The problem is described as follows:  Newton's Law of Universal Gravitation: $F=G \frac{m_1 m_2}{r^2}$ for any point masses $m_1,m_2$ with the distance between them $r$.    The factor $G≈6.67∙10^{-11}  N(m/kg)^2$ is a universal constant.  Denote the mass of Earth as $m_E≈5.972∙10^{24}  kg$,  the mass of the Moon as $m_M≈7.348∙10^{22}  kg$.  The unknown mass of the object will be $m$ and the distance in question will be $R$.  Then the distance between the object and Earth is  $\frac{1}{10} R$ and the distance between the object and the Moon is $\frac{9}{10} R$. The force from the Moon and Earth acting on the object is equal.  Now, I worked on the problem and found that from the law we obtain:  $$G \frac{mm_E}{\left(\frac{9}{10} R\right)^2} =G \frac{mm_M}{\left(\frac{1}{10} R\right)^2}$$  It seems like there is no solution to the question as not enough information is given. However, I was given this question as an assignment in math class and believe that steps should be shown towards a solution whether it exists or not. I fail to see how to solve this problem, at least to an approximate value, which is what is needed. If anybody knows, help will be immensely appreciated. Thank you.
Problem:  $F = \frac{(G m_1 m_2)}{r^2}$  An object of unknown mass is placed directly between the Earth and the Moon such that the forces of gravity acting on the object are equal from both the Earth and the Moon. The distance from the Moon to the object is exactly one-tenth of the distance from the Moon to Earth. Using the above formula, determine the distance from the Moon to the Earth.  Given is:  The factor $G≈6.67∙10−11\mathbf{N(\frac{m^2}{kg^2})}$ is a universal constant.  Mass of Earth $≈5.972∙10^{24}\mathbf{kg}$   Mass of the Moon as $≈7.348∙10^{22}\mathbf{kg}$  Mass of unknown object = $m$  My solution:  $F = \frac{(6.67 *10^-11) m (7.348*10^{24})}{R^2}$  $F = \frac{(6.67 *10^-11) (5.92* 10^22) m}{10R^2}$  As gravitational force acting on the object is equal   $\frac{(5.92\cdot 10^22)}{100r^2} = \frac{(7.348*10^{24})}{r^2}$  $R = 1.24 \cdot 10^4 KM$  Is this a viable solution to the problem? Please provide feedback. Thank you!
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