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Two workers $A$ and $B$ are engaged to do a piece of work. $A$ working alone would take 8 hours more to complete the work than if $A$ and $B$ work together. If $B$ worked alone, it would take $B$ $4\frac{1}2$ hours more than the time required when both of them work together. Find the total time required for $A$ and $B$ to finish the work together.

Please guide me on how should one go about solving this and such other related problems involving time-work constraints.

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    The way the question is written it is unclear what it means. Clean up the answer please. – 2017-02-21
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    I have got the answer 1/x+8 + 1/x+9/2 = 1/x – 2017-02-21
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    There are many typos in your question. I think you mean "move" = "more" right ? Please edit your question. – 2017-02-21
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    sorry I have changed @Zubzub – 2017-02-21
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    please guide it in your own way answer for this question, so for everyone it is useful – 2017-02-21
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    Let $t$ be the time to accomplish the work together, let $a$ the time for A to accomplish the work alone and let $b$ be the time for B. You know that $$ a = 8+t \\ b = 4.5 + t $$ However I'm not sure how we can proceed further since we miss a piece of information to uniquely determine the value of $t$. – 2017-02-21
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    Edited the question to make it concise and readable. – 2017-02-21

2 Answers 2

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In the beginning you want to get some work done by $2$ person in $x$ hours,so you calculate how much work if done by $2$ person per hour your total work will be done in $x$ hours?
Let $x$ hours be required to do some work, then how much work needs to be done per hour?

$x $hour $\to$ total work
$1$ hour $\to$ how much work?
$\Rightarrow$per hour if $1/x$ work is done then the work gets completed...(I)

Then you contact $A$, and $A$ says that he/she will take $(x+8)$ hours to complete the work.
So you calculate if $A$ takes $x+8$ hours to complete the work, that means $A$ does $\frac{1}{x+8}$ work per hour...(II)

Then you contact $B$, and $B$ says that he/she will take $(x+4.5)$ hours to complete the work.
So you calculate if $B$ takes $x+4.5$ hours to complete the work, that means $B$ does $\frac{1}{x+4.5}$ work per hour...(III)

then you decide to hire both of them and they both working together finish the job in $x$ hours(read 1st line bold words), that means their work output combined together was $1/x$ per hour
$\Rightarrow$(II)+(III)=(I)

$\frac{1}{x+8}+\frac{1}{x+4.5}=\frac{1}{x}$

$\frac{2x+12.5}{(x+8)(x+4.5)}=\frac{1}{x}$

$2x^2+12.5x=x^2+12.5x+36$

$x=6$

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Let $A$ does $x$ work in $1$ hour
Let $B$ does $y$ work in $1$ hour
Let they work for $z$ hours to complete the work together

$$\therefore (x+y)\cdot z = (z+8) \cdot x = (z+4.5) \cdot y$$

first and second expression gives $yz=8x$ ... (A)

second and third expression gives $xz+8x = yz + 4.5y$...(B)

but $yz=8x , {\hspace 10pt}\therefore (B)$ becomes ${\hspace 10pt}xz=4.5y\Rightarrow z=4.5\dfrac{y}{x}$

again from $(A) {\hspace 25pt}\dfrac{y}{x}=\dfrac{8}{z}$

$$\therefore z^2=4.5*8=36 \Rightarrow z=6$$

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    your method is clearly understood @Vikram – 2017-02-21
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    But book answer is 1/(x+8) + 1/(x+9/2) = 1/x , i have a doubt regarding on this x be the piece of work, then 1 hour it will be 1/x , but it will take time morethan 8 hours means we have to add like this only - (1/x ) + 8 but how they are adding like 1/x+8 @Vikram please clarify , x work means how they add value 8 with that – 2017-02-21
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    please guide me also @Kanwaljit Singh – 2017-02-21
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    @Learninguser,in the equation that you have written, since x is added with 8 hours and 4.5 hours, x cannot be work, x has to be hours, is it clear or do you want me to post another answer? – 2017-02-21
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    X be the time and what will be the work here? @vikram, please post it another answer, sorry vikram weak in aptitude – 2017-02-21
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    1/x work per hour, total work is 1 – 2017-02-21
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    you helped me a lot, what i will give to you i dont know , only affection is there i have to give you, thank you lot @Vikram – 2017-02-21
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    @Learninguser I am very happy to help you:) – 2017-02-21