Basic speed distance time exercises: need strategies for awkward division without calculator

[LordLongStyle]

So I'm training for an aptitude test in which I need to be able to quickly solve basic speed distance and time and fuel calculation questions without a calculator. I understand how to come to the correct answer using Distance = Speed x Time formula and rearrange it but when I need to divide awkward number 19/570 I cannot do it quickly in my head, I'm looking for any stratergies to help me be divide those more awkward numbers. Any help or suggestions are appreciated, Thanks.

 

[stapel]

So I'm training for an aptitude test in which I need to be able to quickly solve basic speed distance and time and fuel calculation questions without a calculator. I understand how to come to the correct answer using Distance = Speed x Time formula and rearrange it but when I need to divide awkward number 19/570 I cannot do it quickly in my head, I'm looking for any stratergies to help me be divide those more awkward numbers. Any help or suggestions are appreciated, Thanks.

They generally set up these no-calculator exercises so that the numbers work out nicely. For instance, in the first exercise:

You travel 19 miles. How much fuel will you need if you burn 30 kgs per hour and are travelling at 570 mph?

The number of miles, 19, goes neatly into 57: 3*19 = 57.

You know "d = rt", you have been given "d" and "r", and you need to find "t". So plug the known values into the known formula, and get the time t. This will fit nicely into the "kilograms per hour" value.

View attachment 36474
You have used 34 kgs of fuel in the past 3 hours and 24 mins. How much do you burn per hour?

How much of one hour is 24 minutes? (Hint: 6 minutes is 1/10 of an hour; 24 = 6*4)

How much time, then, is 3 hours and 24 minutes? (Hint: Use a decimal)

How does this relate to the amount of fuel?

 

[topsquark]

So I'm training for an aptitude test in which I need to be able to quickly solve basic speed distance and time and fuel calculation questions without a calculator. I understand how to come to the correct answer using Distance = Speed x Time formula and rearrange it but when I need to divide awkward number 19/570 I cannot do it quickly in my head, I'm looking for any stratergies to help me be divide those more awkward numbers. Any help or suggestions are appreciated, Thanks.View attachment 36473View attachment 36474

I'll give you a quick list of advice I give my students. stapel has already given you some of it.

1. As stapel mentioned, if this is a non-calclulator situation, if you have to divide by something ugly, like 17, odds are that whatever is in the numerator has a factor of 17 in it, too. Look for the simplification before you multiply everything out. More likely than not, the numbers won't turn out to be too bad.

2. One of the first things to address in a Physics problem is units: pick a distance unit... if the problem uses speed in mph you probably want to use miles. Ditto for time. It really doesn't matter in the end which units you use, you just need to be consistent. When in doubt, if you have two units in the problem, such as 20 mph and 100 ft, choose the smaller unit and convert mph to feet/hr, or even ft/min. Then change anything else accordingly.

3. Work through as much of the problem as you can using just variables. It probably won't do much for this level of problem, but it's easier to keep track of units, you don't accumulate rounding errors, and in many cases a lot of stuff you didn't really need to know just falls out of the problem. (If you continue in Physics, you will find a number of problems that appear to need a mass do not give you one. That's because, in the end, the mass term cancels out somewhere and you didn't really need it.)

4. Take a moment to ask "Does my answer make sense?". If your question is about how long it takes a car moving with a speed of 60 mph to travel a distance of distance of 60 miles, and your answer is 10 hours, the hairs on the back of your neck should be stiffening! It doesn't take long and you can catch some simple and easy to make mistakes this way.

5. Finally, since this is a timed test, you need to budget your time. My personal favorite strategy is to go through the whole test and answer any questions you can immediately answer, or that are simple enough to you to not require a lot of time to do. Then see how much time is left and divide it up between the rest of the problems. Then go through it again. If you get stuck somewhere, move to the next problem... don't get bogged down on a single problem, you can always come back to it again if you have time. Keep track of the time, but do not panic when time is getting low! Just do the best that you can, you can't do it any better. Never take a personal stake in an exam, it can cripple your ability to think on your feet. As Adidas says "Just do it.". And when you are done, just walk away.

Hope it helps!

-Dan

 

[LordLongStyle]

They generally set up these no-calculator exercises so that the numbers work out nicely. For instance, in the first exercise:



The number of miles, 19, goes neatly into 57: 3*19 = 57.

You know "d = rt", you have been given "d" and "r", and you need to find "t". So plug the known values into the known formula, and get the time t. This will fit nicely into the "kilograms per hour" value.



How much of one hour is 24 minutes? (Hint: 6 minutes is 1/10 of an hour; 24 = 6*4)

How much time, then, is 3 hours and 24 minutes? (Hint: Use a decimal)

How does this relate to the amount of fuel?

Thanks for the response,

24 minutes is 0.4hrs, plus the 3 gives 3.4 hours. 34Kg of fuel divide by 3.4hrs = 10kg/hr?

Back, to the first question, how does knowing 19 neatly goes into 570 or 57 help us when d=rt says it will be the other way around, 19/570 = t. To me 57/19 = 3 or 570/19 = 30 is not d/r=t.

Thanks again.

 

[LordLongStyle]

I'll give you a quick list of advice I give my students. stapel has already given you some of it.

1. As stapel mentioned, if this is a non-calclulator situation, if you have to divide by something ugly, like 17, odds are that whatever is in the numerator has a factor of 17 in it, too. Look for the simplification before you multiply everything out. More likely than not, the numbers won't turn out to be too bad.

