SOLUTION: I would appreciate your help on the following problem: The two digits in the numerator of a fraction whose value is 4/7 are reversed in its demoninator. The reciprocal of the fra
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Question 173693This question is from textbook Algebra Structure and Method Book 1
: I would appreciate your help on the following problem: The two digits in the numerator of a fraction whose value is 4/7 are reversed in its demoninator. The reciprocal of the fraction is the value obtained when 16 is added to the original numerator and 5 is subtracted from the original demonimator. Find the original fraction. I have 10t+u/10u+t=4/7; 10u+t/10t+u=10t+u+16/10u+t-5.
7(10t+u)=4(10u+t), 70t+7u=40u+4t, 66t=33u, t=3/6u. I'm not sure where to go from there. Thank you so much for any help you can give me. I appreciate it so much! This question is from textbook Algebra Structure and Method Book 1
Found 4 solutions by Mathtut, gonzo, stanbon, solver91311:Answer by Mathtut(3670) (Show Source):
You can put this solution on YOUR website! lets call the numbers a and b
:
:
but remember ab can also be written as 10a+b and ba can be written as 10b+a
:
so
:
now cross multiply
:
4(10b+a)=7(10a+b)
:
40b+4a=70a+7b
:
66a-33b=0:.......eq 1---->or revised eq 1
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we also know that
: (the reciprocal of 4/7)
:
so lets cross multiply again
:
4(10a+b+16)=7(10b+a-5)
:
40a+4b+64=70b+7a-35
:
-33a+66b=99.......eq 2
66a-33b=0........eq 1
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multiply eq 1 by 2 and add the equations together
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a's are eliminated and we are left with 99b=198
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:since we know that a=b/2 from revised eq 1
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so the original fraction is ...which reduces to 4/7
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incidently other number that would have worked had it not been for the 2nd equation would be
24/42
36/63
48/84
:
which all reduce to 4/7
You can put this solution on YOUR website! i believe i stumbled into the answer.
your equation of 66t = 33u was good.
that means that u = 2t
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you should have made your second equation equal to 7/4
that would be:
(10t+u+16)/(10u+t-5) = 7/4
then you could have solved as follows:
cross multiply:
7*(10u+t-5) = 4*(10t+u+16)
working this through:
70u + 7t - 35 = 40t + 4u + 64
move things around to get the u and the t on the left side and the constants on the right side.
66u - 33t = 35 + 64
66u - 33t = 99
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substitute 2t for u to get
132t - 33t = 99
99t = 99
t = 1
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since u = 2t, your answers are:
t = 1
u = 2
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to prove:
12/21 = 4/7 if you divide numerator and denominator by 3
12 + 16 / 21 - 5 = 28/16 = 7/4 if you divide numerator and denominator by 4
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You can put this solution on YOUR website! I would appreciate your help on the following problem:
The two digits in the numerator of a fraction whose value is 4/7 are reversed in its denominator.
I have 10t+u/10u+t=4/7
Rearrange:
7(10t+u) = 4(10u+t)
70t + 7u = 40u+4t
66t - 33u = 0
t = (1/2)u
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The reciprocal of the fraction is the value obtained when 16 is added to the original numerator and 5 is subtracted from the original denominator.
(10u+t)/(10t+u) = [10t+u+16]/[10u+t-5]
Rearrange:
(10u+t)(10u+t-5) = (10t+u)(10t+u+16)
[(10u+t)^2 - 5(10u+t)] = [(10t+u)^2 + 16(10t+u)]
[100u^2 + 20tu + t^2 -50u - 5t] = [100t^2 + 20tu + u^2 + 160t + 16u]
[99u^2 - 99t^2 -66u - 165t] = 0
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Substitute t = (1/2)u into the last equation to solve for "u":
Comment: This is a messy problem. I'll leave the substitution to you.
I get u = 2 so t = 1
Number: 12
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Cheers,
Stan H.
But if the value of the original fraction is , the value of the original fraction's reciprocal must be , so for the second relationship I think you should use:
Cross-multiplying and collecting like terms until the equations are in standard form, you should have the following:
Eq. 1: and Eq. 2:
Multiply Eq. 1 by -2: Eq. 3:
Add Eq. 3 to Eq. 2: →
Substitute into Eq. 1: → →
Therefore, the original fraction is which has a value of and further, and the answer checks.
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