Three different coffee blends are formed using the four varieties A, B, C and D. The first blend is formed by taking A and B in the ratio of 2 : 3, the second is formed by taking A and C in the ratio 2 : 5 and the third is formed by taking A and D in the ratio 2 : 7. If the first, second and third blends are taken in the ratio 3 : 4 : 5 to form a new blend, find the ratio of the quantity of A to the quantity of the other three varieties together in the new blend.
Question
Three different coffee blends are formed using the four varieties A, B, C and D. The first blend is formed by taking A and B in the ratio of 2 : 3, the second is formed by taking A and C in the ratio 2 : 5 and the third is formed by taking A and D in the ratio 2 : 7. If the first, second and third blends are taken in the ratio 3 : 4 : 5 to form a new blend, find the ratio of the quantity of A to the quantity of the other three varieties together in the new blend.
Solution
To solve this problem, we first need to find the quantity of A in each blend and the total quantity of each blend.
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In the first blend, the ratio of A to B is 2:3. So, the quantity of A is 2 parts out of a total of 2+3=5 parts. Therefore, the quantity of A in the first blend is 2/5.
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In the second blend, the ratio of A to C is 2:5. So, the quantity of A is 2 parts out of a total of 2+5=7 parts. Therefore, the quantity of A in the second blend is 2/7.
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In the third blend, the ratio of A to D is 2:7. So, the quantity of A is 2 parts out of a total of 2+7=9 parts. Therefore, the quantity of A in the third blend is 2/9.
Next, we need to find the total quantity of each blend in the new blend. The ratio of the first, second, and third blends in the new blend is 3:4:5. So, the total quantity of the new blend is 3+4+5=12 parts.
The quantity of A in the new blend is the sum of the quantities of A in each blend, multiplied by the quantity of each blend in the new blend. So, the quantity of A in the new blend is (2/5)*3 + (2/7)*4 + (2/9)*5 = 6/5 + 8/7 + 10/9 = 1.2 + 1.14 + 1.11 = 3.45 parts.
The quantity of the other three varieties together in the new blend is the total quantity of the new blend minus the quantity of A, which is 12 - 3.45 = 8.55 parts.
Therefore, the ratio of the quantity of A to the quantity of the other three varieties together in the new blend is 3.45 : 8.55. To simplify this ratio, we can multiply both sides by 100 to get 345 : 855. This ratio can be further simplified by dividing both sides by 5 to get 69 : 171.
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