Office Ratio & Probability Question

[Physiona]

QUESTION:
Office

is a small office.

In the morning of day 1 the ratio of male to female workers in the office is 15 : 7.
In the evening of day 1 the unsimplified fraction of male to female workers in the office is

164 males entered the office at lunchtime and stayed for the remainder of the day.
184 females entered the office at lunchtime and stayed for the remainder of the day.
Find the number of females that were in Office

at the start of day 1.

ATTEMPT:
This question has been a bang in the head for me. I have went to my teacher and he took half an hour and still didn't figure it out. I'm not sure where to begin it from, as the unsimplifed fraction is tripping me up.
Can someone help offer guidance for me please?

 

[member 587159]

This is just a question where you have to write everything in symbols and then solve for what is asked.

As a start, you can simplify the fraction...

Denote $M_m$ for the amount of males in the morning, $F_m$ for the amount of females in the morning, $M_e$ for the amount of males in the evening and $F_e$ for the amount of females in the evening.

It is given that $M_m/F_m = 15/7$ and $M_e/F_e = 7/5$

You are also given that $M_m + 164 = M_e$ and $F_m + 184 = F_e$.

Now, solve for $F_m$.

 

[Physiona]

This is just a question where you have to write everything in symbols and then solve for what is asked.

As a start, you can simplify the fraction...

Denote $M_m$ for the amount of males in the morning, $F_m$ for the amount of females in the morning, $M_e$ for the amount of males in the evening and $F_e$ for the amount of females in the evening.

It is given that $M_m/F_m = 15/7$ and $M_e/F_e = 7/5$

You are also given that $M_m + 164 = M_e$ and $F_m + 184 = F_e$.

Now, solve for $F_m$.

Right thanks for that. How do I determine what the value for $F_e$? I've been given the fraction share of it which is 5, how do I put that knowledge into figuring out the answer?

 

[member 587159]

Right thanks for that. How do I determine what the value for $F_e$? I've been given the fraction share of it which is 5, how do I put that knowledge into figuring out the answer?

What have you tried? You have $4$ equations that contain information? Obviously, you will have to combine them: plug equations in others, and try to isolate one variable. Once you found one, you will easily find everything, although only $F_m$ is asked.

 

[Physiona]

What have you tried? You have $4$ equations that contain information? Obviously, you will have to combine them: plug equations in others, and try to isolate one variable. Once you found one, you will easily find everything, although only $F_m$ is asked.

Yes I do have 4 equations. The main tricky thing here is which one to begin with and rearrange it. I do have a few unknowns in one equation itself as well.
If I deduce to find one part of th ratio, $(15+7=22)$ then $(22/15= 1.46)$. I get decimals and this isn't continuous data.

 

[member 587159]

Yes I do have 4 equations. The main tricky thing here is which one to begin with and rearrange it. I do have a few unknowns in one equation itself as well.
If I deduce to find one part of th ratio, $(15+7=22)$ then $(22/15= 1.46)$. I get decimals and this isn't continuous data.

I did a quick calculation and obtained a round answer, so you are doing something wrong. Just play around with the equations a lot. Put one equation in another, and replace the variables until you obtain something where only one variable is left. Don't think you are wrong because it takes a couple of steps.

 

[Physiona]

I did a quick calculation and obtained a round answer, so you are doing something wrong. Just play around with the equations a lot. Put one equation in another, and replace the variables until you obtain something where only one variable is left. Don't think you are wrong because it takes a couple of steps.

Can you at least give me one starting point to start from to rearranging the equations? I seriously don't know where to begin. From the ratio/fraction or the whole number addition

 

[member 587159]

Can you at least give me one starting point to start from to rearranging the equations? I seriously don't know where to begin. From the ratio/fraction or the whole number addition

It really doesn't matter actually. But okay:

$M_e/F_e = 7/5 \implies M_e = 7F_e/5 = 7/5(F_m + 184)$

and continue substituting equations (substitute for example something on the left, and something on the right now and you will have an equation with one variable)

 

[Physiona]

It really doesn't matter actually. But okay:

