I've come across the following question: "Let $6\frac{1}{m}\times n\frac{2}{11}=21$, where m, n are natural numbers. Find m+n."
Outside of actually solving the question, I've tried to guess a value for n (3 in this case) and adjust m to so solve the equation. Is there a faster way to solve this problem? For context, this is on a test where no work is allowed and only the answer must be written.
Thank you for your help.
Assuming from the title that $6\frac1m$ means $6+\frac 1m=\frac {6m+1}m$ I would focus on clearing the denominator of $11$. That requires that $6m+1$ must be a multiple of $11$. Maybe $6 \cdot 9=54$ comes to mind, or maybe you note that $m=1$ makes the numerator $7 \pmod {11}$ and each increase of $2$ in $m$ increases the numerator by $1 \pmod {11}$. Either way we find $m=9$ to be a candidate and if we are to work mentally the numbers must be small so this must be right. Then we can write $n\frac 2{11}=21 \cdot \frac 9{55}=\frac {189}{55}$ and approximate division is enough to give $n=3$
