At a certain committee meeting only associate professors and

Now, Check -

2a + b = 10 ---------> ( I )
and
a + 2b = 11 ---------> ( II )

In ( I ) , 2a will always result in an even number , hence b must be Odd
In ( II ) , 2b will always result in an even number , hence a must be Odd

Plug in the values the correct result satisfying both the equations, a will be 3 and b will be 4

Hence Total number of people is 7 , correct answer will be (B)

Notice that the question asks us to find that TOTAL number of people present.
In other words, it the question does NOT ask us to find the number of associate professors or the number of assistant professors present.

Since each person brings THREE items, and since a total of 21 items were brought (10 pencils and 11 charts), the number of people = 21/3 = 7

Answer: B

Cheers,
Brent

A pencil-bringing rate for associate professors is combined with a pencil-bringing rate for assistant professors to yield a pencil-bringing rate for the MIXTURE of professors.
Since this is a MIXTURE problem, an alternate approach is to solve with ALLIGATION.

Let X = the number of associate professors and Y = the number of assistant professors.

Pencil-bringing rate for X \(= \frac{2-pencils}{3-items-brought} = \frac{2}{3} = \frac{14}{21}\)
Pencil-bringing rate for Y \(= \frac{1-pencil}{3-items-brought} = \frac{1}{3} = \frac{7}{21}\)
Pencil-bringing rate for the MIXTURE of X and Y \(= \frac{10-pencils}{21-items-brought} = \frac{10}{21}\)

Since the fractions have been put over the same denominator, the alligation can be performed with only the numerators.

Step 1: Plot the 3 numerators on a number line, with the numerators for X and Y on the ends and the numerator for the mixture in the middle.
X 14-----------10------------ 7 Y

Step 2: Calculate the distances between the numerators.
X 14-----4-----10-----3-----7 Y

Step 3: Determine the ratio in the mixture.
The ratio of X to Y is equal to the RECIPROCAL of the distances in red:
X:Y = 3:4

Since there are 3 X's for every 4 Y's -- and 3+4 = 7 -- the total number of people must be a multiple of 7.

We can let the number of associate professors = a and the number of assistant professors = n, and create two equations:

Pencil equation:

2a + n = 10

Chart equation:

a + 2n = 11

Let’s add the two equations together:

3a + 3n = 21

a + n = 7

Thus, there were 7 people in the meeting.

Answer: B

Let the number of Associate professors be ‘x’ and the number of Assistant professors be ‘y’. A table can be drawn summarizing the information given, to look like the one below:



From the table, it can be seen that the number of pencils brought by the Associates is 2x and those brought by the Assistants is y. Similarly, the number of charts brought by the Associates is x and those brought by the Assistants is 2y.

Therefore, 2x + y = 10 and x + 2y = 11.
Solving, we get y = 4 and x = 3. 7 people were present at the meeting.

The correct answer option is B.

Hope that helps!
Aravind B T

is this  considered algebra? ratio? number properties? Is there any other ways of solving this?­

­I'd categorize this question under Algebra and Word Problems. Different solution approaches have been presented in the discussion above. If anything remains unclear, please feel free to let me know.

­Keep it simple. System of equations:

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