Proportion questions related fourth proportions

Question

I have a couple of basic Proportion questions that I am unable to solve.

1. 18 men build a wall that is 140m long in 42 days. How long would it take 30 men to build a wall that is 100m long?

2. In a factory, the ratio of male workers to female workers is 5:3. If there are 40 less female workers than male workers, what is the total number of workers in the factory?

Answer 1

1. Solution to Problem 1

Let's say each person does $x$ amount of work each day. Work done by $18$ men in $42$ days will be equal to constructing $140$ m wall. $$18\times42\times x=140$$

$$\Rightarrow x=\frac{140}{18\times42}$$ and if time taken by $30$ men to build $100$m wall is $y$ then $$30\times y \times x=100$$

$$\Rightarrow y=\frac{100\times42\times18}{30\times140}=18\text{ days}$$

2. Solution to Problem 2

Let there be $x$ males and $y$ females Then we have $$5y=3x$$ $$x-y=40$$

Solving these two equations we get $$x=100 , y=60$$

Answer 2

I believe this is correct, but I am not sure:

1.) You need to calculate how many meters per day an 18 man crew can build a 140m wall. This is given by:

$$\frac{140 \text{ m}}{42 \text{ days}} \approx 3.\overline{33} \text{ meters per day}$$

Using this proportion we can set up another proportion: (number of men) per (meters per day). Now we just set up another proportion:

$$ \frac{18 \text{ men}}{3.\overline{33} \text{ meters per day}} = \frac{30 \text{ men}}{x \text{ meters per day}}$$

Solving for $x$ gives us:

$$ x = \frac{30 \text{ men}\cdot 3.\overline{33} \text{ meters / day}}{18 \text{ men}} $$

Simplifying this gives us

$$ x = 5.\overline{55} \text{ meters / day}$$

This means a 30 man crew can build $5.\overline{55}$ meters of a wall a day. With this ratio we can calculate the number of days:

$$ \frac{5.\overline{55} \text{ meters}}{1 \text{ days}} = \frac{100 \text{ meters}}{x \text { days}}$$

Solving for $x$ we get:

$$ x \text{ days} = \frac{100 \text{ meters} \cdot 1 \text{ days}}{5.\overline{55} \text{ meters}} \approx 18 \text{ days}$$

2.) You know that there is a ratio of 5 guys per 3 girls. Also, you know that in the current scenario there are 40 less girls than guys. If we set up a proportion:

$$ \frac{5 \text{ guys}}{3 \text{ girls}} = \frac{x \text{ guys}}{(x - 40) \text{ girls}}$$

and solve for $x$, we should be able to calculate the number of total employees.

$$ \frac{5 \text{ guys}}{3 \text{ girls}} = \frac{x \text{ guys}}{(x - 40) \text{ girls}} \implies (x - 40) \text{ girls} \cdot 5 \text{ guys} = 3 \text{ girls} \cdot x \text{ guys}$$

When solving for $x$ the "units" are going to get weird, but everything will workout.

$$ 5x \text{ guys} \cdot \text{girls} - 200 \text{ guys} \cdot \text{girls} = 3x \text{ guys $\cdot$ girls} $$

Combining like terms yields:

$$ 2x \text{ guys $\cdot$ girls} = 200 \text{ guys} \cdot \text{girls} $$

Solving for $x$ gives

$$ x = 100 $$

However, this only gives us the number of guys. We simply need to add 40 to the number of guys and we will have the number of guys and girl. The total should be 140.