UPTET Paper 2 Math and Science 2017 Solved Paper

Concept:When two numbers have a fixed sum and a fixed sum of reciprocals, we can find their product from the reciprocal sum and then determine the numbers.Explanation:Let the two natural numbers be $a$ and $b$.Given: $a + b = 8$ and $\frac{1}{a} + \frac{1}{b} = \frac{8}{15}$.Rewrite the reciprocal sum as $\frac{a+b}{ab} = \frac{8}{15}$.Substitute $a+b=8$: $\frac{8}{ab} = \frac{8}{15}$. This gives $ab = 15$.Now we need two numbers with sum 8 and product 15. The only pair of natural numbers that satisfy these are 3 and 5.We can verify: $(a+b)^2 - 4ab = (a-b)^2$, so $64 - 60 = 4$ gives $a-b=2$. Solving $a+b=8$ and $a-b=2$ yields $a=5$, $b=3$.Check the options: Option A (1,7) product 7; Option B (4,4) product 16; Option C (2,6) product 12; Option D (3,5) product 15. Only option D fits both conditions.Answer:The numbers are 3 and 5, which corresponds to option D.