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# Arithmetic Sequence (Worded Problem)

Angela takes a job starting with an hourly wage of ₱3.50 and is promised a raise of ₱5.00 per hour per two months for 5 years. At the end of 5 years, what would be Angela’s hourly wage?

This is the problem our teacher gave us in class (although I changed the name to Angela, it doesn't matter much hahaha). I am confused whether the answer is ₱148.50 or ₱153.50.

Our teacher made it a point that the phrase "at the end of 5 years" is important in determining the correct answer in the problem.

What is the correct answer (with solution) to this problem?

Update:

We have finally agreed that it is ambiguous and the answer may be ₱148.50 or ₱153.50 depending on one's understanding of the problem. In her own understanding, she believes it is ₱148.50.

Thank you everyone for your support and cooperation. 'Till the next time :D

Note by Angela Fajardo
1 year, 3 months ago

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Thanks everyone once again for your thoughts. I have sent our teacher some picture of it. Perhaps I'll be able to get a more concrete/direct answer from her tomorrow. ^_^

- 1 year, 3 months ago

153.50

- 1 year, 3 months ago

Thank you everyone for your explanations. I would be asking our teacher tomorrow regarding the problem (that is if I remember it). Perhaps I'll post again tomorrow. Once again, thank you to everyone ^_^

- 1 year, 3 months ago

- 1 year, 3 months ago

Looking forward for it.

- 1 year, 3 months ago

Me too

- 1 year, 3 months ago

I asked our teacher. She first said that the answer is 148.50. Then, I presented some "thoughts/ideas" and told her about this discussion. I'll give her a picture of it. She said she'll study the problem once again and I'll ask her again tomorrow. And she also said that she was confused when she was answering it.

- 1 year, 3 months ago

- 1 year, 3 months ago

Perhaps I'll get a direct answer tomorrow. For now we are currently studying geometric sequences.

- 1 year, 3 months ago

We have finally agreed that it is ambiguous and the answer may be ₱148.50 or ₱153.50 depending on one's understanding of the problem. In her own understanding, she believes it is ₱148.50.

Thank you everyone for your support and cooperation. 'Till the next time :D

- 1 year, 3 months ago

- 1 year, 3 months ago

-_- The answer is not 148.50. It can even be 153.50 as angela's teacher specified. so, no one wins :D

- 1 year, 3 months ago

- 1 year, 3 months ago

:P Yrah i got it. Sorry.

- 1 year, 3 months ago

Eh , that's a complicated thing. The last moment of the year is the first of the other.

- 1 year, 3 months ago

His first comment stated that it's ambiguous. So yeah, he wins. Too bad, there's no prize

- 1 year, 3 months ago

:Pp

- 1 year, 3 months ago

Angela wants to save money so that she can buy the new book of her favourite author. She plans to save a certain amount of money in her weekly allowance for eight weeks. She decides to keep P10 in the first week increasing by P10 every week. How much will her total savings be at the end of 8th week?

What is the answer? Is this also ambiguous? I just want to make sure. Our examination in Maths is tomorrow.

- 1 year, 3 months ago

No question is clear, answer is 360.

- 1 year, 3 months ago

Then what is the difference of that problem with the at the end of 5 years?

- 1 year, 3 months ago

It specifies "at the end" means the 8th week has ended so, the answer is 360, the previous one does not provide any such thing.

- 1 year, 3 months ago

8th week and 8 weeks are different. End of 8th week clearly means the end of that week, which would be around the last day of the 8th week

- 1 year, 3 months ago

Isn't 8 weeks actually equal with the time spend until the last moment of the 8th week including that last moment also ?

In 8 weeks means in the time spent of 8 week which means anyway that the amount of time end at the end of the 8th week but that end is also the beggining of the next week and because they are equal the end being included on one hand in the time spent of 8 weeks while on the other not it is paradoxical.

- 1 year, 3 months ago

and @Ashish Siva Ahhh okay I understand now. Thanks

- 1 year, 3 months ago

All the best for your maths exam :)

- 1 year, 3 months ago

Thanks :)

- 1 year, 3 months ago

A much clearer explanation.

