#### Answer

$20$ gallons of $13\%$ nitrogen plant food and $30$ gallons of $18\%$ nitrogen plant food.

#### Work Step by Step

Step 1. Assume we need $x$ gallons of $13\%$ nitrogen plant food, the nitrogen contained here is given by $0.13x$
Step 2. Assume we need $y$ gallons of $18\%$ nitrogen plant food; the nitrogen contained here is given by $0.18y$
Step 3. The 50 gallons of $16\%$ plant food will contain nitrogen given by $0.16(50)=8$ gallons
Step 4. As the amount of nitrogen is balanced, we have $0.13x+0.18y=8$
Step 5. The total volume is preserved; we have $x+y=50$ or $y=50-x$
Step 6. Using substitution, we have $0.13x+0.18(50-x)=8$, which gives $0.13x+9-0.18x=8$ or $0.05x=1$ and $x=20$ gallons; thus $y=50-x=30$ gallons.
Step 7, We conclude that we need $20$ gallons of $13\%$ nitrogen plant food and $30$ gallons of $18\%$ nitrogen plant food.