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