Question

the numbers which divide 80 in such a way that sum of
their reciprocals is 4/75 are ​

Answer 1

answer is 30,50

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Answer 2

Given,

The numbers divide = 80

The sum of their reciprocals = $\frac{4}{75}$

To Find,

The numbers which divide 80 =?

Solution,

We can solve the question using the following steps:

Let the two numbers which divide 80 be a and b.

Now,

The sum of reciprocals of a and b is = $\frac{4}{75}$

$\frac{1}{a} + \frac{1}{b} = \frac{4}{75}$

Solving the above equation,

$\frac{a + b}{ab} = \frac{4}{75}$

$75(a + b) = 4*ab$.  ---------- (1)

Now, since the numbers divide 80,

$a + b = 80$

$a = 80 - b$

Substituting the above equations in (1),

$75*80 = 4(80 - b)b$

$6000 = 320b - 4b^{2}$

$b^{2} - 80b + 1500 = 0$

Solving the above quadratic equation,

$b^{2} - 30b - 50b + 1500 = 0$

$b(b - 30) - 50(b - 30) = 0\\(b - 50)(b - 30) = 0$

Hence, b = 30,50

Now,

$a + b = 80\\a = 80 - b\\a = 80 - 30, a = 80 - 50\\a = 50, 30$

Hence, the numbers which divide 80 are 50 and 30.