Question

• The sum of two numbers a and b is 15 and
the sum of their reciprocals 1/a and 1/b is
3/10. Find the numbers a and b.​

Answer 1

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Let ,

The two numbers be a and b

Condition (I) : Sum of two number is 15

a + b = 15 ---- (i)

It can be written as ,

a = 15 - b ---- (ii)

Condition (II) : Sum of their reciprocals is

3/10

$\sf \mapsto \frac{1}{a} + \frac{1}{b} = \frac{3}{10} \\ \\ \sf \mapsto</p><p> \frac{b + a}{ab} = \frac{3}{10} \\ \\ \sf \mapsto</p><p> \frac{15}{ab} = \frac{3}{10} \\ \\ \sf \mapsto</p><p>150 = 3ab \\ \\ \sf \mapsto</p><p>ab = 50</p><p>$

From equation (ii) , we get

$\sf \mapsto (15 - b)b = 50 \\ \\ \sf \mapsto</p><p>15b - {(b)}^{2} - 50 = 0 \\ \\ \sf \mapsto</p><p> {(b)}^{2} - 15b + 50 = 0 \\ \\ \sf \mapsto</p><p> {(b)}^{2} -10b-5b+50 = 0 \\ \\ \sf \mapsto</p><p>b(b-10)+5(b-10) = 0 \\ \\ \sf \mapsto</p><p>b = 10 \: or \: b = -5</p><p>$

Putting the values of b in eq (i) , we get

$\sf \mapsto a + 10 = 15 \: or \: a + (-5) = 15 \\ \\ \sf \mapsto </p><p>a = 5 \: or \: a = 20$

Hence , the two numbers are 5 and 10 or 20 and -5

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