# How many men required?

2000 men and 500 women can build a bridge in two weeks. If only 1500 men and 250 women are employed, it will take three weeks to build the same bridge.

How many men will be required if it has to be built in just one week?

• Is it not just $0$? You simply need a whole bunch of women, but that number isn't even required. – Benji Altman Feb 9 '17 at 18:21
• @BenjiAltman: I think the assumption is that the work force in the question consists only of men, but your objection is a fair one; the question as posed is pretty ambiguous. – Brian Tung Feb 9 '17 at 18:23
• It is not realistic to assume that the amount of bridges built varies linearly with the number of men, or with the number of women. – Jonas Meyer Feb 9 '17 at 20:40

[N.B. As Benji Altman points out in the comments to the OP, the question is unclear about the make-up of the work force. The below assumes that the work force for the question consists solely of men.]

Basic approach. We assume, as is typical in these problems, that each person (possibly differentiated by gender) builds a certain fraction of the bridge per week. Obviously, this is not truly realistic, but it's the characteristic fiction of this kind of problem.

If we let $m$ and $w$ stand for this fraction, then we have

$$2000m+500w = \frac{1}{2}$$

since the combined efforts of $2000$ men and $500$ women suffice to build half a bridge in a week. Do the same with the other equation, and solve for $m$ and $w$. Keep in mind that the desired answer—the number of men required to build the bridge in a week—is not $m$ itself, but its reciprocal. That is to say, if $m = \frac{1}{5000}$ (it isn't; I just chose this number for example), then $5000$ men would be required to build the bridge in one week.

Incidentally, it looks to me as though someone was careful to avoid charges of discrimination.

• My comment is not about specifically your answer, but rather complaint about the common approach to teaching solving such problems in mathematics classes. I think that explaining this kind of problems could be more transparent and understandable if dimensions of all values are written explicitly, like they are in problems in Physics. E.g. to write the equation as $2000 {\ workers} \cdot m + 500 {\ workers}\cdot w=\frac 1 2\ \frac{bridges}{week},$ and emphasise that $m$ and $w$ are quantities that measure performance of each kind of workers measured in bridges divided by worker-weeks. – Alexander Rodin Feb 9 '17 at 18:47
• @AlexanderRodin I agree completely. I used to do this on word problems and would lose points. the prof saying something stupid like "you can't multiply [workers] with [men]." Or "[bridges] and [weeks] aren't units of measurement." Quite frankly I always found doing this make every problem as trivial as calculating the acceleration when knowing velocity. I'm glad to see someone else in this world shares my point of view. – user335907 Feb 15 '17 at 0:08

Let's introduce a quantity called productivity and denote it as $P.$ This quantity characterizes the number of bridges that a single worker or a team of workers could build in some unit of time. For example, if one worker could build a bridge in $30$ years then for this worker $P = \frac{1}{30} \frac{bridge}{year}.$

If we have $n$ workers with productivities $P_1, P_2, ..., P_n$ then we assume that their joint productivity is equal to sum of their productivities: $P_{1,2,...,n} = P_1 + P_2 + ... P_n.$

Note that if we wanted to add two productivities measured in different units, for example $P_1 = 1 \frac{bridge}{week}$ and $P_2 = 1 \frac{bridge}{day},$ then we should at first convert them to the same units, in this case for example as $P_2 = 1 \frac{bridge}{day} = 7 \frac{bridge}{week},$ and only then we are allowed to add up their numerical values.

In this problem we are implicitly prompted to assume that the productivity are of all men is similar and productivity of all women is similar too. Then these productivities could be denoted as $P_m$ and $P_w$ respectively. Taking in into account the provided numeric values in this problem it is convenient to measure productivities in bridges per week.

In the first case the joint productivity of a team is equal to $2000 P_m + 500 P_w,$ and the build time is two weeks, so $$2\ weeks \cdot (2000 P_m + 500 P_w) = 1\ bridge.$$

The units of the quantity in the left side of the equation is $week\ \cdot\ \frac{bridge}{week} = bridge,$ that's the same as the units in the right side. If these inits were different it would indicate an error made somewhere, because dimensions in left and right sides of equalities should always correspond.

It is easy to write a similar equation for the second case, when the bridge was build in three weeks.

Using these equations it would be easy to calculate productivities of men and women $P_m$ and $P_w$ respectively, and then make predictions using these productivity values.

Every $500$ women there are $2,000$ men for every two weeks. If we need the bridge built in one week then we should increase the amount of men by $150$ percent.

• Can you please explain this a little further. – Bala Eesan Feb 16 '17 at 2:10