# Time and work . Men and women working.

It takes six days for three women and two men working together to complete a work. Three men would do the same work five days sooner than nine women. How many times does the output of a man exceed that of a woman?

Question?

The statement "Three men would do the same work five days sooner than nine women."

Does this mean that three men would complete "any amount" of the "given type of work" five days earlier than 9 women? By which I mean to say if there are 100 units of work, then three men would complete the work 5 days earlier than 9 women. Also if there are 1000 units of work, then also three men would complete the work 5 days earlier than 9 women.Is this the case?How?

If this is the case then we can write:

Let a man do $m$ units of work/day and a woman do $w$ units of work/day. Let the time taken by 9 women to do the entire work be $t$ days.

$(3w+2m)*6=1$

$3m(t-5)=1$

$9w*t=1$

If this is not the case then we have:

$(3w+2m)*6=1$

$3m(t-5)=9w*t$

In this case where is the 3rd equation that we need to find the variables?

• wow that's a racist question. – Yanko Dec 14 '17 at 10:49
• The unknown you need is $m/w$, so you need to get rid of the $1$ in the first equation. – Mark Bennet Dec 14 '17 at 10:56
• @yanko Because women are a..race? :-) – MathematicianByMistake Dec 14 '17 at 11:07
• @yanko I think you mean misogynistic. – gen-z ready to perish Dec 14 '17 at 13:03
• The far more better question is on what kind of task they are working ... – Michael Hoppe Dec 14 '17 at 16:21

Let one woman complete the job in $w$ days $\implies$ rate of woman’s work is $\frac{1}{w}$ job/day.

Let one man complete the job in $m$ days $\implies$ rate of man’s work is $\frac{1}{m}$ job/day.

Your mistake is in the step: $(3w+2m)\times 6=1$. It should instead be $$(\frac{3}{w}+\frac{2}{m})\times 6 = 1$$

Now, as three men would do the work five days sooner than nine women, we have, $$\frac{m}{3}+5=\frac{w}{9}$$

Solving the quadratic, we have, $m=15$ and thus, $w=90$, giving us, $\frac{w}{m}=6$.

• Thanks. I had assumed m as the no. of units worked per day. I had taken the efficiency as m units/day...and you have taken the no. of days required to complete the work as 'm'....Thanks :) – Soumee Dec 14 '17 at 11:33
• @Soumee You can accept my answer if it helped you. – Rohan Dec 14 '17 at 12:16
• I have upvoted your answer. I am waiting for a few more answers. I want it to be done by the 'efficiency' method. And I have one fundamental question that I can't understand. – Soumee Dec 14 '17 at 12:19
• @Soumee Rohan is correct. You should explain what you do not understand. – N. F. Taussig Dec 14 '17 at 13:05
• @Soumee It will change as the amount of work changes. – N. F. Taussig Dec 14 '17 at 15:15

m - amount of work one man can do per day
w - amount of work one woman can do per day
1 - amount of the work when it has been fully completed (100% of the work is done)
x - number of days it takes 3 men to complete the job (100% of it or 1)

Our conditions:

$$6(2m+3w)=1 \text{ (1)}\\ x\cdot 3m =1 \text{ (2)}\\ (x+5)9w=1 \text{ (3)}$$

In equation 1, let's express the amount of work a man can do per day in terms of the amount of work a woman can do per day:

$$m=\frac{1-18w}{12}$$

Now, in equation 2, let's express the number of days, $x$, in terms of $w$:

$$x\cdot 3m =1\implies x=\frac{1}{3m}\implies x=\frac{1}{3\left(\frac{1-18w}{12}\right)}\implies x=\frac{4}{1-18w}$$

Let's plug in $x$ into equation 3, simplify and solve the resulting quadratic equation for $w$:

$$(x+5)9w=1\implies \\ \left(\frac{4}{1-18w}+5\right)9w=1\implies\\ 810w^2-99w+1=0 \implies \\ w_{1}=\frac{1}{9}\\ w_{2}=\frac{1}{90}$$

Let's now find $x$:

$$x=\frac{4}{1-18\cdot 1/9}=-4 \text{ (meaningless solution)}\\ x=\frac{4}{1-18\cdot 1/90}=5$$

Now, let's find $m$ and $w$:

$$m=\frac{1}{3x}\implies m=\frac{1}{15}\\ w=\frac{1}{9(x+5)}\implies w=\frac{1}{90}$$

And now let's compare their rates of work by diving a man's amount of work done per day by that of a woman: $$\frac{m}{w}=\frac{1/15}{1/90}=6$$

This can be interpreted as saying that a man can accomplish six times as much work per day as a woman.

Answer: The output of a man exceeds the output of a woman $6$ times.