fuel gauge registers one third of a tank then ten gallons is added now the gauge reads three fourths of a tank. How many more gallons is needed to fill the tank?

x/3 + 10 = 3x/4

4x + 120 = 9x
5x = 120
x = 24

The tank is 3/4 full, so 6 more gallons are needed.

To solve this problem, we can break it down into two parts:

1. Determine how much fuel was initially in the tank.
2. Calculate how much more fuel is needed to fill the tank.

Let's start with the first part:

The fuel gauge initially registers one third of a tank. Since the tank is not completely full, we can assume that the tank has a total capacity of 100%. Therefore, one third of a tank is equal to 1/3 * 100 = 33.33%.

Now, let's find out how much fuel was added:

When ten gallons of fuel is added, the fuel gauge reads three fourths of a tank, which is equal to 75%. Therefore, the difference between the initial reading and the reading after adding ten gallons is 75% - 33.33% = 41.67%.

Since ten gallons of fuel represents 41.67% of the tank capacity, we can set up a proportion to find the capacity of the tank:

10 gallons / X = 41.67% / 100%

Simplifying the proportion, we have:

10 / X = 41.67 / 100

Cross-multiply to solve for X:

100 * 10 = 41.67 * X
1000 = 41.67X
X = 1000 / 41.67
X ≈ 23.98

Therefore, the tank capacity is approximately 24 gallons.

Now, let's move on to the second part:

To determine how many more gallons are needed to fill the tank, we subtract the current fuel level (represented by 75% or three-fourths of the tank capacity) from the tank capacity (represented by 100%):

Amount to fill the tank = 100% - 75% = 25%

Using the proportion again:

Y gallons / 24 gallons = 25% / 100%

Simplifying the proportion, we have:

Y / 24 = 25 / 100

Cross-multiply to solve for Y:

100 * Y = 24 * 25
100Y = 600
Y = 600 / 100
Y = 6

Therefore, 6 more gallons are needed to fill the tank.