# 2 men and 3 women take 16 days to complete a task. How long would it take for 4 men and 6 woman?

Dec 14, 2017

8 days

#### Explanation:

We know that the number of days is inversely proportional to the number of men and women, with respect to some constant $k$.

That means,

$d = \frac{k}{m + w}$

Subbing in the values from the first bit of information:

$16 = \frac{k}{5}$

$k = 80$

So, $d = \frac{80}{m + w}$

Substituting the second bits of information into the formula, we get the equation:

$d = \frac{80}{4 + 6}$

$d = 8$

Dec 14, 2017

Please see the steps and process in solving the question above..

#### Explanation:

In this kind of question, you should note that the higher the number of workers the lower the number of days..

So lets start solving..

Let;

$n = \text{Total Workers}$

$m = \text{Total Number of Days}$

Then we have;

$n \to m = {n}_{1} {m}_{1} = {n}_{2} {m}_{2}$

Where;

${n}_{1} = 2 + 3 = 5$

${n}_{2} = 4 + 6 = 10$

${m}_{1} = 16$

m_2 = ?

Therefore;

$5 \times 16 = 10 \times {m}_{2}$

$80 = 10 {m}_{2}$

Divide both sides by $10$

$\frac{10 {m}_{2}}{10} = \frac{80}{10}$

$\frac{\cancel{10} {m}_{2}}{\cancel{10}} = \frac{80}{10}$

${m}_{2} = \frac{80}{10}$

${m}_{2} = 8$

So it will take $8$ days for $10$ workers..

Hope this helps!

Dec 14, 2017

8 days. Full explanation given building the solution one step at a time.

#### Explanation:

$\textcolor{b l u e}{\text{Shorter technique}}$
If you double the work force they will put in double the amount of work. So the task will be completed twice as quickly. That is, you halve the time.

$\frac{16}{2} = 8 \text{ days}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Much deeper look at the problem showing a useful technique.}}$

Let the work rate for a man be ${w}_{m}$
Let the work rate for a woman be ${w}_{w}$

Let time in days be ${t}_{d}$

Let the total amount of work needed to complete the task be ${W}_{T}$

Then we have

$2 {w}_{m} {t}_{d} + 3 {w}_{w} {t}_{d} = {W}_{T}$

Factoring out the time

${t}_{d} \left(2 {w}_{m} + 3 {w}_{w}\right) = {W}_{T}$

Set $2 {w}_{m} + 3 {w}_{w}$ as $x$ giving

$x {t}_{d} = {W}_{T}$
~~~~~~~~~~~~~~~~~~~~~~~~~
Initial condition is that ${t}_{d} = 16 \text{ days}$

Not that $x {t}_{d} = {W}_{T}$ where ${W}_{T}$ is a constant. So if $x$ changes then ${t}_{d}$ must also change in such a way that we still get ${W}_{T}$.

This is the same principle as that of measuring power used in your home verses time -> kilowatt hours.

Both the count of men and women have doubled

So instead of just $x$ we have $2 x$
However as we have changed $x {t}_{d}$ to $2 x {t}_{d}$ it will no longer equal ${W}_{T}$.

The thing is that the total amount of work needed( ${W}_{T}$ )to complete the task will not change. It is constant.

So if ${W}_{T}$ is fixed we must change $2 x {t}_{d}$ is such a way that it changes back to $x {t}_{d}$. We can do that by including the correction of $\times \frac{1}{2}$

So we end up with $2 x \times {t}_{d} / 2 = {W}_{t}$

In other words we changed $x {t}_{d}$ one way then changed it back again.

We know that ${t}_{d} = 16 \text{ days}$ so ${t}_{d} / 2 = \frac{16}{2} = 8 \text{ days}$