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Your friend Veer wants to participate in a 200 m race. He can currently run that distance in 51 seconds and with each day of practice it takes him 2 seconds less. He wants to do in 31 seconds. What is the minimum number of days he needs to practice till his goal is achieved?

My question is what should be the first term of this AP? 51 or 49? Subsequently, what would be the answer? 11 or 10?

I would tell my reasoning but I don’t want to influence the answer.

EDIT: Okay, so my reasoning is this: since the question is asking how many days he needs to practice: we’ll start the AP, when he starts practising. And on his first day of practice, he can finish the race in 49s. Therefore, the AP should start with 49.

Atul Anand
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    Please do share your reasoning. Part of asking a good question on Mathematics Stack Exchange is sharing your own reasoning/methodology before asking for help – 冥王 Hades Dec 15 '22 at 17:06
  • Okay, so my reasoning is this: since the question is asking how many days he needs to practice: we’ll start the AP, when he starts practising. And on his first day of practice, he can finish the race in 49s. Therefore, the AP should start with 49. – Atul Anand Dec 15 '22 at 17:10
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    What you choose as your starting times depends on your definition of "first". Let $t_n$ be the amount of time it takes to complete the race after $n$ days of practice. Veer is able to run it in $51$ seconds without any practice, so $t_0=51$. On the other hand, you can start measuring his progress after $1$ day of practice, so $t_1=49$. The only practical difference is the precise definition of $t_n$, either$$\begin{cases}t_0=51\t_n=t_{n-1}-2&n\ge1\end{cases}\text{ or }\begin{cases}t_1=49\t_n=t_{n-1}-2&n\ge2\end{cases}$$ – user170231 Dec 15 '22 at 17:18
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    I would advise to always count from $0.$ The progression would start on day $0$ at $51$ seconds. And model his times on each day as $t_n= 51-2n.$ – user317176 Dec 15 '22 at 18:02

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You should start with $51$, since "What is the minimum number of days he needs to practice till his goal is achieved?" Questions like this usually assume that the start is "now" and we are given that he "currently" could run in $51$ seconds. So the answer should be $11$. I think what is wrong with your reasoning is that you thought that the AP should start when he starts practicing for one day. I will try to make a visualization:

Correct counting: $|\space\space\space |\space\space\space |...$

Your counting: $\space\space\space|\space\space\space|\space\space\space|$ You start with a space (doing the routine) and not a bar (end of previous routine and start of new routine). I hope you could understand what I'm trying to say.

Kamal Saleh
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  • Can you tell what is wrong with my reasoning? Since the question is asking how many days he needs to practice: we’ll start the AP, when he starts practising. And on his first day of practice, he can finish the race in 49s. Therefore, the AP should start with 49. – Atul Anand Dec 15 '22 at 17:12
  • @AtulAnand I edited the answer. I don't blame you if you couldn't understand. – Kamal Saleh Dec 15 '22 at 17:18
  • Hmmm, I get what you say. But shouldn’t the AP start when he starts practising? Since, everything he did before that is worthless. We don’t care if he was finishing the race in 51s for the past 2 years. We care about what happens AFTER the practice starts. – Atul Anand Dec 15 '22 at 17:36
  • @AtulAnand I will edit my visualization again, I left some things out. – Kamal Saleh Dec 15 '22 at 17:39
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I'm familiar with these questions. Trademark of lower level maths in India. According to my experience and the question a(initial term)=51 seconds

Then you have already solved the questions I guess.

P.S. These types of questions are very general in AP of India, and the examiners expect us to blindlessly apply general term of AP formula. Taking the given value as first term. According to the question the time given doesn't clearly specify what the starting point is.

Banaj Mahajan
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  • Can you tell what is wrong with my reasoning? Since the question is asking how many days he needs to practice: we’ll start the AP, when he starts practising. And on his first day of practice, he can finish the race in 49s. Therefore, the AP should start with 49. – Atul Anand Dec 15 '22 at 17:11
  • Actually your first part of reasoning is correct but you are misinterpreting it, at the beginning of the first day he can finish the race only at 51 seconds, that means lets say at the 0th day he can finish the race in 51 seconds. That means that at the end of the first day only, he will be able get his time down to 49 secs. So you will have to count this day as well. – Banaj Mahajan Dec 15 '22 at 17:19
  • I still don’t think I get it. Let’s say his aim was to run the distance in 49s. He can currently run the distance in 51s. How many days should he practice? Quite obviously, the answer will be 1 day (unless I’m misinterpreting this as well). Since, after 1 day of practice, his time would decrease by 2s. – Atul Anand Dec 15 '22 at 17:34
  • Yes that is right that it will take 1 day to decrease time to 49 seconds, but it will take the whole day. Lets say Veer decides on Monday he wants to decrease his time to 31 seconds, and says I will start practicing from tomorrow, then let Monday be the 0th day. So Tuesday will be a1 or the first day, at whose end the time will decrease to 49 seconds. Then wednesday will be a2 where time will decrease to 47 seconds and then so on.. – Banaj Mahajan Dec 15 '22 at 17:43
  • Firstly, I don’t think he is practising for the whole day (that’d be torturous LOL). And secondly, even if it is at the end of a1 when he reaches his goal, he still only practiced for 1 day and not 2 days. He may achieve his goal on the second day but he practices for 1 day. If this is still wrong, can you ELI5? – Atul Anand Dec 15 '22 at 17:47
  • @AtulAnand I think I understand your confusion now, this is a poorly stated questions probably from CBSE. You can just blindly apply the general term of AP formula here. Taking a1 as 51 seconds, as in this question it isn't clearly stated what the first term is. – Banaj Mahajan Dec 15 '22 at 17:58