Question

the sum of two numbers is 15 and the sum of their reciprocals is 3/10 then the numbers are

Answer 1

let the required number be X and Y.
X+Y = 15
1/X + 1/Y = 3/10.
add the fractions, (X+Y)/XY = 3/10.
so, X+Y = 3/10 * XY.
if we equate the equation then

15 = 3/10 * XY.
so, XY = 50.
so, X= 50/Y.
now, X+Y = 15
so, 50/Y +Y = 15
so, Y^2 -15Y + 50 = 0.
if we solve the quadratic equation, we get Y = 5 or 10.
now, X+Y = 15
so, X = 15 -5 = 10
or, x = 15- 10 = 5 .
so, answer is X = 5 or 10
Y = 5 or 10.
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Answer 2

$\huge \sf \color{purple}{\underline{\underline{Answer}}} :$

Two numbers are 5 and 10

$\huge \sf \color{purple}{\underline{\underline{Solution}}}:$

✯ Let the number be x .

✯ Then the other number is 15 - x.

$\sf \color{brown}{Using \: the \: given \: information \: , we\: get }$

$\sf \frac{1}{x}+\frac{1}{15-x}=\frac{3}{10} \\ \\ \sf \frac{15-x+x}{x(15-x)}=\frac{3}{10} \\ \\ \sf \frac{15}{(15x-x²)}=\frac{3}{10} \\ \\ \sf 150=45x-3x² \\ \\ \sf 3(x²-15+50)=0 \\\\ \sf x²-15+50=0 \\ \\ \sf x²-5x-10x+50 =0 \\ \\ \sf (x-10)(x-5)=0 \\ \\ \sf x-10=0 \: \: \ \ \ OR \ \ \ \ \sf x-5 \\ \\ \sf\boxed{ \colorbox{lightgreen}{x=10}} \ \ \ \ or \ \ \ \ \boxed{\colorbox{lightgreen}{ x = 5 }}$

When x = 10 , then the other number is 15-5 = 10

When x = 5, then the other number is 15-5 = 10

$\color{red}━━━━━━━━━━━━━━━━━━$

$\sf \color{blue}{Hence, \ the \ two \ numbers \ are \ 5 \ and \ 10 }$

$\color{red}━━━━━━━━━━━━━━━━━━$

$\color{red}━━━━━━━━━━━━━━━━━━$