Boys and girls in a class are writing letters.

Hi GMAT01,

If you had included the 5 answer choices, then I could show you a really easy way to answer this question. However, since you have NOT included them, we're forced to do the Algebra (which will likely take longer). This question is based on "system" Algebra. With that approach, you have to create the necessary equations, then do all of the math needed to get to the solution.

Based on the prompt, here's what we know:
1) There are TWICE as many Girls as Boys in a class
2) Each Girl wrote 3 MORE letters than each Boy
3) Boys wrote 24 letters (out of 90 total letters)

We're asked for the number of letters that EACH BOY wrote.

Here are the equations that you can create from the above information....

G = number of girls
B = number of boys
X = number of letters written by each Boy
(X+3) = number of letters written by each Girl

G = 2B
(B)(X) = 24
(G)(X+3) = 66

Now we have 3 variables and 3 unique equations, so we CAN solve for the value of each variable. We're asked to solve for X....

From here, I'm going to let you try to do the Algebra. If you get stuck, then write back and I'll walk you through the math.

To answer your initial question though, this is an example of a 3-variable System question. They're relatively rare on Test Day, but if you're doing well in the Quant section, then there's a decent chance that you'll see one.

GMAT assassins aren't born, they're made,
Rich

Actually, there isn't much you need to do and no equations/variables you need to use if you just utilize what is given. The question is like a puzzle and certainly something GMAT could give:

Boys write total 24 letters
Girls write total 90 - 24 = 66 letters

66 = 2*3*11
Number of girls must be an even number since it is twice of number of boys. The number of letters each writes should be more than 3. So number of girls can be 2 (which means 1 boy), or 6 (which means there are 3 boys) but not 22 (since then each girl would have written 3 letters)

If number of girls is 6, each writes 11 letters.
In that case, number of boys is 3 and each writes 24/3 = 8 letters.
Girls write 3 letters more than boys so it satisfies all our conditions.

Merging topics.

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This is really easy to solve, but I feel as if most people get lost in the algebra here.

You know the sum of the letters 90, so the ratio of G:B must be able to produce a sum of 90, given the letter writing constraints.

Tip: start writing out the different ratios that satisfy the initial condition: ratio of boys to girls is 2:1. These are 2:1, 4:2, 6:3, 8:4, etc....writing these out will give you a great "feel" for the answer.

Working with what you're given:

If boys write 24 letters, you know the ratio of boys must be factor of 24. Looking through the ratios you listed out, all of these work, however oddly enough, 6:3 sums to 9, and the total number of letters is 90. Mmhhh, suspicious.

Using the 6:3 ratio, the multiple you have for boys is 8, which means the multiple used for girls is 11 (8+3). This leaves you with 90 total letters, 66 written by girls, 24 written by boys.

However, you don't need to solve for the ratio of letters written by girls, this information doesn't need to be solved for.

The crux of this problem is reading through and comprehending what is being stated/asked. My personal favorite approach to ratio problems is to just start writing down ratios that fit the given constraint. This gives me a great "feel" for the problem.

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Given that B = 2G.
Also, each girl writes 3 more letters than each boy.
And, boys write 24 letters out of 90 => Girls write 66 letters.

Let us simply use the options to get the answer.

Option 1: If each boy writes 3 letters => 24/3 = 8 boys in the class => 16 girls in the class.
Since each boy writes 3 letters => Each girl writes 6 letters.
So, (8 boys * 3 letters) + (16 girls * 6 letters) = 120. This is NOT EQUAL to 90.

Option 2: If each boy writes 4 letters => 24/4 = 6 boys in the class => 12 girls in the class.
Since each boy writes 4 letters => Each girl writes 7 letters.
So, (6 boys * 4 letters) + (12 girls * 7 letters) = 108. This is NOT EQUAL to 90.

Option 3: If each boy writes 6 letters => 24/6 = 4 boys in the class => 8 girls in the class.
Since each boy writes 6 letters => Each girl writes 9 letters.
So, (4 boys * 6 letters) + (8 girls * 9 letters) = 96. This is NOT EQUAL to 90.

Option 4: If each boy writes 8 letters => 24/8 = 3 boys in the class => 6 girls in the class.
Since each boy writes 8 letters => Each girl writes 11 letters.
So, (3 boys * 8 letters) + (6 girls * 11 letters) = 90
Hence option D.

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b*l=24

2b(l+3)=66 => 2bl+6b=66 =>b=3

24/3=8

D

At first glance, this problem may seem unsolvable. However, the problem requires three equations to solve for three variables. Step-by-Step,
g=number of boys
b=number of boys
y=number of letters written by each boy
g=2b
bx=24
g(x+3)=66 (If the total is 90, then 90-24=66 to find the number of letters written by girls.
gx+g3=66
It's convenient to find the variable b , so , if bx=24, then b=24/x.
So, 2b(24/b)+2b=66
b=3
3x=24
x=8

Let's first set the different variables we will be using in the problem:

G: Number of girls in the class
B: Number of boys in the class

And we are given: G=2B (equation 1)

LG: Number of letters written by each girl
LB: Number of letters written by each boy

And we are given: LG=LB+3 (equation 2)

TG: Total letters written by girls
TB: Total letters written by boys

And we are given: TB+TG= 24+66=90 (equation 3)

The question asks; how many letters does each boy write?
We know that: LB (number of letters written by each boy)= TB (total letters written by boys)/ B (number of boys)
And TB= LBxB= 24
And TG=LGxG= 66

We will start from equation 3 and work our way to the answer:

TB + TG=90
(LBxB) + (LGxG)= 90
(LBxB) + ((LB+3) x 2B)= 90 (substitute LG by LB+3 (equation 2) and G by 2B (equation 1))
LBxB + 2LBxB +6B=90
3LBxB + 6B=90
3x24 + 6B=90 (we were given that TB= LBxB= 24)
Therefore the number of boys, B= 18/6 =3

TB= 24 and B= 3
Hence, LB=TB/B= 24/3 =8

Correct answer is D