# a+b=5, a*b=10 S= (a/b)+(b/a). Calculate S.a+b=5, a*b=10 S= (a/b)+(b/a). Calculate S.

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a+b = 5

ab= 10

S = (a/b) + (b/a)

Let us rewrite using the common denomonator:

==> S = (a^2 + b^2) / a*b

==> S = [(a+b)^2 - 2ab] / ab

Now let us substitute with values from (1) and (2):

==> S = (5^2 - 2*10)/ 10

= (25-20)/10 = 5/10 = 1/2

==> S = 1/2

a+b=5 and ab =10 . To find S - (a/b)+(b/a)

Solution:

S = a/b+b/a = a^2/ab+b^2/ab = (a^2+b^2/ab = {(a+b)^2-2ab}/ab. Now substitute the given valeue for a+b =5 and an =10.

S = (5^2-2*10)/10 = (25-10)/10 = 5/10 =1/2.

a+b=5, ab=10 S= (a/b)+(b/a)

s=(a^2+b^2)/ab

s=([a+b]^2-2ab)/ab

so s=(5^2-2*10)/10

s=5/10

s=1/2

To calculate S, we need to make the denominator of (a/b) and (b/a) the same.

When this is done and the denominator is made the same,i.e a*b, we get:

S=((a*a)/ab)+(b*b)/ab)= (a^2+b^2)/ab

Now a^2+b^2=(a+b)^2-2*a*b=5^2-2*10=25-20=5.

So the numerator is 5.

As a*b=10, the denominator is 10.

Therefore S=5/10=1/2

To calculate the sum of 2 ratios, we'll have to calculate the least common denominator.

LCD = a*b

(a/b)+(b/a)=a*(a/b)+b*(b/a)=(a^2 +b^2)/a*b

We'll add and subtract the same amount, to the numerator, namely 2*a*b, and we'll combine the terms in a convenient way:

(a^2+2*a*b +b^2-2*a*b)/a*b

But a^2+2*a*b +b^2 = (a+b)^2

S=(a+b)^2/a*b - 2a*b/a*b

S = 5^2/10 - 2*5/10 = 25/10 - 1 = (25-10)/10 = 15/10

**S = 3/2**