Filling the tank | Scoins.net | DJS

## Filling the tank

This page reviews the issues that fit with the objective of putting fuel into your car when there are two petrol stations, one nearby and a cheaper one further away. I offer a rule of thumb, that when the extra mileage to add to reach the further one equates to the pennies difference in price, there is gain to be had from going further once the spend exceeds £35.

That rule of thumb assumes, as I have done initially, that you have a car that generally gives 45 miles to the gallon but with the understanding that consumption varies with road conditions.

I also assume that the price of fuel is approaching £2 per litre.

First, let us take a concrete example and explore.

Q1. Show that 45 miles to the gallon (mpg) is ten miles to the litre and give the appropriate precision. Reversing that, indicate what ten miles to the litre (to the nearrest mile) is in mpg.

We will assume that the vehicle achieves ten miles per litre.

Let us suppose that the nearby garage, that you pass every time you drive home, is offering £1.93 per litre of fuel and that the supermarket three miles away is offering £1.89. If you were to combine filing the tank with shopping, this is a no-brainer; you would fill up if appropriate whenever at the distant supermarket.

Let us assume that the supermarket is in the opposite direction to your commute so that a specific fuel trip adds six miles of use. We should cost the difference.

Six miles (three each way) represents a simple fuel cost of 0.6 litre or around £1.13. Let's call the two fuel points the local and the supermarket.

Q2.  Suppose you aim to put £60 worth of fuel into the tank, how much more fuel do you get at the supermarket? By the time you have reached home again, what is your money gain to the nearest penny? Is this worthwhile?

Q3.  Repeat Q2 but aim to put £90 worth of fuel into the tank.

Q4.  Suppose the price difference is 6p (say, 1.88 and 1.94 in £/litre) and continue to assume the extra distance is twice three miles. Find the gain at the supermarket.

Q5. Repeat Q4 for an extra distance of ten miles (five each way).

Now we have explored some numbers,we're ready to use some algebra. The pump prices are P and Q, with P>Q, in pence per litre. The additional distance to travel is d, or 2d in total. The total spend on fuel is T in pence. Assume for the time being that the local garage is very local indeed.

Q6. Write two equations that show the number of litres gained by filling up in either place. When these volumes are the same, you can equate them and write this as T in terms of P, Q and d. Check the units of your result to confirm you have a money result.

I say this formula is        T = 2d/10  *  P.Q / (P-Q).          My units check.

The T produced by this formula is the minimum spend at the supermarket to show any gain at all.

Q7.  Apply the formula given to

(a)  P=194, Q=188, d=3         and       (b) to P=198, Q=188, d=5.

Q8.  Explore how T varies with different values of P, Q and d and make some comment.

I suggest to you that differences under £1 are not worthwhile unless other shopping occurs. If money is tight and there are shops within convenient walking distance, then the shopping price difference of local and supermarket might well represent the cost of travel to the distant supermarket. If you were to price the car on a per mile basis for any year, you might well find that not only is the car expensive, but that either you don't need one at all or you need to find good (efficient) use for it to justify its ownership, which means turning every trip into covering several reasons for travel.

DJS 20220628

A1. 4.54 litres = 1 gallon. 45 miles to the gallon = 9.912 miles per litre, which is 10 to 1 sig fig.   10 miles to the litre ± 0.5 mile is 43.13  to 47.67 mpg. This is 45.4±2.27.

A2.  £60/£1.93 = 31.088 litres. 60/1.89=31.746, so a gain of 0.658 litres, less the 0.6 litres for the journey, so this gain is proiced at £1.89, about 11p gain. This is hardly worthwhile, unless you do some significant shopping at prices that beat what you can achieve more locally.

A3. 90/1.89 - 90/1.93 - 0.6 = 0.3869. 0.387 x 189 = 73p. Again, hardly worth it unless there are other advantages.

A4.  (90 (1/1.88 - 1/1.94) -0.6 ) * 1.88 = 1.6655 so a gain of £1.66.

A5.   Ten miles represents a litre used, so  (90 (1/1.88 - 1/1.94) - 1 ) * 1.88 = 0.903 so a gain of 90p..

A6.  T/P = litres = T/Q - 2d/10    I say T = 2d/10  *  P.Q / (P-Q).

d (miles) 10 miles/litre, P and Q are p/litre. so miles cancel out, litres cancel out and the money leaves an answer in pence.

A7. (a)  £36.47      (b)   £37.22

A8.  I made a headline comment, that while P is near £1.90 per litre, then the pennies difference in price (P-Q) is similar to the extra distance to the cheaper fuel and that this makes the value of T around £35.

When P=215, and Q is up to 10p less, the figures lie around £45. At 203, around £40. At 176, £30.

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