SAT Data - Tables

Probability Level 1

High School Middle School Total Boys a 2 b X Girls 2 a b Y Total M N T \begin{array}{|c|c|c|c|c|} \hline & \text{High School} & \text{Middle School} & \text{ Total}\\ \hline \text{Boys} & a & 2b & X\\ \hline \text{Girls} & 2a & b & Y\\ \hline \text{Total} & M & N & T\\ \hline \end{array}

In the table above, each letter represents the number of students in that category. Which of the following equals M − N ? M-N?

(A) X \ \ X

(B) Y \ \ Y

(C) 3 ( Y − X ) \ \ 3(Y-X)

(D) T 3 \ \ \frac{T}{3}

(E) X + Y \ \ X+Y

A B C D E

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Tatiana Georgieva by Tatiana Georgieva

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3 solutions

Tatiana Georgieva Staff
Mar 18, 2015

Correct Answer: C

Solution:

According to the table,

M = a + 2 a = 3 a M = a+2a=3a
N = 2 b + b = 3 b N = 2b+b=3b
M − N = 3 a − 3 b M - N = 3a-3b

We analyze each of the answer choices:

(A) X = a + 2 b ≠ 3 a − 3 b . X = a+2b \neq 3a-3b. Wrong choice.
(B) Y = 2 a + b ≠ 3 a − 3 b . Y = 2a+b \neq 3a-3b. Wrong choice.
(C)

3 ( Y − X ) = 3 [ 2 a + b − ( a + 2 b ) ] = 3 ( 2 a + b − a − 2 b ) = 3 ( a − b ) = 3 a − 3 b = M − N . \begin{array}{ l c l} 3(Y-X) &=& 3[2a+b - (a+2b)]\\ &=& 3(2a+b -a-2b)\\ &=& 3(a-b) = 3a-3b = M-N. \end{array}

Correct answer.

(D) T 3 = M + N 3 = 3 a − 3 b 3 = a − b . \frac{T}{3} = \frac{M+N}{3} = \frac{3a-3b}{3} = a-b. Wrong answer.
(E) X + Y = a + 2 b + 2 a + b = 3 a + 3 b . X+Y = a+2b + 2a+b = 3a+3b. Wrong answer.

Incorrect Choices:

(A) , (B) , (D) , and (E)
The Solution explains why these choices are wrong.

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Nguyen Tr Hien
Apr 12, 2015

M-N = 3a-3b We have: 3X = 3a+6b and 3Y = 6a+3b. So we eventualy have 3(Y - X) = 6a + 3b - 3a - 6b = 3a - 3b = M - N

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Andrew Tawfeek
Mar 18, 2015

We can start off by excluding the choices below for the provided reasons, given it is understood that we're looking for the total high school students minus the total middle school students:

  • *A * X represents the total boys in high school and middle school.

  • *B * Y represents the total girls in high school and middle school.

  • *E * X+Y represents the total boys and girls in high school and middle school, or in other words, T.

This leaves us with C and D .

*Choice D: * Here, T 3 { \frac { T }{ 3 } } represents the total divided by three. To check if this is equal to M-N, let us start off by evaluating M-N.

M = a + 2 a = 3 a N = 2 b + b = 3 b M − N = 3 a − 3 b M=a+2a=3a\\ N=2b+b=3b\\ M-N=3a-3b

Next, lets evaluate T 3 { \frac { T }{ 3 } } and see if they're equal to one another.

T 3 = ( M + N ) 3 M + N = ( a + 2 a ) + ( 2 b + b ) = 3 a + 3 b T 3 = 3 a + 3 b 3 = a + b a + b ≠ 3 a − 3 b \frac { T }{ 3 } =\frac { (M+N) }{ 3 } \\ M+N=(a+2a)+(2b+b)=3a+3b\\ \frac { T }{ 3 } =\frac { 3a+3b }{ 3 } =a+b\\ \\ a+b\neq 3a-3b

*Choice C: * So, lastly, we're left with the final choice. Before getting cocky and picking it right away, we'll evaluate it.

3 ( Y − X ) 3 [ ( 2 a + b ) − ( a + 2 b ) ] 3 ( 3 a − 3 b − a − 2 b ) 3 ( a − b ) 3 a − 3 b = 3 a − 3 b 3(Y-X)\\ 3[(2a+b)-(a+2b)]\\ 3(3a-3b-a-2b)\\ 3(a-b)\\ 3a-3b=3a-3b

There we have it, D is correct.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I think you meant choice C.

Loki Yelnats - 6 years ago

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