All science is essentially applied mathematics.
This post takes a short detour into the world of mathematics, because I have been pondering the following for a few weeks.
In New Zealand, there is a magical card* you can get from a popular motoring organisation. The magic card gives you a discount on motor fuel if you use it at particular refuelling stations. The rules are as follows:
Maths equations are just sentences, with numbers instead of words. The equation for working out the amount of discount you get per litre is the numerical equivalent of the sentence "The amount you would have paid if you had paid full price for a particular amount of fuel, minus the amount you did pay for that particular amount of fuel, then divided by the number of litres in the amount of fuel". It's quicker to replace "The amount you would have paid if you had paid full price for a particular amount of fuel" with "40A+50X", and replace "The amount you did pay for that particular amount of fuel" with "40A+50(X-b)". You can also replace "The number of litres in the amount of fuel" with "A(40/X)+50". So the full equation would be "(40A+50X)-(40A+50(X-b))/A(40/X)+50".
I'll explain.
You have to buy a minimum amount of $40 of fuel to claim the discount, but you can use the discount on up to 50L. So if X is the price of one full-price litre of fuel, then the price of 50L of full-price fuel is 50X. The number of litres you get for $40 is 40/X. The amount you pay for putting $40 of fuel in your tank, then putting 50L of fuel in your tank, but not claiming the discount, would be $40 plus 50X.
If you are putting $40 of fuel in your tank A times, then subsequently using the discount on 50L, then the total amount you would save would be (40A+50X) - (40A+50(X-b)), where b is the discount per litre (6c or 10c).
To calculate your savings per litre, you need to divide the total savings by the number of litres you bought (A(40/X)+50). Again, this has to include the litres you bought full price (those $40 purchases) plus the 50 litres you claimed the discount on.
Note: "Savings per litre" is just another way of saying "Total savings divided by total litres".
A couple of days ago, petrol at my local refuelling station (which accepts the magic card) was $2.35 without a discount. So assuming the discount was 6c every time you refuel, and you are refuelling at a station which sells fuel at $2.35 per litre, here is a graph showing the relationship between the amount you save and the number of times you save the points from $40 of fuel.
I am a little surprised. I thought the line would plateau and then begin to drop after a much smaller number of $40 purchases. Perhaps that is the cynic in me, but also I still approach maths with a childlike sense of wonder because I don't understand it very well. The more $40 purchases you make, at least up to 15 (but I ran the calculations a little further and it seemed to go into the hundreds), the more you save. This is probably a logarithmic curve due to its shape, which means it should keep going up.
But don't all rush out to put hundreds of $40 fuel purchases into your car.
Most of the savings are made at the beginning of the curve, where you buy $40 of fuel several times. After about 3 $40 loads, the savings gained by subsequent fuellings drop to about 1c per litre. After 7 $40 loads, you are getting 14c off per litre, but it takes another 3 loads to get to 15c. Bear in mind, though, that each cent you get is a cent off every litre that you have purchased. So if you put in $40 (a smidgen over 17 litres) seven times, then put in 50 litres and claim the discount, you put in 169 litres and get a little over 14c off each one, saving you a whopping $24 (note I have rounded at each calculation here, so this is approximate). If you put $40 in ten times, then put in 50 litres and claim the discount, you get 220 litres at a saving of 15 cents each, saving you $33 in total.
Is the extra $11 worth you refuelling three more times? This is where I am on firmer footing, due to my background in ecology. In ecology, there are always other factors to consider that mess up your calculations. How much does it cost you to put in that $40 of fuel? Sure, there's the $40, but do you have to travel out of your way to get to the fuel station? How much fuel do you use driving into the station, parking, restarting your engine, then driving out? How much is your time worth? How much would you pay not to have to go to the refuelling station every few days?
And how many times do you normally refuel in two months? Fifteen is a lot! If you are using that much fuel, you may want to look at your vehicle's efficiency or reducing your mileage (unless you're a driver by trade, in which case I'll shut up - except, have you considered switching to electric?). There's also the element of risk: if you forget, or are unable, to put in that last 50 litres and claim your discount by the time the discounts start to expire, you risk losing the fruits of all your hard work. Note as well that my model assumes that fuel prices stay the same, which they don't: it is usually better to buy more petrol when it is cheaper, so you might be better off putting more than $40 in if the price is low. Finally, there may be cheaper petrol stations near you, and it may be worth your while using them instead if they aren't too far out of your way.