- 1 year, 3 months ago

Our teacher told me to meet her tomorrow after classes to explain her idea of the problem. She still sticks to the answer being ₱148.50.

I now believe that the problem is ambiguous. Anyone thinks otherwise?

- 1 year, 3 months ago

Somehow and I say somehow because it is not the wording which is ambigous but the concept of what is "end of some period of time" concretely and rigorously.

It is jsut one end of some period of time and therefore is just a correct answer but the problem leads to a paradoxical case in which the end equals the beginning anyway.

- 1 year, 3 months ago

:P Again

- 1 year, 3 months ago

In this problem:

Angela bought a lot at the beginning of 2000 for ₱50, 000. If it increased ₱1, 000 in value each year, how much is it worth at the end of 2025?

- 1 year, 3 months ago

For this case, then yes, it is 75,000. We are still in 2025, so technically, the addition in value for the 26th year is not added yet.

Though when you think about this logically, the value of land does not suddenly increase once a year. I would like to say that the price increase in the lot is distributed evenly throughout the year, and that on average, it increases about P2.74 a day. Therefore, at the end of 2025, the price of the lot should be somewhere around P75,990.

Though this is totally out of topic, it's something to ponder on. For this kind of questions, what we would usually say is

Angela bought a lot in the year 2000 for P50,000. If the value of the lot increases by P1,000 annually, what is the value of the lot in the year 2025?

A better, simpler way to phrase this question

- 1 year, 3 months ago

Yh the word $$\text{at}$$ the end of 5 years defines it properly.

- 1 year, 3 months ago

At the end of 5th year and at the end of 5 years. There's a difference :D

- 1 year, 3 months ago

I have another question somehow similar to this.

Angela takes a job starting with an hourly wage of ₱10 and is promised a raise of ₱5 per hour per 12 hours. At the end of the day, what would be Angela’s hourly wage?

- 1 year, 3 months ago

This is a flawed problem. "End of the day" can mean anything.

When math questions are created, they should be specific enough to avoid such ambiguity

- 1 year, 3 months ago

So end of the 5 years is also ambiguous?

- 1 year, 3 months ago

For the question above, it's not just the end that is vaguely defined, the start time is also not defined at all. For this case, I cannot calculate anything at all.

For the original question, to be honest with you, yes, "end of 5 years" is rather ambiguous. We found two different interpretations of the phrase, which led to two different possible answers. Now that I think about it, both are actually acceptable. If this was an exam question, then it is horribly flawed. However, as a discussion question on Brilliant, it is actually quite fun.

However, I still stick by my original answer of P153.50. From a perspective of teaching and learning math, what we want to do here is to test your knowledge on arithmetic progressions. We aren't here to test your language understanding. Therefore, when it is stated "end of 5 years", I would presume that the question is saying "5 years later".

In this case, note that there are $$30$$ pay raises in 5 years. The pay after the first raise, $$a_1=8.5$$. Therefore,

$$a_{30}=8.5+(30-1)(5) = 8.5 + 145 = 153.5$$

- 1 year, 3 months ago

Yes!

- 1 year, 3 months ago

Hmm, what is the time Angela started working?

- 1 year, 3 months ago

It is not given. But let's just assume that she works non-stop and started working at 6:00 AM on January 1, 2016.

- 1 year, 3 months ago

Then at the end of the day(11.59pm jan 1) she woukd have an hourly wage of 15 which was increased at 6.00pm(jan 1)

- 1 year, 3 months ago

a1 = 10 d = 5 n = 24hrs / 12hrs = 2

an = 10 + (2-1)(5) an = 15

- 1 year, 3 months ago

So, following this solution, the answer to the problem above would be 148.50 since it would be like

a1 = 3.50 d = 5 n = 12 months x 5 years / 2 = 30

an = 3.50 + (30-1)(5) an = 3.50 + 145 an = 148.50

- 1 year, 3 months ago

Ahh you are deviating from what I and Hung Woei Neoh meant. Sure using AP you might get 148.50 or 153.50 based on how you approach the problem. If you considered the time of 5 yeard has fully passed then you would get 153.59 but if you consider it as the last few moments of the end of 5 years you would get 148.50 i.e. the difference of 0.0000000000000000.....000001 sec between 11.59pm and 12.00am makes a huge difference

