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Home » 2. The sum of 2 numbers is 18. If the sum of their reciprocals is ¼, find the numbers.
2. The sum of 2 numbers is 18. If the sum of their reciprocals is ¼, find the numbers.
CBSE | Class 10 | Class 10 | Frank | Maths | Problems Based On Quadratic Equations
2. The sum of 2 numbers is 18. If the sum of their reciprocals is ¼, find the numbers.

Solution:-

Let us assume the numbers be P and Q.

As per the condition given in the question,

The sum of 2 numbers is 18, P + Q = 18 … [equation i]

the sum of their reciprocals is ¼, 1/P + 1/Q = ¼ … [equation ii]

Consider equation (i), P + Q = 18

P = 18-Q

Now, substitute the value of P in equation (ii) we get,

1/(18-Q) + 1/Q =1/4

(Q +(18-Q))/((18-Q)Q) =1/4

By cross multiplication,

4(Q + 18-Q) = (18-Q)Q

    \[4\left( 18 \right)=18Q-{{Q}^{2}}\]

    \[72=18Q-{{Q}^{2}}\]

Now, transposing we get,

    \[{{Q}^{2}}18Q+72=0\]

    \[{{Q}^{2}}-12Q-6Q-72=0\]

    \[{{Q}^{2}}-12Q-6Q-72\text{ }=\text{ }0\]

Take out common in each terms,

Q(Q - 12) - 6(Q - 12) = 0

(Q - 12) (Q - 6) = 0

Equate both to zero,

Q - 12 = 0

Q - 6 = 0

Q = 12

Q = 6

Therefore, the numbers are 6 and 12.

i

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