I hope I have armed you with the tools you need to decide on your strategy, and I hope you also now have an appreciation of how maths can be extremely useful in day-to-day life (if you hadn't already had that impression drilled into you by a maths teacher). Good luck, and drive safely.
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| XKCD.com explaining the various scientific fields, and how they all relate to each other. Everything is based on maths. |
In New Zealand, there is a magical card* you can get from a popular motoring organisation. The magic card gives you a discount on motor fuel if you use it at particular refuelling stations. The rules are as follows:
- The discount is normally 6c per litre (NZD 0.06).
- On occasional days, the discount goes up to 10c per litre for one day.
- The discount only applies to the first 50 litres of fuel you put in.
- You can choose to take the discount every time you refuel, or let the discounts build up.
- A discount saved in one month expires at the end of the following month.
The last two rules have lent themselves to the development of strategy. To clarify, if you put 60 litres of petrol in your car, you only get the discount for 50 of those litres. More importantly, you can put $40 worth of petrol in, pay full price and save the discount, then put 50 litres in and get double the discount (most likely 12c) for every one of those 50 litres. However, a discount acquired in April will expire at the end of May: you can't save discounts forever.
The magic card came in a few years ago, and to begin with there was much speculation about the ideal strategy. Do you redeem the discount every time you refuel? Or do you save it until the end of next month, then get a whopping great discount on 50 litres?
The discussion died down. People settled on their individual strategies. I used the card and took the discount each time because I wasn't sure whether saving up discounts was a good strategy or not, given that you have to pay full-price every time you save the discount.
Then, about three weeks ago, I redeemed my points at my local refuelling station, and the cashier struck up a conversation that went something along the lines of:
Cashier: Did you know that you can save the discount every time, and it builds up?
Me: Yeah, I did.
Cashier: Well, you can put in $40 of fuel three times and save the discount, and then put in 50 litres and get 40c off each litre. (It was one of those special 10c off days.)
Me: Right.
Cashier: So, next time you come, put in $40, hang up the pump, pick it up, put in another $40, and do it again and then the last time put in 50 litres. Then when you come in, ask the cashier to run each transaction through separately.
Me: Doesn't that annoy you and the other people in the queue?
Cashier: Well, I don't mind!
I couldn't get this strategy out of my head. It sounded like such a good idea, getting 40c a litre off, but what about the three times you have to pay full price? What is the overall discount?
I decided to use the power of mathematics to figure out how to make the best use of my fuel card. I hypothesised that there would be an optimum number of times you should save up the discount on $40 of fuel before using the discount.
I created spreadsheets of savings accumulated under a 6c discount regime, or a 10c discount regime. You can view them in Google Drive, or use any spreadsheet program, such as Microsoft Excel, or Apache OpenOffice Calc, to read them.
Then, about three weeks ago, I redeemed my points at my local refuelling station, and the cashier struck up a conversation that went something along the lines of:
Cashier: Did you know that you can save the discount every time, and it builds up?
Me: Yeah, I did.
Cashier: Well, you can put in $40 of fuel three times and save the discount, and then put in 50 litres and get 40c off each litre. (It was one of those special 10c off days.)
Me: Right.
Cashier: So, next time you come, put in $40, hang up the pump, pick it up, put in another $40, and do it again and then the last time put in 50 litres. Then when you come in, ask the cashier to run each transaction through separately.
Me: Doesn't that annoy you and the other people in the queue?
Cashier: Well, I don't mind!
I couldn't get this strategy out of my head. It sounded like such a good idea, getting 40c a litre off, but what about the three times you have to pay full price? What is the overall discount?
I decided to use the power of mathematics to figure out how to make the best use of my fuel card. I hypothesised that there would be an optimum number of times you should save up the discount on $40 of fuel before using the discount.
I created spreadsheets of savings accumulated under a 6c discount regime, or a 10c discount regime. You can view them in Google Drive, or use any spreadsheet program, such as Microsoft Excel, or Apache OpenOffice Calc, to read them.
