Question 1204074: Given that A:B = 2:3, B:C=3:1, C:D=4:5, D:E = 2:1 and A+B+C+D+E = 18900. What is the value of E?
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
D:E = 2:1
D/E = 2/1
D/E = 2
D = 2E
C:D = 4:5
C/D = 4/5
5C = 4D ...... cross multiply
5C = 4(2E) .... plug in D = 2E
5C = 8E
C = 8E/5
B:C = 3:1
B/C = 3/1
B/C = 3
B = 3C
B = 3*(8E/5) ...... plug in C = 8E/5
B = 24E/5
A:B = 2:3
A/B = 2/3
3A = 2B ...... cross multiply
A = (2/3)*B
A = (2/3)*(24E/5) ...... plug in B = 24E/5
A = 16E/5
Each block of steps shown above has us isolate variables A, B, C, and D such that E is the only variable on the right hand side.
A = 16E/5
B = 24E/5
C = 8E/5
D = 2E
This will allow us to do substitutions in the next set of steps to let us solve for E.
A+B+C+D+E = 18900
16E/5+24E/5+8E/5+2E+E = 18900
5*(16E/5+24E/5+8E/5+2E+E) = 5*18900
16E+24E+8E+10E+5E = 94500
63E = 94500
E = 94500/63
E = 1500
Use this to find the other variables
A = 16E/5 = 16*1500/5 = 4800
B = 24E/5 = 24*1500/5 = 7200
C = 8E/5 = 8*1500/5 = 2400
D = 2E = 2*1500 = 3000
To summarize
A = 4800
B = 7200
C = 2400
D = 3000
E = 1500
I'll let the student verify each of those values.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
All the cross multiplying in the solution from the other tutor seems like far more work than is necessary to solve the problem.
It seems to me the easiest way to solve the problem is to combine the given ratios into a single ratio comparing all 5 quantities, scaling the ratio up where necessary to keep the ratio in whole numbers.
A:B = 2:3 and B:C = 3:1 gives us
A:B:C = 2:3:1 [1]
C:D = 4:5, so scale [1] up to make C=4:
A:B:C = 8:12:4
Then A:B:C = 8:12:4 and C:D = 4:5 gives us
A:B:C:D = 8:12:4:5 [2]
D:E = 2:1; scale both this and [2] up to where D is a whole number in both ratios:
A:B:C:D = 16:24:8:10 and D:E = 10:5 gives us
A:B:C:D:E = 16:24:8:10:5 [3]
Now we have a single ratio statement relating all 5 numbers, so we can solve the problem knowing that the sum of the five numbers is 18900.
Using the extended ratios we have found...
A = 16x
B = 24x
C = 8x
D = 10x
E = 5x
A+B+C+D+E = 16x+24x+8x+10x+5x = 63x = 18900
x = 18900/63 = 300
ANSWER: E = 5x = 5(300) = 1500
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