- 1 year, 3 months ago

What do you mean by deviating? Sorry about that

- 1 year, 3 months ago

Deviation means getting away. :P why sorry? Its always good to ask doubts. Btw who knows the real answer maybe 148.50. Its just my viewpoint that the answer is 153.50

- 1 year, 3 months ago

But isn't this problem same with the problem above? The only differences are the value of salary and the time.

PS. How about changing the question into At the end of 24 hours, what would be her hourly wage? Is the answer still the same, is it still 15?

- 1 year, 3 months ago

No at the end of 24 hours would give the answer 20. In the initial problem if you start at 6.00, your next raise would be at 18.00 when it changes to 16. The next raise would be at 6.00 tomorrow. Its not the same day.

- 1 year, 3 months ago

When is the end of 24 hours?

- 1 year, 3 months ago

Since angela started working at 6.00 on jan 1, the end of 24 hours is on 6.00 of jan 2.

- 1 year, 3 months ago

Ahhhh, one more thing:

The problem with the 12 hours is different from the problem with the 5 years right? Since for the 5 years the answer may be 148.50 or 153.50 depending on how you understand what the at the end of 5 years meant. Your answer is 153.50 because it means the time when 5 years has passed meaning if I started working Jan. 1 2001, the 5 years passed is on Jan 1, 2006? Then, @A A he understood it as the time before the 5 years has fully passed so It may be Dec 31, 2005 at 11:59:59 PM? That's how I understood it.

- 1 year, 3 months ago

Yh thats because we can see end by two viewpoints one by iy has ended, and something new has begun and one by at the end period. So, it plain terms its subjective.

- 1 year, 3 months ago

Even if we do it manually??

e.g. Jan 1 2001= 3.50

March 1 2001= 5.50...

Nov 1, 2015 = 148.50

Jan 1 2016 = 153.50

- 1 year, 3 months ago

Yes, even if we do it manually-

- 1 year, 3 months ago

So the final answer, the problem is ambiguous and required better wording in order for it to be solved with a definite answer?

- 1 year, 3 months ago

It's not subjective , there is only one right answer of what "end of x period of time" means and at most is as I mentioned paradoxical anyway.

The problem asks about the interval of time of exactly 5 years which doesn't include anything after so to say the 5 year span of time , even if it implies an after period.

Look at it this way. The period considered here of 5 years does include just the amount of time of 5 years therefore jsut the elements belonging to the 5 years.

They therefore do not include anything outside that span of time , not even the smallest unit after it and though when it ends means exactly after the last moment of the last year which is included in the 5th year that after moment is not included and by this the rise doesn't happen for a 30th time.

For example take the "end of 1 day" , it is on one hand the last moment of the 1 day and has to be included in the last day but implies the moment after it too therefore the moment of time which is not included so the infinitisemal end should both be included and not be included in the set of 1 day which is a contradiction.

For the problem this paradox isn't important.

In 5 years means exactly just the moments included in the span of 5 years and therefore doesn't imply anything after it to be conceived , as such the proper answer is just the one of the last moment of the 5th year in which you don't get a rise because the moment after it , when you get a rise , is not included anyway.

At best , the problem can be stated with the condition of breaks between the days of jobs of some hours because this avoids completely the paradox.

Do that manually on a calendar and you find that in the last day of the last year of work you don't receive an increase

- 1 year, 3 months ago

@A A So your answer is 148.50 since the "30th month" is after the 5 years and not included in what the problem is asking?

- 1 year, 3 months ago

Haha , yea that's my opinion but it's not the 30th month but the 30th rise which is not included in the 5 years.

Yet , I think it's more than that taken in a rigurous look at it , I think it's a paradox involved for which we don't agree with one another.

I mean we see part of the same thing. I see one side , Ashish , Prince , Hung Woei see the different part of it and I'm still not sure who is right on it either.