$M_e/F_e = 7/5 \implies M_e = 7F_e/5 = 7/5(F_m + 184)$

and continue substituting equations (substitute for example something on the left, and something on the right now and you will have an equation with one variable)

Thank you.
I've reached up to M_e 7/5(M_m/15/7 +184)
The M_m/15/7 I got by rearranging M_m\F_m=$15/7$

 

[member 587159]

Thank you.
I've reached up to M_e 7/5(M_m/15/7 +184)
The M_m/15/7 I got by rearranging M_m\F_m=$15/7$

Good job. Now, use that $M_m + 164 = M_e$ on the left side! Post me if you got the solution for $M_m$.

 

[Physiona]

Good job. Now, use that $M_m + 164 = M_e$ on the left side!

Oh I think I'm getting it now! Do I collect the like terms on one side? So I collect M_m on the left hand side and rearrange the rest?

 

[member 587159]

Oh I think I'm getting it now! Do I collect the like terms on one side? So I collect M_m on the left hand side and rearrange the rest?

Yes! Let me know what your answer is, then I can confirm that you are right!

 

[Physiona]

Yes! Let me know what your answer is, then I can confirm that you are right!

So far I've got M_m= M_m/15/7+20 I then rearranged to M_m*15/7=M_m+20. Am I correct so far?

 

[member 587159]

So far I've got M_m= M_m/15/7+20 I then rearranged to M_m*15/7=M_m+20. Am I correct so far?

No, sorry. You must have made a mistake somewhere.

 

[Physiona]

No, sorry. You must have made a mistake somewhere.

Alright hang on. Retreating back is M_m+164=7/5(M_m/15/7 +184) correct?

 

[member 587159]

Alright hang on. Retreating back is M_m+164=M_m/15/7 +184 correct?

Nope. Post all your steps. Even tiny ones. Then I can see where you go wrong.

 

[Physiona]

Nope. Post all your steps. Even tiny ones. Then I can see where you go wrong.

Right okay.
M_e/F_e=7/5
M_e=F_e$(7/5)$
M_e= 7/5(F_m+184) [F_e=F_m+184]
M_m\F_m=15/7 (REARRANGING IS 7M_m/15 =F_m)
So 7/5(7M_m/15 +184)
Then M_m+164=7/5(7M_m/15+184)
I'm up to here, where's my mistakes?

 

[member 587159]

Right okay.
M_e/F_e=7/5
M_e=F_e$(7/5)$
M_e= 7/5(F_m+184) [F_e=F_m+184]
M_m\F_m=15/7 (REARRANGING IS 7M_m/15 =F_m)
So 7/5(7M_m/15 +184)
Then M_m+164=7/5(7M_m/15+184)
I'm up to here, where's my mistakes?

Now you are correct. The first time you didn't add parentheses, so it was false. Continue to find $M_m$ and then the result you are looking for. I think I'm really spoiling you by immediate answers :P

 

[Physiona]

Now you are correct. The first time you didn't add parentheses, so it was false. Continue to find $M_m$ and then the result you are looking for. I think I'm really spoiling you by immediate answers :P

Right okay. I'll post my answer when I've found it. (Actually, you're not. You are guiding me too much though, just let me know if I've gone wrong and let me fix it myself it'll be better that way)

 

[Physiona]

Now you are correct. The first time you didn't add parentheses, so it was false. Continue to find $M_m$ and then the result you are looking for. I think I'm really spoiling you by immediate answers :P

I'm now on /frac($M_m$+164)=7$M_m$\15+184
Is this right..?

 

[member 587159]

I'm now on /frac($M_m$+164)=7$M_m$\15+184
Is this right..?

You say me I'm guiding you too much. Your equation isn't clear (Parentheses?). Solve the equation for $M_m$ and I'll tell you if you are right or not. Have some confidence!

 

[Physiona]

You say me I'm guiding you too much. Your equation isn't clear (Parentheses?). Solve the equation for $M_m$ and I'll tell you if you are right or not. Have some confidence!

