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16 Aug 2017, 01:57
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A tiara is studded with a mixture of gems. The ratio of sapphires to emeralds is 3:1. If 6 emeralds are added, the tiara will contain an equal number of sapphires and emeralds. How many emeralds must be added to the original tiara so that the ratio between emeralds and sapphires is 3:1?
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27
16 Aug 2017, 06:52
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Let the sapphires be S and emeralds be E, S:E = 3:1
If 6 emeralds are added, the number of sapphires = number of emeralds.
3x = x+6 => x =3
Therefore, initially the tiara contains 9 sapphires and 3 emeralds.
Now, we need to add emeralds, such that the ratio of emeralds to sapphires becomes 3:1
So, since we have 9 sapphires, we must have 27 emeralds.
Since we already have 3 emeralds, we need 24 more emeralds(Option D)
If 6 emeralds are added, the number of sapphires = number of emeralds.
3x = x+6 => x =3
Therefore, initially the tiara contains 9 sapphires and 3 emeralds.
Now, we need to add emeralds, such that the ratio of emeralds to sapphires becomes 3:1
So, since we have 9 sapphires, we must have 27 emeralds.
Since we already have 3 emeralds, we need 24 more emeralds(Option D)
16 Aug 2017, 08:50
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Bunuel
Two stages, because there are little twists IMO:
1. Original ratio of sapphires to emeralds is 3:1
\(\frac{S}{E}\) = \(\frac{3x}{x}\)
If 6 emeralds are added, ratio is \(\frac{1}{1}\)
\(\frac{3x}{(x+6)}\) = \(\frac{1}{1}\)
3x = x + 6
2x = 6
x = 3, which is the multiplier for the original ratio.
There are 3x or (3*3) = 9 sapphires and x or (1*3) = 3 emeralds in the tiara
2. How many emeralds must be added to the tiara to make the ratio of emeralds to sapphires 3:1?
We need \(\frac{S}{E}\) = \(\frac{1}{3}\)
\(\frac{9}{(3 + x)}\) = \(\frac{1}{3}\)
3 + x = 27
x = 24
Check the numbers. There are 9 sapphires. There are 3 emeralds; 24 have been added. There are now 27 emeralds.
\(\frac{S}{E}\)= \(\frac{9}{27}\) = \(\frac{1}{3}\)
Correct. Answer D
