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At the end of a school year, the 7th grade teachers came to a meeting with a specified number of parents, where a total of $31$ people have participated.

Sixteen parents of students put questions to the Latin teacher, with the German teacher spoke $17$, with the English teacher $18$, etc., til to the teacher of mathematics, to which all present parents turned.

How many were they exactly?

$$$$

When it is said "Sixteen parents" is it meant $16$ people or $16$ couples ans do $32$ people?

Since there is a total number of $31$ people that participated, I suppose that it is meant $16$ people. Is that correct?

Could you give me a hint how we could solve the above problem? I haven't really understood what we looking for. Do we want to determine the number of teacher? Then since we have successive numbers of people at each teacher, do we have $31-16+1$ teacher?

Or have I understood the problem wrong?

DanielV
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Mary Star
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1 Answers1

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Let $P$ be the number of parents and $T$ be the number of teachers. Call the parents the teachers spoke to $s_1, s_2, \dots s_T$

a total of 31 people have participated

$$P + T = 31$$

Sixteen parents of students put questions to the Latin teacher, with the German teacher spoke 17, with the English teacher 18, etc., til to the teacher of mathematics, to which all present parents turned.

$$\begin{array} {rcll} s_1 &=& 16 \\ s_2 &=& 17 \\ &\vdots& \\ s_T &=& 16 + T - 1 &= P \end{array}$$

Can you finish?

DanielV
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  • Could you explain to me the last equation? Why does it hold that $s_T=16+T-1$ ? And why is this equal to $P$ ? – Mary Star Jul 01 '19 at 17:13
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    The $16 + T - 1$ is because they are increasing in arithmetic progression, 16, 17, 18, 19, etc. The $P$ is because "to which all present parents turned". – DanielV Jul 01 '19 at 17:26
  • Ahh ok!! So, we have the system $P+T=31$ and $16+T-1=P \Rightarrow -T+P=15$ and the solution is $P=23$, $T=8$, i.e. there are $23$ parents and $8$ teacher. Is that correct? – Mary Star Jul 01 '19 at 17:34
  • You got it. Well done. – DanielV Jul 01 '19 at 17:37
  • Great!! Thank you so much!!! :-) – Mary Star Jul 01 '19 at 21:48
  • @MaryStar If this answers you question then you should accept it so it won't stay open. Open means you are still looking for an answer. – DanielV Jul 01 '19 at 23:04