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# Bottle X is 2/3 full of water and has half the capacity of Bottle Y.

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Math Expert
Joined: 02 Sep 2009
Posts: 43849
Bottle X is 2/3 full of water and has half the capacity of Bottle Y. [#permalink]

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11 Jan 2018, 22:20
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Difficulty:

35% (medium)

Question Stats:

72% (01:32) correct 28% (03:26) wrong based on 29 sessions

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Bottle X is 2/3 full of water and has half the capacity of Bottle Y. Bottle Y is 1/2 full of water and has twice the capacity of Bottle Z, which is 1/4 full of water. If Bottle X and Bottle Z were poured into Bottle Y, Bottle Y would then be filled to what fraction of capacity?

A. 23/24

B. 11/12

C. 7/8

D. 2/3

E. 1/2
[Reveal] Spoiler: OA

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Re: Bottle X is 2/3 full of water and has half the capacity of Bottle Y. [#permalink]

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12 Jan 2018, 08:17
According to the capacity ratios, let the cap of bottles X,Y& Z be 12, 24 & 12 litres.
Then vol of water in X, Y & Z is 8L, 12L and 3L.
If X and Z are poured in Y, water in Y would be 23L.
Thus the fraction of water filled= 23/24.
Ans - A, IMO.
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Joined: 22 May 2016
Posts: 1336
Bottle X is 2/3 full of water and has half the capacity of Bottle Y. [#permalink]

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12 Jan 2018, 08:38
Bunuel wrote:
Bottle X is 2/3 full of water and has half the capacity of Bottle Y. Bottle Y is 1/2 full of water and has twice the capacity of Bottle Z, which is 1/4 full of water. If Bottle X and Bottle Z were poured into Bottle Y, Bottle Y would then be filled to what fraction of capacity?

A. 23/24

B. 11/12

C. 7/8

D. 2/3

E. 1/2

This question is not hard, but the language is tricky. Assigning values is efficient.

Break the setup (and hence the language) into two parts: capacity, then actual amount of water in each.

Use the LCM of the denominators for the biggest number: 2*3*4 = 24

Capacity
X has $$\frac{1}{2}$$ of Y's capacity
Y has twice Z's capacity
Y is the largest
X and Z have equal capacity
Let Y = 24
Let X = 12
Let Z = 12

Amount of water in each bottle
(fraction full)*(capacity) = amount of water
Y is $$\frac{1}{2}$$ full: $$\frac{1}{2}$$ * 24 = 12 units of water in Y

X is $$\frac{2}{3}$$ full: $$\frac{2}{3}$$ * 12 = 8 units of water in X

Z is $$\frac{1}{4}$$ full: $$\frac{1}{4}$$ * 12 = 3 units of water in Z

Bottle Y is filled to what fraction of its capacity?
All the water in X and Z is poured into Y
X + Z: (8 + 3) = 11 units
Y has 12 units
Y has (11 + 12) = 23 units of water in it

Bottle Y is filled to what fraction of capacity?

$$\frac{Part}{Whole}=\frac{AmtOfWater}{Capacity}=\frac{23}{24}$$

Algebraically
Define X and Z in terms of Y.
Water in X: $$\frac{1}{2}Y * \frac{2}{3}=\frac{1}{3}Y$$

Water in Z: $$\frac{1}{2}Y*\frac{1}{4}=\frac{1}{8}Y$$

Water in Y: $$\frac{1}{2}Y$$

Bottle Y is filled to what capacity?

$$\frac{1}{3}Y + \frac{1}{8}Y +\frac{1}{2}Y =$$

$$(\frac{8}{24}Y +\frac{3}{24}Y+\frac{12}{24}Y) = \frac{23}{24} Y$$

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Bottle X is 2/3 full of water and has half the capacity of Bottle Y.   [#permalink] 12 Jan 2018, 08:38
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