Two alloys A and B are composed of two basic elements. The ratios of the compositions of the two basic elements in the two alloys are 5:3 and 1:2 respectively. A new alloy X is formed by mixing the two alloys A and B in the ratio 4:3. what is the ratio of the composition of the basic elements in alloy X?
A. 1 : 1
B. 2 : 3
C. 5 : 2
D. 4 : 3
Solution (By Examveda Team)
Proportion of 1st element in A = 5/8
Proportion of 1st element in B = 1/3
let, the proportion of 1st element in mixture = x
Using allegations:
x - (1/3) /(5/8) -x = 4/3
On solving, we will get,
x = 1/2
Thus, proportion of two elements in the mixture is 1:1
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Let the actual amount of A be 4y and actual amount of B be 3y since A and B are in the ratio 4/3.
Amount of first basic element in the new alloy X,
(5/8)*4y + (1/3)*3y = (7y)/2
Amount of second basic element in the new alloy X,
(3/8)*4y + (2/3)*3y= (7y)/2
So, ratio of first basic element to second basic element,
[(7y)/2 ] / [ 7y/2] = 1/1 = 1:1
Let F = the first element and S = the second element.
Alloy A:
Since F:S = 5:3, and 5+3=8, F/total = 5/8.
Alloy B:
Since F:S = 1:2, and 1+2 = 3, F/total = 1/3.
Mixture X:
A:B = 4:3.
If we mix 8 units of A with 6 units of B, we get:
Amount of F in alloy A = (5/8)8 = 5 units
Amount of F in alloy B = (1/3)6 = 2 units.
(Total F)/(Mixture X) = (5+2)/(8+6) = 7/14 = 1/2.
Since 1/2 of the mixture is composed of F, F:S = 1:1.
Step 1: Convert to FRACTIONS the ratios attributed to the two INGREDIENTS.
A:
Since (element 1) : (element 2) = 5:3, (element 1)/total = 5/8.
B:
Since (element 1) : (element 2) = 1:2, (element 1)/total = 1/3.
Step 2: Put these fractions over a COMMON DENOMINATOR.
A = 5/8 = 15/24.
B = 1/3 = 8/24.
Step 3: Plot the 2 fractions at the ends of a number line, with the unknown goal fraction (X) in the middle.
A(15/24)----------------X--------------------B(8/24)
Step 4: The distances between the fractions = the RECIPROCAL of the ratio of A:B in the mixture.
A(15/24)-------3y-------X---------4y---------B(8/24)
Step 5: Solve for y.
Since the total distance between 15/24 and 8/24 = 7y, we get:
y = (15/24 - 8/24)/7 = 1/24.
Step 6: Calculate the value of the goal fraction.
T = 15/24 - 3y = 15/24 - 3(1/24) = 12/24 = 1/2.
Since, element 1 = 1/2 of alloy X, (element 1) : (element 2) = 1:1.
Consider the weights of alloys be 8 and 6
=> weight of element 1 in A = 5 & in B=2
=> weight of ele 2 in A = 3 & in B = 4
=> the ratio = 7:7=1:1