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.
The question is contradictory. As the earth's mass is higher than the moon's mass and the object is $9$ times closer to the earth, the earth's attraction will be $9^2\frac {m_E}{m_M}\approx 81 \cdot 81.3$ It appears the object should be about $9$ times closer to the moon than to the earth to make the gravitational forces (almost) equal. You don't say what the "distance in question $R$" is. I suspect it is the distance from the earth to the moon and you are to find the location where the gravitational forces are equal. It will be about $1/10$ of the way from the moon to the earth, but maybe you are expected to get a more precise answer. You don't have anything to set $R$ in the data you have presented. You also don't say what you are asked to solve for. What is the question?