Monday

The tank by the Glass House door holds 20 liters, and Comet is reading the plant-food label with a frown.

"It says to use a 10 percent solution for seedlings. Ours is 10 percent now. But the new trays want it weaker."

Wren lifts the measuring pitcher. "So we add plain water. What do you notice about the plant food itself?"

"It does not change. Two liters of plant food, however much water we pour in."

Nova hovers over the tank and projects a fraction on the glass. "Would you like a hint? Write the percent as food over total."

"Two over twenty plus x," Comet says. "Times 100. So 200 over x plus 20."

"A fraction with x on the bottom," Wren says. "A rational expression. Add 5 liters and see what it says."

Comet works it out. "Eight percent. Let me check with the pitcher."

The crew's tank holds 20 liters at 10%, so it holds 2 liters of plant food. That amount never changes.

Add x liters of plain water and the total is x + 20 liters. The percent is 2 ÷ (x + 20) × 100.

Written as one fraction: C(x) = (200)/(x + 20). A polynomial over a polynomial is a rational expression.

One value is not allowed: x = -20 would make the bottom zero. A fraction with a zero bottom has no value.

Every row is the crew's own reading from Nova's log, made up for the Glass House. The rule and the pitcher agree.

A rational expression can be split into a whole part and a leftover fraction, just like 7/2 = 3 + 1/2.

Divide the top by the bottom: (x + 5) ÷ (x + 2) gives quotient 1 and remainder 3.

So (x + 5)/(x + 2) = 1 + 3/(x + 2). The quotient tells you what the fraction is close to for large x.

The same move works on bigger tops: (x² + 3x + 5)/(x + 1) = x + 2 + 3/(x + 1). Divide, then write quotient plus remainder over the divisor.

Strong start. Tomorrow you simplify, multiply and add rational expressions, two ways each.