# Two brigades had to build a house. The first brigade works alone and builds the house in 15days. The second brigade builds it in 30days. How long will it take to build the house when both brigades work together?

Mar 12, 2016

10 days.

#### Explanation:

The combined effort is sum of the efforts.
Effort1/day = $\frac{1}{15}$ unit.
Effort2/day= $\frac{1}{30}$ unit.
Combined effort is $\left(\frac{1}{15} + \frac{1}{30}\right)$ unit =\ $\frac{1}{10}$ unit.
So, when both work together, they finish one unit in 10 days.

Mar 12, 2016

10 days

#### Explanation:

As each person is assumed to work at the same rate and the second brigade takes twice as long as the first; it implies that brigade 2 has 1/2 the membership as brigade 1
( 1/2 as many people means that they work twice as long)

So combining the two give $1 \frac{1}{2}$ times as many as in brigade 1

Consider brigade 1 as being unit the size man days.
Let the number of days be $d$

Then color(brown)(1 ("man days") xx d_1=15" days Where "d_1 = 15" days")

Adding the two groups give $1 \frac{1}{2} \text{man days}$

Thus color(blue)(1 1/2 "man days") xx d_2=15" days Where "d_2" is unknown"

So $\textcolor{g r e e n}{{d}_{2} = 15 \div 1 \frac{1}{2} = 10 \text{ days}}$

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Further explanation
Choosing random numbers just for this demonstration:

Job 1
3 people working 6 days gives $\left(3 \times 6\right) = 18$ man days of work

Job 2
5 people working 10 days gives $\left(5 \times 10\right) = 50$ man days of work

So if 5 people did job 1

Then $3 \times 6 = 5 \times x = 18$

$x = \frac{18}{5}$days to complete the task