At the end of their paddling trip, Natalie and Herb drive back in separate vehicles. Natalie has 8 gallons of gas left in the tank of her vintage 1969 Volkswagen Beetle, which gets 30 miles to the gallon. Herb has 12 gallons left in the tank of his 1998 Jeep Grand Cherokee, which gets only 18 miles to the gallon.

After how many miles will Natalie and Herb have the same amount of gas in their tanks, and what is that amount?

After x miles, you want

8 - x/30 = 12 - x/18

To find out after how many miles Natalie and Herb will have the same amount of gas in their tanks, we need to determine the point at which the total amount of gas used by both vehicles is equal. We can start by calculating the total miles each vehicle can travel with the given amount of gas.

For Natalie's Volkswagen Beetle:
Natalie has 8 gallons of gas, and her car gets 30 miles per gallon. So, Natalie can travel 8 * 30 = 240 miles.

For Herb's Jeep Grand Cherokee:
Herb has 12 gallons of gas, and his car gets 18 miles per gallon. So, Herb can travel 12 * 18 = 216 miles.

Now, let's calculate how many miles they need to travel for the gas in their tanks to be equal:
Let's assume they travel x miles.

For Natalie's Beetle, the amount of gas used is given by (x / 30) gallons.
For Herb's Grand Cherokee, the amount of gas used is given by (x / 18) gallons.

We can now set up an equation:
8 - (x / 30) = 12 - (x / 18)

Now, we can simplify and solve for x:
Multiply through by 30 and 18 to eliminate fractions:
240 - x = 360 - (5/3)x

Multiply through by 3 to eliminate the fraction:
720 - 3x = 1080 - 5x

Combine like terms:
2x = 360

Divide by 2 to isolate x:
x = 180

Therefore, Natalie and Herb will have the same amount of gas in their tanks after traveling 180 miles, and that amount will be:
8 - (180 / 30) = 8 - 6 = 2 gallons.

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