Tiger Gongs

[kartikshah]

In an ensemble of gongs, all gongs have a diameter of either ten inches, or twelve inches or fifteen inches. In the collection there are 18 ten inch gongs. Half of the gongs in the collection are Tiger gongs. Of the Tiger gongs, there are equal numbers of ten inch, twelve inch and fifteen inch gongs. Half of the twelve inch gongs are not Tiger gongs, and half of all gongs are fifteen inches in diameter. How many gongs are there in the collection?

1. 18
2. 54
3. 72
4. 90
5. 108

Source: Master GMAT

OA is [spoiler]Option 4 (108)[/spoiler]

I got it right but it took me 3 mins 38 secs (!!) Anyone who solved it without forming an equation or plugging numbers is welcome to share his/her method ? Thanks!

[eagleeye]

In an ensemble of gongs, all gongs have a diameter of either ten inches, or twelve inches or fifteen inches. In the collection there are 18 ten inch gongs. Half of the gongs in the collection are Tiger gongs. Of the Tiger gongs, there are equal numbers of ten inch, twelve inch and fifteen inch gongs. Half of the twelve inch gongs are not Tiger gongs, and half of all gongs are fifteen inches in diameter. How many gongs are there in the collection?

1. 18
2. 54
3. 72
4. 90
5. 108

Source: Master GMAT

OA is [spoiler]Option 4 (108)[/spoiler]

I got it right but it took me 3 mins 38 secs (!!) Anyone who solved it without forming an equation or plugging numbers is welcome to share his/her method ? Thanks!

You can do it using a matrix, well under 2 minutes. Just keep on filling the matrix as you encounter the statements in the thing.

Then, refer to the attached figure.

Step 1: Let the total number be x.
Step 2: In the collection there are 18 ten inch gongs.
Step 3: Half of the gongs in the collection are Tiger gongs.
Step 4: Of the Tiger gongs, there are equal numbers of ten inch, twelve inch and fifteen inch gongs.
Step 5: Half of the twelve inch gongs are not Tiger gongs,
Step 6: and half of all gongs are fifteen inches in diameter.

Step 7: The total in 15inch column must be x/2. Hence (Not Tiger AND 15 inch) = x/2 -x/6
Step 8: The total in Not tiger row must be x/2. Hence (10 Inch AND not tiger) = 0
Step 9: 18 = x/6 + 0 => x= 18*6 = 108.

Let me know if this helps

[kartikshah]

I did it the same way. First tabulating the data and then equating variables to constant values.
But it took too long!

[niketdoshi123]

Start with option C
# of gongs in the collection = 72
# of 15-inch gongs = 36 (half of all gongs are fifteen inches in diameter)
# of 12-inch gongs = 72 - 18 - 36 = 18
1)# of 12-inch tiger gongs = 9 (Half of the twelve inch gongs are not Tiger gongs)
# of tiger gongs = 36 ( Half of the gongs in the collection are Tiger gongs.)
2)# of 12-inch tiger gongs = 36/3 = 12 ( Of the Tiger gongs,there are equal numbers of ten inch, twelve inch and fifteen inch gongs.)

Clearly values of 12-inch tiger gongs from 1) & 2) are not equal.
Hence eliminate options A,B,and C, since the # of 12-inch gongs has to be greater than 18 to satisfy the conditions.

Now, consider option D

# of gongs in the collection = 90
# of 15-inch gongs = 45 (half of all gongs are fifteen inches in diameter)
# of 12-inch gongs = 90 - 18 - 45 = 27
1)# of 12-inch tiger gongs = 27/2 not an integer value (Half of the twelve inch gongs are not Tiger gongs)
Eliminate option D

Hence the remaining option E is correct.

You can check the option to verify your answer

# of gongs in the collection = 108
# of 15-inch gongs = 54 (half of all gongs are fifteen inches in diameter)
# of 12-inch gongs = 108 - 18 - 54 = 36
1)# of 12-inch tiger gongs = 18 (Half of the twelve inch gongs are not Tiger gongs)
# of tiger gongs = 54 ( Half of the gongs in the collection are Tiger gongs.)
2)# of 12-inch tiger gongs = 54/3 = 18 ( Of the Tiger gongs,there are equal numbers of ten inch, twelve inch and fifteen inch gongs.)
Both the values are equal. Hence verified.

[GMATGuruNY]

In an ensemble of gongs, all gongs have a diameter of either ten inches, or twelve inches or fifteen inches. In the collection there are 18 ten inch gongs. Half of the gongs in the collection are Tiger gongs. Of the Tiger gongs, there are equal numbers of ten inch, twelve inch and fifteen inch gongs. Half of the twelve inch gongs are not Tiger gongs, and half of all gongs are fifteen inches in diameter. How many gongs are there in the collection?

1. 18
2. 54
3. 72
4. 90
5. 108

An efficient approach is to plug the answers -- which represent the total number of gongs -- into a GROUP GRID.
When the correct answer is plugged in, the total number of 10-inch gongs = 18.

Answer choice C: 72


Half of the gongs in the collection are Tiger gongs.


Of the Tiger gongs, there are equal numbers of ten inch, twelve inch and fifteen inch gongs.
Half of the twelve inch gongs are not Tiger gongs.


Half of all gongs are fifteen inches in diameter.


In the grid above, the total number of 10-inch gongs = 12.
Since the total number of 10-inch gongs must be 18 -- 50% greater -- the total number of gongs must increase by 50%:
72 + .5(72) = 72+36 = 108.

The correct answer is E.