In a class of 36 students, some boys and some girls, exactly 1/3 of the boys and 1/4 of the girls walk to school. What is the greatest possible number of students who walk to school?
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jayhawk2001
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We essentially need to find
max (1/3 * boys + 1/4 * girls)
We also know that boys + girls = 36. For max value, the boy count needs
to be higher (as girl count is divided by 4). We also know that girl count
is not zero. So...
1/3 * 33 + 1/4 * 3 --> number of girls is not divisible by 4
1/3 * 30 + 1/4 * 6 --> same...
...
1/3 * 24 + 1/4 * 12 --> both are divisible
Hence total count = 8+3 = 11
max (1/3 * boys + 1/4 * girls)
We also know that boys + girls = 36. For max value, the boy count needs
to be higher (as girl count is divided by 4). We also know that girl count
is not zero. So...
1/3 * 33 + 1/4 * 3 --> number of girls is not divisible by 4
1/3 * 30 + 1/4 * 6 --> same...
...
1/3 * 24 + 1/4 * 12 --> both are divisible
Hence total count = 8+3 = 11
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aficionado
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let no: of girls be x.
then by equation we have: (36-x)/3 + x/4 = no: of students who walk.
therefore: 12 - x/12 = no: of students who walk.
the above expression will be maximum when x=12.
hence, no: of boys = 36-12=24, girls=12., students who walk = 11.
then by equation we have: (36-x)/3 + x/4 = no: of students who walk.
therefore: 12 - x/12 = no: of students who walk.
the above expression will be maximum when x=12.
hence, no: of boys = 36-12=24, girls=12., students who walk = 11.
All the best
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Cybermusings
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In a class of 36 students, some boys and some girls, exactly 1/3 of the boys and 1/4 of the girls walk to school. What is the greatest possible number of students who walk to school?
9
10
11
12
13
Since the number of boys who walk to school is greater than the number of girls who walk to school. Maximise the number of boys in the total class population. Greatest possible number boys is 36. but the class has some boys and some girls.
33/3 = 11 but girls = 3/4 = .75. Hence not possible
30/3 = 10 and girls 6/4 = 1.5. Hence not possible
27/3 = 9 and girls 9/4 = 2.5. Hence not possible
24/3 = 8 and girls 12/4 = 3 girls. Hence possible
I know this is trial and error but this just took me 1 minute. fair enough
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Since the number of boys who walk to school is greater than the number of girls who walk to school. Maximise the number of boys in the total class population. Greatest possible number boys is 36. but the class has some boys and some girls.
33/3 = 11 but girls = 3/4 = .75. Hence not possible
30/3 = 10 and girls 6/4 = 1.5. Hence not possible
27/3 = 9 and girls 9/4 = 2.5. Hence not possible
24/3 = 8 and girls 12/4 = 3 girls. Hence possible
I know this is trial and error but this just took me 1 minute. fair enough
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BTGmoderatorRO
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Exactly $$\frac{1}{3}$$ of boys
and $$\frac{1}{4}$$ of girls. Out of 36 students. This means that the total number of boys must be EXACTLY DIVISIBLE by 3 and the total number girls must be EXACTLY DIVISIBLE by 4.
Possible sum of numbers that fall into this category and adds up to 36 are
12 + 24
and
24 + 12
If number of boys is 12 and girls is 24,
$$\frac{1}{3}$$ * 12 + $$\frac{1}{4}$$ * 24 = 4+6 =10
If the number of boys is 12 and girls is 12,
$$\frac{1}{3}$$ * 24 + $$\frac{1}{4}$$ * 12 = 8+3 = 11
Therefore, the greatest possible number of students who walk to school is 11.
and $$\frac{1}{4}$$ of girls. Out of 36 students. This means that the total number of boys must be EXACTLY DIVISIBLE by 3 and the total number girls must be EXACTLY DIVISIBLE by 4.
Possible sum of numbers that fall into this category and adds up to 36 are
12 + 24
and
24 + 12
If number of boys is 12 and girls is 24,
$$\frac{1}{3}$$ * 12 + $$\frac{1}{4}$$ * 24 = 4+6 =10
If the number of boys is 12 and girls is 12,
$$\frac{1}{3}$$ * 24 + $$\frac{1}{4}$$ * 12 = 8+3 = 11
Therefore, the greatest possible number of students who walk to school is 11.
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Brent@GMATPrepNow
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The important thing here to recognize here is that the number of girls and the number of boys must be positive INTEGERS. For example, we can't have 5 1/3 boys.In a certain class consisting of 36 students, some boys and some girls, exactly 1/3 of the boys and exactly 1/4 of the girls walk to school. What is the greatest possible number of students in this class who walk to school?
