Questions: The gasoline gauge on a van initially read 1/8 full. When 13 gallons were added to the tank, the gauge read 2/3 full. How many more gallons are needed to fill the tank? To fill the tank, more gallons are needed.

Solution Steps

Let $T$ represent the total capacity of the gasoline tank in gallons.

Initially, the tank is $\frac{1}{8}$ full, which means it contains $\frac{1}{8}T$ gallons of gasoline. After adding 13 gallons, the tank becomes $\frac{2}{3}$ full. This can be expressed as: $$\frac{1}{8}T + 13 = \frac{2}{3}T$$

Subtract $\frac{1}{8}T$ from both sides to isolate the variable: $$13 = \frac{2}{3}T - \frac{1}{8}T$$ To combine the terms on the right-hand side, find a common denominator (which is 24): $$13 = \frac{16}{24}T - \frac{3}{24}T$$ Simplify: $$13 = \frac{13}{24}T$$ Now, solve for $T$: $$T = 13 \times \frac{24}{13} = 24$$ So, the total capacity of the tank is 24 gallons.

The tank is currently $\frac{2}{3}$ full, which means it contains: $$\frac{2}{3} \times 24 = 16 \text{ gallons}$$ To fill the tank completely, the remaining gallons needed are: $$24 - 16 = 8 \text{ gallons}$$

Final Answer