Every object in a box is either a sphere or a cube

jjjinapinch wrote:Every object in a box is either a sphere or a cube, and every object in the box is either red or green. How many objects are in the box?
(1) There are six cubes and 5 green objects in the box.
(2) There are two red spheres in the box.

Official Guide question
Answer: E

Hi jjjinapinch,

Hope this picture tells all.



The correct answer: E

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jjjinapinch wrote:Every object in a box is either a sphere or a cube, and every object in the box is either red or green. How many objects are in the box?
(1) There are six cubes and 5 green objects in the box.
(2) There are two red spheres in the box.

We are given that we have red or green spheres or cubes in a box. We need to determine the total number of objects.

Statement One Alone:

There are six cubes and 5 green objects in the box.

Although we know the number of cubes in the box, we do not know the number of spheres, and thus we cannot determine the total number of objects in the box. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

There are two red spheres in the box.

Since we do not know how many spheres or cubes are in the box, we cannot determine the total number of objects. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statements one and two together, although we know there are 6 cubes and 2 red spheres, we do not know the shapes of the green objects and thus cannot determine the total number of cubes and spheres. For example, if there is only one green sphere, then there are 5 - 1 = 4 green cubes, and thus 6 - 4 = 2 red cubes. In this case, there are 2 + 2 = 4 red objects and 5 green objects; thus, the total number of objects is 9. On the other hand, if there are two green spheres, then there are 5 - 2 = 3 green cubes, and thus 6 - 3 = 3 red cubes. In this case, there are 3 + 2 = 5 red objects and 5 green objects; thus, the total number of objects is 10. So, we cannot determine with certainty the total number of objects in the box.

Answer: E

jjjinapinch wrote:Every object in a box is either a sphere or a cube, and every object in the box is either red or green. How many objects are in the box?
(1) There are six cubes and 5 green objects in the box.
(2) There are two red spheres in the box.

Official Guide question
Answer: E

We are given that we have red or green spheres or cubes in a box. We need to determine the total number of objects.

Statement One Alone:

There are six cubes and 5 green objects in the box.

Although we know the number of cubes in the box, we do not know the number of spheres, and thus we cannot determine the total number of objects in the box. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

There are two red spheres in the box.

Since we do not know how many spheres or cubes are in the box, we cannot determine the total number of objects. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statements one and two together, although we know there are 6 cubes and 2 red spheres, we do not know the shapes of the green objects and thus cannot determine the total number of cubes and spheres. For example, if there is only one green sphere, then there are 5 - 1 = 4 green cubes, and thus 6 - 4 = 2 red cubes. In this case, there are 2 + 2 = 4 red objects and 5 green objects; thus, the total number of objects is 9. On the other hand, if there are two green spheres, then there are 5 - 2 = 3 green cubes, and thus 6 - 3 = 3 red cubes. In this case, there are 3 + 2 = 5 red objects and 5 green objects; thus, the total number of objects is 10. So, we cannot determine with certainty the total number of objects in the box.

Answer: E

jjjinapinch wrote:Every object in a box is either a sphere or a cube, and every object in the box is either red or green. How many objects are in the box?
(1) There are six cubes and 5 green objects in the box.
(2) There are two red spheres in the box.

Official Guide question
Answer: E

Target question: How many objects are in the box?

Given: Every object in a box is either a sphere or a cube, and every object in the box is either red or green.
We can solve this using the Double Matrix Method.
This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions)..
Here, we have a population of objects, and the two characteristics are:
- sphere or cube
- red or green

So, we can set up our matrix as follows:


From here, I'll jump straight to . . .

Statements 1 and 2 COMBINED
When we combine the statements, we get the following matrix:


There are several scenarios that satisfy BOTH statements. Here are two:
Case a:

In this case, the total number of objects = 3 + 3 + 2 + 2 = 10
So, the answer to the target question is there are 10 objects in the box

Case b:

In this case, the total number of objects = 5 + 1 + 2 + 4 = 12
So, the answer to the target question is there are 12 objects in the box

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

This question type is VERY COMMON on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video: https://www.youtube.com/watch?v=jK-tiBrrf04
Here's a practice question too: https://www.youtube.com/watch?v=DsbPUSH_Wu4