Solve for r 103.7=(1/(1+r))*30.4/r

Math
103.7=(11+r)⋅30.4r
Rewrite the equation as (11+r)⋅30.4r=103.7.
(11+r)⋅30.4r=103.7
Factor each term.
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Multiply 11+r by 30.4r.
11+r⋅30.4r=103.7
Multiply 11+r and 30.4r.
30.4(1+r)r=103.7
30.4(1+r)r=103.7
Solve for r.
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Multiply each term by 1+r and simplify.
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Multiply each term in 30.4(1+r)r=103.7 by 1+r.
30.4(1+r)r⋅(1+r)=103.7⋅(1+r)
Cancel the common factor of 1+r.
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Factor 1+r out of (1+r)r.
30.4(1+r)(r)⋅(1+r)=103.7⋅(1+r)
Cancel the common factor.
30.4(1+r)r⋅(1+r)=103.7⋅(1+r)
Rewrite the expression.
30.4r=103.7⋅(1+r)
30.4r=103.7⋅(1+r)
Simplify 103.7⋅(1+r).
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Apply the distributive property.
30.4r=103.7⋅1+103.7r
Multiply 103.7 by 1.
30.4r=103.7+103.7r
30.4r=103.7+103.7r
30.4r=103.7+103.7r
Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
r,1,1
Since r,1,1 contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part 1,1,1 then find LCM for the variable part r1.
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
1
The factor for r1 is r itself.
r1=r
r occurs 1 time.
The LCM of r1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
r
r
Multiply each term by r and simplify.
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Multiply each term in 30.4r=103.7+103.7r by r in order to remove all the denominators from the equation.
30.4r⋅r=103.7⋅r+103.7r⋅r
Cancel the common factor of r.
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Cancel the common factor.
30.4r⋅r=103.7⋅r+103.7r⋅r
Rewrite the expression.
30.4=103.7⋅r+103.7r⋅r
30.4=103.7⋅r+103.7r⋅r
Multiply r by r by adding the exponents.
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Move r.
30.4=103.7r+103.7(r⋅r)
Multiply r by r.
30.4=103.7r+103.7r2
30.4=103.7r+103.7r2
30.4=103.7r+103.7r2
Solve the equation.
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Rewrite the equation as 103.7r+103.7r2=30.4.
103.7r+103.7r2=30.4
Move 30.4 to the left side of the equation by subtracting it from both sides.
103.7r+103.7r2-30.4=0
Factor the left side of the equation.
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Factor 0.1 out of 103.7r+103.7r2-30.4.
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Factor 0.1 out of 103.7r.
0.1(1037r)+103.7r2-30.4
Factor 0.1 out of 103.7r2.
0.1(1037r)+0.1(1037r2)-30.4
Factor 0.1 out of -30.4.
0.1(1037r)+0.1(1037r2)+0.1(-304)
Factor 0.1 out of 0.1(1037r)+0.1(1037r2).
0.1(1037r+1037r2)+0.1(-304)
Factor 0.1 out of 0.1(1037r+1037r2)+0.1(-304).
0.1(1037r+1037r2-304)
0.1(1037r+1037r2-304)
Reorder terms.
0.1(1037r2+1037r-304)
Replace the left side with the factored expression.
0.1(1037r2+1037r-304)=0
0.1(1037r2+1037r-304)=0
Divide each term by 0.1 and simplify.
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Divide each term in 0.1(1037r2+1037r-304)=0 by 0.1.
0.1(1037r2+1037r-304)0.1=00.1
Cancel the common factor of 0.1.
1037r2+1037r-304=00.1
Divide 0 by 0.1.
1037r2+1037r-304=0
1037r2+1037r-304=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1037, b=1037, and c=-304 into the quadratic formula and solve for r.
-1037±10372-4⋅(1037⋅-304)2⋅1037
Simplify.
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Simplify the numerator.
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Raise 1037 to the power of 2.
r=-1037±1075369-4⋅(1037⋅-304)2⋅1037
Multiply 1037 by -304.
r=-1037±1075369-4⋅-3152482⋅1037
Multiply -4 by -315248.
r=-1037±1075369+12609922⋅1037
Add 1075369 and 1260992.
r=-1037±23363612⋅1037
r=-1037±23363612⋅1037
Multiply 2 by 1037.
r=-1037±23363612074
r=-1037±23363612074
The final answer is the combination of both solutions.
r=-1037-23363612074,-1037+23363612074
r=-1037-23363612074,-1037+23363612074
r=-1037-23363612074,-1037+23363612074
The result can be shown in multiple forms.
Exact Form:
r=-1037-23363612074,-1037+23363612074
Decimal Form:
r=0.23698936…,-1.23698936…
Solve for r 103.7=(1/(1+r))*30.4/r

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