᴀɴᴅ ʟᴇᴛ ʏ ʙᴇ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ʟᴇғᴛ ʜᴀɴᴅᴇᴅ ᴍᴀʟᴇ sᴛᴜᴅᴇɴᴛs.
ᴛʜᴇɴ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ʀɪɢʜᴛ ʜᴀɴᴅᴇᴅ ғᴇᴍᴀʟᴇ sᴛᴜᴅᴇɴᴛs ᴡɪʟʟ ʙᴇ 5x ᴀɴᴅ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ʀɪɢʜᴛ ʜᴀɴᴅᴇᴅ ᴍᴀʟᴇ sᴛᴜᴅᴇɴᴛs ᴡɪʟʟ ʙᴇ 9ʏ. sɪɴᴄᴇ ᴛʜᴇ ᴛᴏᴛᴀʟ ɴᴜᴍʙᴇʀ ᴏғ ʟᴇғᴛ ʜᴀɴᴅᴇᴅ sᴛᴜᴅᴇɴᴛs ɪs 18 ᴀɴᴅ ᴛʜᴇ ᴛᴏᴛᴀʟ ɴᴜᴍʙᴇʀ ᴏғ ʀɪɢʜᴛ ʜᴀɴᴅᴇᴅ sᴛᴜᴅᴇɴᴛs ɪs 122, ᴛʜᴇ ғᴏʟʟᴏᴡɪɴɢ sʏsᴛᴇᴍ ᴏғ ᴇǫᴜᴀᴛɪᴏɴs ᴍᴜsᴛ ʙᴇ sᴀᴛɪsғɪᴇᴅ.
x+ʏ=18 ᴀɴᴅ 5x+9ʏ=122
sᴏʟᴠɪɴɢ ᴛʜɪs sʏsᴛᴇᴍ ɢɪᴠᴇs x=10 ᴀɴᴅ ʏ=8.
ᴛʜᴜs, 50 ᴏғ ᴛʜᴇ 122 ʀɪɢʜᴛ ʜᴀɴᴅᴇᴅ sᴛᴜᴅᴇɴᴛs ᴀʀᴇ ғᴇᴍᴀʟᴇ. ᴛʜᴇʀᴇғᴏʀᴇ, ᴛʜᴇ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ᴛʜᴀᴛ ᴀ ʀɪɢʜᴛ ʜᴀɴᴅᴇᴅ sᴛᴜᴅᴇɴᴛ sᴇʟᴇᴄᴛᴇᴅ ᴀᴛ ʀᴀɴᴅᴏᴍ ɪs ғᴇᴍᴀʟᴇ ɪs ᴛʜᴇ ғʀᴀᴄᴛɪᴏɴ
122
50
, ᴡʜɪᴄʜ ᴛᴏ ᴛʜᴇ ɴᴇᴀʀᴇsᴛ ᴛʜᴏᴜsᴀɴᴅᴛʜ ɪs 0.410.
ʙᴇ ʜᴀᴘᴘʏ .. :)
Answer:
Let x be the number of left handed female students and let y be the number of left handed male students.
Then the number of right handed female students will be 5x and the number of right handed male students will be 9y. Since the total number of left handed students is 18 and the total number of right handed students is 122, the following system of equations must be satisfied.
x+y=18 and 5x+9y=122
Solving this system gives x=10 and y=8.
Thus, 50 of the 122 right handed students are female. Therefore, the probability that a right handed student selected at random is female is the fraction
/frac122
50
, which to the nearest thousandth is 0.410.