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
What is the right solution for this problem? Need help from experts.
OA D
A tiara is studded with a mixture of gems
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The ratio of sapphires to emeralds is 3:1.lheiannie07 wrote: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
Options for S and E:
S=3, E=1
S=6, E=2
S=9, E=3
S=12, E=4.
If 6 emeralds are added, the tiara will contain an equal number of sapphires and emeralds.
Adding 6 to the value of E in each of the options above, we get:
S=3, E=1+6=7
S=6, E=2+6=8
S=9, E=3+6=9.
We can stop here.
The option in blue yields 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?
In the blue option, the original values are as follows:
S=9, E=3.
After some emeralds are added, the resulting number of emeralds must be 3 times the original number of sapphires:
3S = 3*9 = 27.
Since the value of E must increase from 3 to 27, the number of emeralds that must be added = 27-3 = 24.
The correct answer is D.
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Thanks a lot!GMATGuruNY wrote:The ratio of sapphires to emeralds is 3:1.lheiannie07 wrote: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
Options for S and E:
S=3, E=1
S=6, E=2
S=9, E=3
S=12, E=4.
If 6 emeralds are added, the tiara will contain an equal number of sapphires and emeralds.
Adding 6 to the value of E in each of the options above, we get:
S=3, E=1+6=7
S=6, E=2+6=8
S=9, E=3+6=9.
We can stop here.
The option in blue yields 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?
In the blue option, the original values are as follows:
S=9, E=3.
After some emeralds are added, the resulting number of emeralds must be 3 times the original number of sapphires:
3S = 3*9 = 27.
Since the value of E must increase from 3 to 27, the number of emeralds that must be added = 27-3 = 24.
The correct answer is D.
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Since the original ratio of sapphires to emeralds is 3:1, we can represent the number of sapphires by 3x and the number of emeralds by x, where x is a positive integer.BTGmoderatorDC wrote: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
Since adding 6 emeralds results in an equal number of sapphires and emeralds, we have:
x + 6 = 3x
6 = 2x
x = 3
So, currently there are 3*3 = 9 sapphires and 3 emeralds on the tiara. Let y be the number of additional emeralds to bring the ratio of emeralds to sapphires to 3:1 (from 1:3). Then,
(3 + y)/9 = 3/1
3 + y = 27
y = 24
Answer: D
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