Let the number is x.
reciprocal of the number is 1/x.
It is given the sum of the number and it's reciprocal is 10/3.
x+1/x=10/3
x²+1/x=10/3
3(x²+1) =10x
3x²+3=10x
3x²-10x+3=0 by solving
3x²-x-9x+3
x(3x-1) -3(3x-1)
(3x-1)(x-3)
3x-1=0 or x-3=0
3x=1 or x=3
x=1/3
so x=3 or 1/3 both are right answer
Proof
Number is 3 and reciprocal 1/3
or number 1/3 &reciprocal 3
3+1/3 =10/3
9+1/3=10/3
10/3=10/3

Problem Analysis:
Let's assume the natural number is "x". According to the given information, the sum of the natural number and its reciprocal is 10/3. We can represent this as an equation:

x + 1/x = 10/3

To find the value of "x", we need to solve this equation. We can do this by multiplying both sides of the equation by "x" to eliminate the fraction:

x^2 + 1 = (10/3)x

Now, we have a quadratic equation. We can rearrange it to the standard form:

x^2 - (10/3)x + 1 = 0

Solving the Quadratic Equation:
To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -10/3, and c = 1. Plugging these values into the quadratic formula, we get:

x = [-(10/3) ± √((-10/3)^2 - 4(1)(1))] / (2(1))

Simplifying further:

x = [-(10/3) ± √(100/9 - 4)] / 2

x = [-(10/3) ± √(100/9 - 36/9)] / 2

x = [-(10/3) ± √(64/9)] / 2

x = [-(10/3) ± (8/3)] / 2

Finding the Solutions:
Now, we have two possible solutions for "x":

1. x = (-(10/3) + (8/3)) / 2 = -1/3
2. x = (-(10/3) - (8/3)) / 2 = -3

However, since the question states that the number is a natural number, we can ignore the negative solution (-3) and consider the only valid solution as x = -1/3.

Conclusion:
The natural number that satisfies the given condition is -1/3.