Data Sufficiency

Of the birds in a particular aviary, 30 are both spotted and crested and 10 are neither spotted nor crested. What is the total number of birds in the aviary?

(1) The number of birds that are spotted but not crested is 10 greater than the number of birds that are crested but not spotted.

(2) More than 3/5 of the birds in the aviary are spotted.

j_shreyans wrote:Of the birds in a particular aviary, 30 are both spotted and crested and 10 are neither spotted nor crested. What is the total number of birds in the aviary?

(1) The number of birds that are spotted but not crested is 10 greater than the number of birds that are crested but not spotted.

(2) More than 3/5 of the birds in the aviary are spotted.

This is an EITHER/OR group problem.
Every bird is EITHER spotted OR not.
Every bird is EITHER crested OR not.
For an EITHER/OR group problem, we can use a GROUP GRID (also known as a double-matrix) to organize the data.

Let S = spotted, NS = not spotted, C = crested, NC = not crested.
In the grids below, the entries in any given row or column must add up to the TOTAL of that row or column.

Prompt:
30 are both spotted and crested and 10 are neither spotted nor crested.
What is the total number of birds in the aviary?
The following grid is yielded:

We need the value represented by the question mark.

Statement 1: The number of birds that are spotted but not crested is 10 greater than the number of birds that are crested but not spotted.
The following grid is yielded:

In the resulting grid, the total number of birds = 2x+50.
Since the value of x is unknown, INSUFFICIENT.

Statement 2: More than 3/5 of the birds in the aviary are spotted.
Many grids are possible.
Two examples:

Here, spotted/total = 30/40 = 3/4.


Here, spotted/total = 40/50 = 4/5.

Since the total number of birds can be different values, INSUFFICIENT.

Statements combined:
The grid in Statement 1 indicates that total spotted = x+40 and that total birds = 2x+50.
Since Statement 2 indicates that total spotted is more than 3/5 of the total number of birds, we get:
(x+40)/(2x+50) > 3/5
5x + 200 > 6x + 150
50 > x.
Since x can be different values, the total number of birds cannot be determined.
INSUFFICIENT.

The correct answer is E.

j_shreyans wrote:Of the birds in a particular aviary, 30 are both spotted and crested and 10 are neither spotted nor crested. What is the total number of birds in the aviary?

(1) The number of birds that are spotted but not crested is 10 greater than the number of birds that are crested but not spotted.

(2) More than 3/5 of the birds in the aviary are spotted.

Another way to do this is to use the formula Total = A + B - Both + Neither.

In this example Total = Spotted(S) + Crested(C) - Both + Neither.

From the question we can fill in some of the equation.

Total = S + C - 30 + 10

So to get Total, we need to get S and C or, alternatively, S + C.

Statement 1 tells us that S - Both = C - Both + 10 and therefore S = C + 10. This does not give us any way to connect S or C to the hard numbers we have. So we can't determine S and C or S + C from this, and Statement 1 is insufficient.

Statement 2 tells us that S > 3/5 * Total. All this means is that S has to be greater than some minimum that makes it 3/5 given what we know about Both and Neither. So there's no way to figure out what S, C, or S + C actually is. So Statement 2 is insufficient.

Combined, the statements tell us the relationship between S and C and tell us that S has to be big enough to be at least 3/5 of Total given what we know about Both and Neither. If S were exactly 3/5 of Total, we could narrow it all down to one variable and figure out the answer, but even give the additional constraint from Statement 1, there are still multiple values of S and C that work, and create multiple values for Total.

This can be confirmed by testing values. With a little playing around I found these two that work.

If S = 60 and C = 50 then Total = 60 + 50 - 30 + 10 = 70 and S > (3/5)70

If S = 40 and C = 30 then Total = 40 + 30 - 30 + 10 = 50 and S > (3/5)50

So even combined the statements are insufficient.

Choose E.