To illustrate the paradox , I think we are confused by the what "end of 5 years" means , of when that end actually happens or when the end of something is. Anyway , I don't think that we have a problem of when the rise happens and everyone agrees it happens in 2 months time , but it's a problem when the end is though.

Let's take it systematically to see the problem and understand better what the end means if you want.

I propose to look at the 5 years as made from a number of continuous moment of time which are included in the set of 5 years , to formalize and make it concrete.

Of course the end of the 5 years should be once the last moment of the set of moments of time of the 5 years ends. Everyone agrees with this statement or are there objections ?

My proof for the statement above is : the 5 year is made of a number of elements which belong to the 5 years and at the end of the 5 years , which means in the last moment of the 5th (or last) year , all those portions of time which compose the number of 5 years should be passed therefore for 5 years passed the last moment should also already be passed because otherwise it belongs to the set and there is a portion of time not passed which would contradict that all the moments have passed.

So the end of 5 years is when the last moment (of the last year) ends.

Nonetheless here is where comes the disagreement between us I think because some people see that when they try to conceive the end of this 5 years feel you have to include the next moment which doesn't belong to the 5 years , while some other (or maybe until now just I) think you don't have to include it anyway.

In other words for the 2 points of view , some people feel that when they try to conceive anyway such a thing as 5 years ended they observe that for the exactly last moment of the year to have ended you have to include another moment which doesn't belong to the set of 5 years because otherwise 5 years didn't have ended or formally that the last moment of the 5 years implies automatically when it ends the next moment of the next year while the other doesn't agree with that.

Suppose the last moment of the last year is m and the immediately next moment after the last moment of the 5th year is m1.

There seem to be 2 paradoxical things to consider when you try to riourously think of this infinitely so to say small instances of time.

I suppose everyone agrees also that when I speak of m and m1 they are at an instance from one another , they are immediately one after another with no other smaller distance between them which would lead to some infinitesimal paradoxes similar with the Zeno's arrow paradox and would be impossible since for any value , no matter how small I find , for the distance between the 2 instance of time anyway there is some other value even smaller and so on in an infinite manner so to say.

There would also be no m or m1 because there is always some smaller time unit but suppose that there is such a unit of time anyway.

Then , when you think end of m , you should also think start of m1 that is m = m1 actually. So anyway so to say the end of the last year is the beginning of the next , which means that that infinitesimal end of the year and that infinitesimal start of the next year are the same and therefore that the element of the next year belongs to the set of the last year which nonetheless is anyway contradictory because would mean that one element which doesn't belong to the set of 5 years belongs and in this way conceived to talk rigorously about some end of the last year is impossible.

Nonetheless when I think "period of 5 years" if m wouldn't be equal with m1 then nonetheless it is possible to conceive just that period excluding the moment after. To conceive this imagine that the Universe would not exist in exactly 5 years. After that moment there would be something which our mind can't conceive and therefore can't be talked about that and if the last moment of the 5th year would imply the next moment of the next year then there are not 5 years passed. But a complete period of 5 years must have passed at some point and this can be done by conceiving that interval of time without the first moment of the next year taken into consideration and thinking just at the elements which belong to the 5 years.

Another way to illustrate that would be by considering a paper which is colored half green and half yellow and the infinitesimal moment when the green passes to yellow. It can be conceived anyway that when the green ends is not yellow excluding the yellow once and for all in that infinitesimal moment of the green coloring I think.

Edit: in short , thisproblem considered like this would enter in some paradoxical things which are hardly solved and made the formal considerations of calculus , Cantor , Zenon , some greeks a little crazy anyway.

The paradoxes can be avoided for changing the formulation of the problem a little.

- 1 year, 3 months ago

@A A What are your thoughts? @Ashish Siva @Prince Loomba @Hung Woei Neoh ?

- 1 year, 3 months ago

I think the other way. :P

- 1 year, 3 months ago

Same here. My opinion is still the same

- 1 year, 3 months ago

But why in the problem above, your answer is ₱153.50? And also what would the solution be using the arithmetic sequence formula?

