SOLUTION: The ratio of freshmen to sophomores to juniors to seniors of a certain high school is 5:5:4:2. Which of the following CANNOT be the total number of the students? A. 560 B. 680

Question 1117924: The ratio of freshmen to sophomores to juniors to seniors of a certain high school is 5:5:4:2. Which of the following CANNOT be the total number of the students?
A. 560
B. 680
C. 1,024
D. 1,600
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
.

The numbers under possible options must be multiple of the number of 5+5+4+2= 16. It is the necessary and sufficient condition for each option to be true. The value 680 under the option B is not multiple of 16, so this option CANNOT be the total number of the students. The other three options are possible.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your ratio is 5:5:4:2

if you sum up the ratio, then you get a total of 16.

that means the ratio of freshman, sophomores, juniors, seniors is equal to 5/16, 5/16, 4/16, 2/16 of the total number of students who attend the school.

what this says is, that for every 16 students, 5 are freshmen, 5 are sophomores, 4 are juniors, 2 are freshmen.

the total number of students needs to be an integer since you are dealing with counting the number of students who attend the school.

take your possible answers and divide them by 16.

A. 560 / 16 = 35
B. 680 / 16 = 42.5 ***** not an integer
C. 1,024 / 16 = 64
D. 1,600 / 16 = 100

since the total count of students is an integer in all but selection B, then selection B is the solution where the total number of students is not possible, since the total number of students needs to be an integer.

when you have a ratio, given in the form that you see, you can convert it into a proportion of the total by adding up the ratios and using that as a divisor.

your ratio is 5:5:4:2.

add it up and you get 16.

the proportion of the total becomes:

5:16, 5:16, 4:16, 2:16.

this can also be shown as 5/16, 5/16, 4/16, 2/16 in fractional form.

note that the proportion of the total and the ratio of each element to the total means the same thing.

here's a table that shows the results for selection A, C, D.
you can't do B, because the number of students doesn't add up to be an integer.

in the table, the number of each type is based on their proportion of the total.

freshman = 5/16
junior = 5/16
sophomore = 4/16
senior = 2/16

selection total Freshman sophomore junior senior sum A 560 175 175 140 70 560 B 1024 320 320 256 128 1024 C 1600 500 400 400 200 1600

total is what the total number of students is expected to be.
sum is what they add up to after calculating the proportions of each.
total = sum as it should.

another way to look at it is as follows:

let a = number of freshmen
let b = number of sophomores
let c = number of juniors
let d = number of seniors.

the ratio of a:b:c:d is equal to 5:5:4:2

this means a/b = 5/5, a/c = 5/4, a/d = 5/2.

from a/b = 5/5, you solve for b to get b = a.
from a/c = 5/4, you solve for c to get c = 4/5 * a
from a/d = 5/2, you solve for d to get d = 2/5 * a

since a + b + c + d must be equal to the total number of students, you get:

a + b + c + d = total of students.

from the equivalencies you just found, you replace b and c and d with their equivalent values of a to get:

a + a + 4/5 * a + 2/5 * a = total number of students.

put everything under a common denominator to get 5/5 * a + 5/5 * a + 4/5 * a + 2/5 * a = total number of students.

sum them up and you get 16/5 * a = total number of students.

solve for a to get a = 5/16 * total number of students.

since you know that b = a, then b = 5/16 * total number of students.

since you know that c = 4/5 * a, then c = 4/5 * 5/16 * total number of students.
simplify this to get c = 1/4 * total number of students.
this is equivalent to 4/16 * total number of students.

since you know that d = 2/5 * a, then d = 2/5 * 5/16 * total number of students.
simplify this to get d = 1/8 * total number of students.
this is equivalent to 2/16 * total number of students.

there you have it.

a = 5/16 * total number of students.
b = 5/16 * total number of students.
c = 4/16 * total number of students.
d = 2/16 * total number of students.

for every 16 students, you have 5 freshmen, 5 sophomores, 4 juniors, 2 seniors.

the total number of students divided by 16 has to be an integer.

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