Question

The ratio of the amount of tomato plants in Inessa’s garden to the number of tomato plants in Ralph’s garden was 5:6. After 1/2 of the tomato plants from Inessa’s garden were replanted into Ralph’s garden, Ralph’s garden had 850 tomato plants. How many tomato plants did Ralph’s garden have in the first place?
Plz help thanks

Answer 1

$\bf \begin{cases} I=\textit{Inessa's tomatoes amount}\\ R=\textit{Ralph's tomatoes amount} \end{cases}~\hspace{4em} \stackrel{\textit{their ratio is 5:6}}{\cfrac{I}{R}=\cfrac{5}{6}}\implies I=\cfrac{5R}{6} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{\textit{half of Inessa's tomatoes}}{\cfrac{5R}{6}\cdot \cfrac{1}{2}\implies \cfrac{5R}{12}}~\hspace{4em}\stackrel{\textit{Ralph gets Inessa's half}}{R+\cfrac{5R}{12}}~~=~~\stackrel{\textit{Ralph's new amount}}{850}$

$\bf ~\dotfill\\\\ \stackrel{\textit{multiplying both sides by the }\stackrel{LCD}{12}}{12\left( R+\cfrac{5R}{12} \right)=12(850)}\implies 12R+5R=10200 \\\\\\ 17R=10200\implies R=\cfrac{10200}{17}\implies \boxed{R=600}$

if you wonder why we multiplied by the LCD, is just to do away with the denominators.