Question

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?A. 18B. 54C. 72D. 90E. 108

Answer 1

Answer: Option 'E' is correct.

Step-by-step explanation:

Since we have given that

Let the total number of gongs be 'g'.

Number of ten inch gongs = 18

Let the number of 12 inch gongs be 'x'.

Let the number of 15 inch gongs be 'y'.

so, equation would be

$18+x+y=g$

Half of the gongs are tiger gongs is given by

$t=\dfrac{g}{2}\\\\g=2t$

Half of the 12 inch gongs are not tiger gongs

so, number of tiger gongs which is of 12 inch = $\dfrac{x}{2}$

Among the tiger gongs, there are equal numbers of ten inch, twelve inch and fifteen inch.

$\dfrac{x}{2}+\dfrac{x}{2}+\dfrac{x}{2}=t\\\\\dfrac{3x}{2}=t$

Half of the gongs are of 15 inch is given by

$y=\dfrac{g}{2}\\\\2y=g$

Since we have

$g=2t\\\\g=2\times \dfrac{3x}{2}\\\\g=3x\\\\\dfrac{g}{3}=x$

so, According to question, it becomes,

$18+\dfrac{g}{2}+\dfrac{g}{3}=g\\\\18+\dfrac{5g}{6}=g\\\\18=g-\dfrac{5g}{6}\\\\18=\dfrac{6g-5g}{6}=\dfrac{g}{6}\\\\18\times 6=g\\\\108=g$

Hence, Option 'e' is correct.