Super easy
Let a b = 1 and define
P = a + 1 a + b + 1 b and Q = a + 1 1 + b + 1 1 .
What can we say about P and Q ?
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19 solutions
p=a/(a+1)+b/(b+1) =(b/b) a/(a+1)+(a/a) b/(b+1) because a/a=b/b=1 =ab/(ab+b)+ab/(ab+a) =1/(1+b)+1/(1+a) because (ab=1) =Q
take L.C.M. of P and Q and compare them.they will be equal.
but given question is different
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Qi Huan Tan has merely substituted ab for 1 in the expression for Q
okay i understood
asume that they are equal :p
P = a + 1 a + 1 − 1 + b + 1 b + 1 − 1
P = 2 − Q
P > Q
Where is the mistake?
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Mmmm but if p=1 q=1 then
p=2-q
1=2-1
You should take into account that ab=1 (Sorry for my bad english)
what if a=-1 ?
you have given the solution for a different question!!!!!!!!!
Let ab=1. Define ...it should be Let a and b = 1, Define to make the answer P=Q
Beyond the quick ways to show that P and Q are equal, you can show that both P and Q are 1 making the equality is trivial.
P = a + 1 a + b + 1 b = a + 1 a + 1 / a + 1 1 / a = a + 1 a + a + 1 1 = a + 1 a + 1 = 1
Q = a + 1 1 + b + 1 1 = a + 1 1 + ( 1 / a ) + 1 1 = a + 1 1 + a + 1 a = a + 1 a + 1 = 1
P= (1\ 1+1/a )+1/ (1+1/b). Now 1+1/a may be greater than or equal to less than a+1. similarly 1+1/b may be greater than or equal to or less than 1+b. Hence the answer is wrong
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I don't follow your explanation. I've double checked my calculations and can't see any mistakes. Could you point out a step in my first calculation that you think is incorrect?
There is no presumption that ab=1 in this problem Write a comment or ask a question...
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The first words at the top of the problem are "Let ab = 1"...
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you are right as there is the misunderstanding of the word "let" and "if"
What about when a=0 and b=0 then P=0 and Q=2 . This also shows the given answer P= Q is wrong for any set of values of a and b. Write a comment or ask a question...
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Since ab = 1 we know that a and b can't be 0.
b=1/a ; substitute b =1/a in P and Q . We will get same equation.
ab=1 or a=1/b; or a+1= (1+b)/b; or b/(b+1) = 1/(a+1);
Therefore , P = a/(a+1) + 1/(a+1); (Substituting second term) or P = 1 Similarly from ab = 1 we can get 1/(b+1) = a/(1+a);
So Q = 1/(a+1) + a / (1+a) (Substituting the second term) Simplifying Q=1 Therefore P=Q
We just need to substitute the variables to solve the question.
a b = 1
P = a + 1 a + b + 1 b
Q = a + 1 1 + b + 1 1
Expanding P :
P = a + 1 a + b + 1 b
Substituting 1 = a b :
P = a + a b a + b + a b b
P = a ( 1 + b ) a + b ( 1 + a ) b
P = 1 + b 1 + 1 + a 1 = Q
Therefore, P = Q ( P and Q are same).
Thus, the answer is: P = Q
Taking, Q=(b/b) (1/a+1)+(a/a)(1/b+1), using the fact that ab=1 we find that P=Q. I first liked the problem and then reshared it but as i researched the problem i find that it's cursed with the fact that it can easily be solved by putting values of a and b. So i un- shared it then.
just change it to... PROVE that P=Q
by solving using LCM we can get the answer easily.
take P/Q=(a(b+1)+b(a+1))/b+1+a+1 =ab+a+ab+b/a+b+2 =a+b++2ab/a+b+2 =a+b+2(1)/a+b+2 =a+b+2/a+b+2 =1 so P/Q=1 =>P=Q
P and Q are equal because they have the same product.
P=a/(a+1) +b/(b+1) =a(b+1)+b(a+1)/(a+1)(b+1) =ab+a+ab+b/ab+a+b+1 =1+a+1+b/1+a+b+1 =a+b+2/a+b+2 =1 now, Q=1/(a+1)+1/(b+1) =b+1+a+1/(a+1)(b+1) =a+b+2/ab+a+b+1 =a+b+2/a+b+1+1 =a+b+2/a+b+2 =1 so, P=Q
P= (1)/ (1)+1 + (1)/(1)+1 Q= 1/(1)+1 + 1/(1)+1
For P, p=a/(a+1) + b/(b+1) , p=a(b+1)+b(a+1)/(a+1)(b+1) , p=ab+a+ab+b/(a+1)(b+1) , p=ab+ab+a+b/(a+1)(b+1) , p=2ab+a+b/(a+1)(b+1) , since , ab =1 put the value of ab =1 in given equation ... p=2(1)+a+b/(a+1)(b+1),
then Q, Q= 1/(a+1) + 1(b+1) , Q=b+1+a+1/(a+1)(b+1) , Q=2+a+b/(a+1)(b+1) , so the answer is .. P=Q
just assuming the value a and b....
The most simple way is to substitute any values of a and b in both equations...in this case you get same answer of both...
you should take care that your answer doesn't come infinity
Since ab =1 therefore value of a & b are 1( 1X1 for a & b +1) Substituting these vvalue for expressions P & Q gives values as 1 therefore P = Q nswer.
K.K.GARG.india
p=q= (a+b+2)/(a+1)(b+1)
after solve p=\frac { 2+a+b }{ (a+1)(b+1) }
Q=\frac { 2+a+b }{ (a+1)(b+1) } so P=Q
just multiply the first component of expression of P by "b" in numerator and denominator and the second component of P by "a" in numerator and denominator and you will get an expression which is equal to Q
P = a + a b a + b + a b b = 1 + b 1 + 1 + a 1 = Q