Maths equations are just sentences, with numbers instead of words. The equation for working out the amount of discount you get per litre is the numerical equivalent of the sentence "The amount you would have paid if you had paid full price for a particular amount of fuel, minus the amount you did pay for that particular amount of fuel, then divided by the number of litres in the amount of fuel". It's quicker to replace "The amount you would have paid if you had paid full price for a particular amount of fuel" with "40A+50X", and replace "The amount you did pay for that particular amount of fuel" with "40A+50(X-b)". You can also replace "The number of litres in the amount of fuel" with "A(40/X)+50". So the full equation would be "(40A+50X)-(40A+50(X-b))/A(40/X)+50".
I'll explain.
You have to buy a minimum amount of $40 of fuel to claim the discount, but you can use the discount on up to 50L. So if X is the price of one full-price litre of fuel, then the price of 50L of full-price fuel is 50X. The number of litres you get for $40 is 40/X. The amount you pay for putting $40 of fuel in your tank, then putting 50L of fuel in your tank, but not claiming the discount, would be $40 plus 50X.
If you are putting $40 of fuel in your tank A times, then subsequently using the discount on 50L, then the total amount you would save would be (40A+50X) - (40A+50(X-b)), where b is the discount per litre (6c or 10c).
To calculate your savings per litre, you need to divide the total savings by the number of litres you bought (A(40/X)+50). Again, this has to include the litres you bought full price (those $40 purchases) plus the 50 litres you claimed the discount on.
Note: "Savings per litre" is just another way of saying "Total savings divided by total litres".
A couple of days ago, petrol at my local refuelling station (which accepts the magic card) was $2.35 without a discount. So assuming the discount was 6c every time you refuel, and you are refuelling at a station which sells fuel at $2.35 per litre, here is a graph showing the relationship between the amount you save and the number of times you save the points from $40 of fuel.
I am a little surprised. I thought the line would plateau and then begin to drop after a much smaller number of $40 purchases. Perhaps that is the cynic in me, but also I still approach maths with a childlike sense of wonder because I don't understand it very well. The more $40 purchases you make, at least up to 15 (but I ran the calculations a little further and it seemed to go into the hundreds), the more you save. This is probably a logarithmic curve due to its shape, which means it should keep going up.
But don't all rush out to put hundreds of $40 fuel purchases into your car.
Most of the savings are made at the beginning of the curve, where you buy $40 of fuel several times. After about 3 $40 loads, the savings gained by subsequent fuellings drop to about 1c per litre. After 7 $40 loads, you are getting 14c off per litre, but it takes another 3 loads to get to 15c. Bear in mind, though, that each cent you get is a cent off every litre that you have purchased. So if you put in $40 (a smidgen over 17 litres) seven times, then put in 50 litres and claim the discount, you put in 169 litres and get a little over 14c off each one, saving you a whopping $24 (note I have rounded at each calculation here, so this is approximate). If you put $40 in ten times, then put in 50 litres and claim the discount, you get 220 litres at a saving of 15 cents each, saving you $33 in total.
Is the extra $11 worth you refuelling three more times? This is where I am on firmer footing, due to my background in ecology. In ecology, there are always other factors to consider that mess up your calculations. How much does it cost you to put in that $40 of fuel? Sure, there's the $40, but do you have to travel out of your way to get to the fuel station? How much fuel do you use driving into the station, parking, restarting your engine, then driving out? How much is your time worth? How much would you pay not to have to go to the refuelling station every few days?
And how many times do you normally refuel in two months? Fifteen is a lot! If you are using that much fuel, you may want to look at your vehicle's efficiency or reducing your mileage (unless you're a driver by trade, in which case I'll shut up - except, have you considered switching to electric?). There's also the element of risk: if you forget, or are unable, to put in that last 50 litres and claim your discount by the time the discounts start to expire, you risk losing the fruits of all your hard work. Note as well that my model assumes that fuel prices stay the same, which they don't: it is usually better to buy more petrol when it is cheaper, so you might be better off putting more than $40 in if the price is low. Finally, there may be cheaper petrol stations near you, and it may be worth your while using them instead if they aren't too far out of your way.
I hope I have armed you with the tools you need to decide on your strategy, and I hope you also now have an appreciation of how maths can be extremely useful in day-to-day life (if you hadn't already had that impression drilled into you by a maths teacher). Good luck, and drive safely.
*I'm not going to name the magical card or organisation because I don't want to get accused of working for or against them. The maths is the important thing here, not the branding.
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