How about changing the question into At the end of 24 hours, what would be her hourly wage? Is the answer still the same?

- 1 year, 3 months ago

Wordings are wrong. Its ambiguous, at the end of 5 years. Clarify what this means!

- 1 year, 3 months ago

Suppose you receive the expression "end of 2 days" from the moment it is told , what does it mean ?

Well , it means when the 2 days from when you are told ends , so the wording is not ambigous and you just have to count when 2 days end anyway.

If you don't believe me , observe thinks like this to understand the increased amount of time span considered , that is to count it in the proper manner anyway so to say. Suppose you have a hourglass which measures 2 days , when the hourglass ends 2 days had passed.

Time span is just like that , with every second passed the amount of time increases just like a grain of sand in an hourglass passes , that is in a cumulative manner. Then considering this time span conceived articulately like the cumulation of every second just like the cumulation of every grain of sand you get the idea of 5 years end.

As another example. Suppose you are going to run 5 km anyway.

Asking when the 5 km end is an analogy of asking when 5 years pass because it's invovled the same cumulative perception of things , the same sum of smaller parts. The wording is therefore no vague when you ask , when 5 km passed at all therefore the wording is not vague also for asking when 5 years pass , if (and only if maybe) you perceive the passing of time in the right way when you think at it , that ios as a sum of parts (cumulation) thing , right ?

To move to the problem , one way to prove the problem is to take a calendar and suppose there are break between the days of work.

In this way certainly you see when you get your last increase that being that in the last day of the 60th month you have the 29 increase.

Another wayis by induction , the answer certainly is 148.50 if you try to thing of it attentively and I'm almost completely convinced of it right now anyway.

- 1 year, 3 months ago

I think end of 2 days means after 2 days i.e. 2 days ended

- 1 year, 3 months ago

Hmmm , but this rises an interesting thing to consider.

Suppose your pulse rises with m heartbeats every 2 km of non-stoped , continuous running. Suppose you traveled 2km , does your pulse rise ? Because you end 2km it should rise but when does it rise ? In , may be said , the last moment of the 2th km of course or after the 2nd km ended. Now , does that point of rise of heartbeat includes the last point of the 2nd km or doesn't ? Seen in the cumulative way described upwards , as increase of smallest units in my opinion it doesn't. Your heartbeat rises just after the 2nd km ended , once it is passed. That means the 2km must be already passed for heartbeat to rise , therefore the last point of the km is not included in the first moment when your beat rises because that moment is not included by definition in the phrase so the answer seems no after all anyway.

Edit : this is rather paradoxical at this point , really dilematic I mean not just a misunderstanding.

The paradox consists in the fact that if you say that it happens in 2 km you should include the end of the 2nd km but if you do then 2km haven't already passed and therefore ended so it's quite impossible. Hmmmm maybe Cantor would have tell something anyway.

Or the mathematicians who formalised infinitesimal and made rigorous calculus anyway.

In other words the pulse (or the salary) can't rise in 2 km/months because if it rises the infinitisimal small end of the km month should be included in the moment of the rise so the problem is not consistent in this form anyway.

The only way of making it consistent is by eliminating the continuous set and transforming it in a set with breaks between the members of the set , that is conceiving there are some gaps between them anyway.

- 1 year, 3 months ago

And end of 2km ? Take the interval 0 2 , does it include the point 2km or doesn't it ? End of 2km means in the last point of 2km. Exactly before the 3th starts I think.

- 1 year, 3 months ago

In what way is it ambiguous?

- 1 year, 3 months ago

End of 5 years may mean just before or after completion. So answer can be 148.5 or 153.5

- 1 year, 3 months ago

So, what could be a better wording?

- 1 year, 3 months ago

If you want 148.5 as answer, write just before the end of 5 years. 153.5 when after 5 years

- 1 year, 3 months ago

Doesn't before the end of 5 years mean any point of time before it actually ends like 10 months before it

- 1 year, 3 months ago

Just before. See my words properly

- 1 year, 3 months ago

Oops sorry. Okay thank you very much ^_^

- 1 year, 3 months ago

Change the wordings

- 1 year, 3 months ago

Hate to break it to you, but "just before" isn't that much clearer

- 1 year, 3 months ago

No problem. Tell your teacher also to clear the wordings.