(Not about confidence, about going on the right track!)
Right I don't know but I've tried attempts I keep getting decimals and then I've reached a whole number however I'm not sure if it's right. $M_m$=104

 

[member 587159]

(Not about confidence, about going on the right track!)
Right I don't know but I've tried attempts I keep getting decimals and then I've reached a whole number however I'm not sure if it's right. $M_m$=104

Nope. Post working.

 

[Physiona]

$M_m$+164=7/5(7$M_m$\15+184)
$M_m$+164/7/5=7$M_m$\15+184
15$M_m$=49/5$M_m$+20
26/5$M_m$=20
$M_m$=104

 

[Ray Vickson]

$M_m$+164=7/5(7$M_m$\15+184)
$M_m$+164/7/5=7$M_m$\15+184
15$M_m$=49/5$M_m$+20
26/5$M_m$=20
$M_m$=104

These expressions are incomprehensible: what do you mean by $M_m$\15? I have never seen something like that (although the backslash "\" is common in LaTeX commands and in some computer languages).

Is the second-last line supposed to be $\frac{26}{5M_m},$ or is it $\frac{26}{5} M_m?$ If it is the first, you can write it in-line as $26/(5 M_m)$, and if it is the second you can write it in-line as $(26/5) M_m,$ both of which are absolutely clear and unambiguous. However, no matter which you mean your "answer" $M_m = 104$ is clearly wrong. You don't need anybody to tell you that; you can substitute $M_m = 104$ into the formula to check if it works. (That is a habit you should develop now, starting today----checking your work. After all, in an exam there will be nobody available to check your work for you.)

 

[member 587159]

$M_m$+164=7/5(7$M_m$\15+184)
$M_m$+164/7/5=7$M_m$\15+184
15$M_m$=49/5$M_m$+20
26/5$M_m$=20
$M_m$=104

What you write is ambiguous, and has a lot of mistakes.

Do you know the distributivity law?

$a(b+c)= ab + ac$

 

[Physiona]

What you write is ambiguous, and has a lot of mistakes.

Do you know the distributivity law?

$a(b+c)= ab + ac$

Yes I do know the law.
Tell me this, is the final answer 126? (As I got that)

 

[Physiona]

These expressions are incomprehensible: what do you mean by $M_m$\15? I have never seen something like that (although the backslash "\" is common in LaTeX commands and in some computer languages).

Is the second-last line supposed to be $\frac{26}{5M_m},$ or is it $\frac{26}{5} M_m?$ If it is the first, you can write it in-line as $26/(5 M_m)$, and if it is the second you can write it in-line as $(26/5) M_m,$ both of which are absolutely clear and unambiguous. However, no matter which you mean your "answer" $M_m = 104$ is clearly wrong. You don't need anybody to tell you that; you can substitute $M_m = 104$ into the formula to check if it works. (That is a habit you should develop now, starting today----checking your work. After all, in an exam there will be nobody available to check your work for you.)

Thank you for clarifying that as I already know. I'm not entirely familiar with the Latex programming as I'm just a beginner starting to develop my knowledge of it.

 

[Ray Vickson]

Thank you for clarifying that as I already know. I'm not entirely familiar with the Latex programming as I'm just a beginner starting to develop my knowledge of it.

LaTeX had nothing to do with it: I am talking about clarity, which is where parentheses come into use. You can type it out without LaTeX or any other fancy tools. You could type (26/5)M_n or (26/5)Mn or (25/5)*Mn and that would be perfectly clear and readable.

At your stage of learning it is important to grasp the "parsing" rules for reading and writing mathematical expressions, so you should never, ever, type something like a/b+c or a+b/c if you mean $\frac{a}{b+c}$ or $\frac{a+b}{c}$; instead, type a/(b+c) or (a+b)/c. These are both clear and say exactly what you intend, and they take about 1/2 second longer to type than the original expressions. The other two expressions I wrote would mean $\frac{a}{b}+c$ and $a+\frac{b}{c}$ when read according to the official rules for mathematical formulas.

 

[member 587159]

Yes I do know the law.
Tell me this, is the final answer 126? (As I got that)

Correct! Well done.

 

[Physiona]

Correct! Well done.

Thank you!