2. One of the first things to address in a Physics problem is units: pick a distance unit... if the problem uses speed in mph you probably want to use miles. Ditto for time. It really doesn't matter in the end which units you use, you just need to be consistent. When in doubt, if you have two units in the problem, such as 20 mph and 100 ft, choose the smaller unit and convert mph to feet/hr, or even ft/min. Then change anything else accordingly.

3. Work through as much of the problem as you can using just variables. It probably won't do much for this level of problem, but it's easier to keep track of units, you don't accumulate rounding errors, and in many cases a lot of stuff you didn't really need to know just falls out of the problem. (If you continue in Physics, you will find a number of problems that appear to need a mass do not give you one. That's because, in the end, the mass term cancels out somewhere and you didn't really need it.)

4. Take a moment to ask "Does my answer make sense?". If your question is about how long it takes a car moving with a speed of 60 mph to travel a distance of distance of 60 miles, and your answer is 10 hours, the hairs on the back of your neck should be stiffening! It doesn't take long and you can catch some simple and easy to make mistakes this way.

5. Finally, since this is a timed test, you need to budget your time. My personal favorite strategy is to go through the whole test and answer any questions you can immediately answer, or that are simple enough to you to not require a lot of time to do. Then see how much time is left and divide it up between the rest of the problems. Then go through it again. If you get stuck somewhere, move to the next problem... don't get bogged down on a single problem, you can always come back to it again if you have time. Keep track of the time, but do not panic when time is getting low! Just do the best that you can, you can't do it any better. Never take a personal stake in an exam, it can cripple your ability to think on your feet. As Adidas says "Just do it.". And when you are done, just walk away.

Hope it helps!

-Dan

Thanks for the response Dan,

I find alot of the time when I am praticing and I review my answers alot of the problems I get wrong logically dont make sense. The unit conversion has not and should not be much of a problem for me as the test will be standard in a consistent unit. Thanks for the advice on the mindset for the exam, your response helped heaps.

Cheers.

 

[Steven G]

Don't multiply any numbers!!!!! Then you get \(\displaystyle \dfrac{19*30}{19}\ =\ 30\)
I do this all the time and it makes life easier for me.

 

[Dr.Peterson]

So I'm training for an aptitude test in which I need to be able to quickly solve basic speed distance and time and fuel calculation questions without a calculator. I understand how to come to the correct answer using Distance = Speed x Time formula and rearrange it but when I need to divide awkward number 19/570 I cannot do it quickly in my head, I'm looking for any stratergies to help me be divide those more awkward numbers. Any help or suggestions are appreciated, Thanks.View attachment 36473View attachment 36474

Combining various ideas that have been mentioned, I would treat the whole thing as one big fraction multiplication problem, and be sure to cancel before multiplying.

For the first problem, [math]19\text{ mi}\times \frac{1\text{ hr}}{570\text{ mi}}\times \frac{30\text{ kg}}{1\text{ hr}}=\frac{19\times30}{570}\text{ kg}=\frac{19\times3\times10}{10\times19\times3}\text{ kg}=1\text{ kg}[/math]For the second, [math]\frac{34\text{ kg}}{3\times60+24\text{ min}}\times \frac{60\text{ min}}{1\text{ hr}}=\frac{34\times60}{204}\text{ kg/hr}=\frac{2\times17\times60}{2\times2\times3\times17}\text{ kg/hr}=10\text{ kg/hr}[/math]
In both, you could cancel by dividing rather than by factoring. The factoring or dividing (on paper, not in your head) can be time-consuming in the worst case, but you'll be dividing by small numbers.

Of course, you could also go ahead and multiply [imath]19\times30=570[/imath] and [imath]34\times60=2040[/imath] to get the answer, because they are set up to have nice answers in which the last step is easy. Problems meant to be solved without a calculator often work that way. Also, if you ever did have to divide 19/570, I'd divide 570/19 instead, yielding 30, so the answer would be 1/30. Avoid decimals.

 

[vikky patel]

So I'm training for an aptitude test in which I need to be able to quickly solve basic speed distance and time and fuel calculation questions without a calculator. I understand how to come to the correct answer using Distance = Speed x Time formula and rearrange it but when I need to divide awkward number 19/570 I cannot do it quickly in my head, I'm looking for any stratergies to help me be divide those more awkward numbers. Any help or suggestions are appreciated, Thanks.View attachment 36473View attachment 36474

To handle awkward divisions in speed/distance/fuel questions without a calculator, use simple ratio tricks instead of trying to divide big numbers directly.

1. Make the numbers easier:
19 ÷ 570 looks bad, but divide both by 19 → 1 ÷ 30 = 0.033 hours.
Fuel = 30 × 0.033 ≈ 1 kg.

2. Turn minutes into decimals:
3 hours 24 mins = 3.4 hours.
34 ÷ 3.4 → multiply both by 10 → 340 ÷ 34 = 10 kg/hour.

3. Always scale to remove decimals:
Multiply or divide both numbers by the same thing to make the math easy.

These shortcuts avoid long division and are fast enough for aptitude tests.

 

[Dr.Peterson]

To estimate a division, it's usually a good idea to change both numbers in the same direction. Here, I would increase both 19 and 570 to nice numbers, 20 and 600; and 20/600 simplifies easily to 1/30 -- the exact value, as it happens! Your attempt is both harder and less accurate!

But students develop a knack for making good choices only with experience. One way to gain useful experience is to always rethink a problem, in this case comparing afterward to the correct answer and seeing how close you came.

Of course, it's also a good idea to know whether you are expected to get an exact or approximate answer to a given problem, so you can choose an appropriate approach. If exact is expected, then you can expect nice simplifications! But you might first estimate, and then guess that maybe 570 is exactly 30 times 19, which can be checked more easily than the division itself.