A) 9
B) 10
C) 11
D) 12
E) 13
Also recognize that we're told that we have some boys and some girls
Since "some" means 1 OR MORE, we cannot have zero boys or zero girls.
Okay, now onto the question...
We want to MAXIMIZE the number of students who walk to school. Since a greater proportion of boys walk to school, we want to MAXIMIZE the number of boys in the class.
The greatest number of boys is 35 (since 36 boys would mean 0 girls, and we must have at least 1 girl)
35 boys
This is no good, because 1/3 of the boys walk to school, and 35 is not divisible by 3.
So, let's try ...
34 boys
This is no good, because 1/3 of the boys walk to school, and 34 is not divisible by 3.
As you can see, we need only consider values where the number of boys is divisible by 3. So, that's what we'll do from here on...
33 boys
If 1/3 of the boys walk to school, then 11 boys walk. Fine.
HOWEVER, if there are 33 boys, then there must be 3 girls.
If 1/4 of the girls walk to school, then there can't be 3 girls, since 3 is not divisible by 4.
30 boys
This means there are 6 girls
If 1/4 of the girls walk to school, then there can't be 6 girls, since 6 is not divisible by 4.
27 boys
This means there are 9 girls
If 1/4 of the girls walk to school, then there can't be 9 girls, since 9 is not divisible by 4.
24 boys and 12 girls
1/3 of the boys walk to school, so 8 boys walk
1/4 of the girls walk to school, so 3 girls walk
PERFECT - it works!!
So, a total of 11 children walk
Answer: C
Cheers,
Brent
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GMATGuruNY
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Of the girls, the fraction who walk = 1/4 = 9/36.pj94z wrote:In a class of 36 students, some boys and some girls, exactly 1/3 of the boys and 1/4 of the girls walk to school. What is the greatest possible number of students who walk to school?
9
10
11
12
13
Of the boys, the fraction who walk = 1/3 = 12/36.
Of the MIXTURE of girls and boys, the fraction who walk = x/36.
The fraction for the MIXTURE (x/36) must be BETWEEN the fractions attributed to the two INGREDIENTS in the mixture (9/36 and 12/36).
Thus, x/36 must be equal to one of the following:
10/36 or 11/36.
Implication:
Of the 36 students, the number who walk must be 10 or 11.
Eliminate A, D and E.
Since the question stem asks for the GREATEST possible number of walkers, use ALLIGATION to test whether it's possible that x/36 = 11/36 (implying 11 walkers out of 36 students).
Step 1: Plot the 3 fractions on a number line, with the fractions for the girls and the boys (9/36 and 12/36) on the ends and the desired fraction for the whole group (11/36) in the middle.
G 9/36-----------------11/36----------------12/36 B
Step 2: Calculate the distances between the fractions.
G 9/36-----2/36-----11/36-----1/36-----12/36 B
Step 3: Determine the ratio in the mixture.
The ratio of G to B is equal to the RECIPROCAL of the distances in red.
G : B = 1/36 : 2/36 = 1:2.
Success!
The resulting ratio indicates that -- of every 3 students -- 1 will be a girl and 2 will be boys, implying that 1/3 of the 36 students will be girls (for a total of 12 girls) and that 2/3 of the 36 students will be boys (for a total of 24 boys).
The correct answer is C.
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Scott@TargetTestPrep
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Solution:jayhawk2001 wrote: ↑Fri Mar 23, 2007 6:12 pmWe essentially need to find
max (1/3 * boys + 1/4 * girls)
We also know that boys + girls = 36. For max value, the boy count needs
to be higher (as girl count is divided by 4). We also know that girl count
is not zero. So...
1/3 * 33 + 1/4 * 3 --> number of girls is not divisible by 4
1/3 * 30 + 1/4 * 6 --> same...
...
1/3 * 24 + 1/4 * 12 --> both are divisible
Hence total count = 8+3 = 11
We can let n = the number of boys; thus 36 - n = the number of girls. We are given that (1/3)n boys walk to school and (1/4)(36 - n) = 9 - (1/4)n girls walk to school.
Since (1/3)n and 9 - (1/4)n must be an integer, we see that n must be divisible by 3 and 4. In other words, n must be divisible by 12. Thus n can be either 12 or 24 (we exclude 0 and 36 since if n = 0, there will be no boys in the class, and if n = 36, there will be no girls in the class).
If n = 12, then (1/3)(12) = 4 boys and 9 - (1/4)(12) = 9 - 3 = 6 girls walk to school. That is, a total of 4 + 6 = 10 students walk to school.
If n = 24, then (1/3)(24) = 8 boys and 9 - (1/4)(24) = 9 - 6 = 3 girls walk to school. That is, a total of 8 + 3 = 11 students walk to school.
Thus the greatest possible number of students in this class who walk to school is 11.
Answer: C