- 1 year, 3 months ago

Well , I still believe the answer is 148.50 but here is my reasoning.

Firstly let's make clear what the teacher is talking about. The teacher can either ask about the salary in the 5 years span of time or about it after 5 years have passed. I interpreted the question as meaning what is the hour wage becoming in a 5 year span. The difference anyway between them is that after 5 years includes the first moment of time in which already 5 years are passed and I will argue about the fact that no matter how you count the 2 months you will always have that the first 2 months should have 0 increase so to say.

The question therefore can be interpreted as follows. You start working on the job at some time , t0 which is included in the span of 5 years and the question is how much starting from t0 you will have your salary increased in this passing of 5 years , that is how many times do the salary increases over the period of 5 years from the initial moment you started working.

Considering this anyway the period of 5 years , or 60 months , posses some problems apparently on when that period of time passes but registering that amount of 60 months based on the increase in salary every 2 months concretely you have the following way of representing the increase by showing precisely what increase correspond to what month : months 0 - 2 (when 2 months pass or until the 3th month out of 60) 3.5 2 - 4th 3.5+5 and so on. By this reasoning 58 - 60 will be read as until the 61th month of 60 which means by following and counting it that the increase in the representation is 3.5+5*29 in it.

The same representation can be read as 0 - 2 meaning when 2 months pass including the first day of the 3th month by which therefore 58 - 60 means that the last 2 months of the time you work for 5 years should include the 1st day of the 61 month but I find it not to be correct since that is a moment of time which happens once the 5 years passed.

In other words you receive in the period of 5 years until the 5 years are close just 29 increases not 30 according to this reasoning.

In Ashish Siva's terms the arithmetic progression starts with 3.5 since the first 2 months anyway are included in the span and do not have an increase by this reasoning.

- 1 year, 3 months ago

The answer I think is ₱153.50 because she starts of with ₱3.5 which is going to become ₱8.5 in the next year i.e. at the end of the first pair of two months.
Now we know that 5 years = 60 months = 30 two months.

So, the hourly wages follow an arithmetic progression starts with 8.5 i.e. $$8.5, 13.5, 18.5, \cdots$$.

We have to find the $${30}^{\text{th}}$$ term of this progression which is $$8.5 + (30-1)×5 = \color{blue}{\boxed{₱153.50}}$$.

- 1 year, 3 months ago

The thing is, some people say "at the end of 5 years" meant somewhere before 5 years are fully complete.

For example, say Angela began work on Jan 1 2011

I'd say that "at the end of 5 years" meant after 5 years has fully past (meaning 1 Jan 2016)

Some people argue that "at the end of 5 years" meant the last few moments of 2015

- 1 year, 3 months ago

Yeah that some margin of 0.000000000000....001 sec makes a difference in their salaries! How sad :( I would defined a specific range of time when my salary raises if I was working :P

- 1 year, 3 months ago

Our teacher's answer is ₱148.50 and I don't know why it is not ₱153.50 though @A A and @Hung Woei Neoh helped me a lot, I am still a bit confused and with their suggestion, made this note :)

- 1 year, 3 months ago

I personally feel that "at the end of 5 years" means the moment 5 years has fully passed. Therefore, I'd say that the answer is 153.50.

@Ashish Siva @Rishabh Tiwari @Rishabh Cool what do you guys think?

- 1 year, 3 months ago

Do you still believe the answer is ₱153.50?

- 1 year, 3 months ago

Yes. I feel that "at the end of 5 years" means right after 5 years has fully ended, which means the answer is 153.50

- 1 year, 3 months ago

Its ambiguous. It can mean both.

- 1 year, 3 months ago

It can, so for this case, I think it means after 5 years ended. Because it would be actually rather silly to ask about the salary at the last moments of the 5th year. We're testing AP knowledge, not grammar

- 1 year, 3 months ago

I also thought the answer as 153.5. Its better acc to me

- 1 year, 3 months ago