Work Rate Problem: Adding Women To Finish Faster

by Luna Greco 49 views

Hey guys! Let's dive into a classic work-rate problem. These problems might seem tricky at first, but once you break them down, they're totally manageable. We've got a scenario where we need to figure out how many more women are needed to complete a project faster, given the work rates of men and women. Let's break it down step by step!

Understanding the Problem

In this work problem, the core challenge revolves around understanding how different groups of people contribute to completing a task within specific timeframes. We start with the baseline information: 12 men OR 18 women can complete a job in 30 days. This tells us something crucial about the relative work rates of men and women. It means 12 men work at the same pace as 18 women. This relationship is the key to unlocking the rest of the problem.

The problem throws a curveball by asking us to scale up the task. We need to complete a job that is three times as difficult, but we only have 36 days to do it. This introduces another layer of complexity, as we need to adjust our workforce accordingly. We already have 8 men on the team, but the big question is: how many additional women do we need to bring in to meet this new deadline and increased workload? This is where our problem-solving skills come into play.

To solve this, we need to think about the total amount of work that needs to be done and the combined work rate of the men and women involved. We'll need to translate the given information into equations and use them to figure out the missing piece – the number of additional women. Don't worry, we'll break it down into smaller, more digestible steps to make it crystal clear. So, let's get started and see how we can tackle this challenge!

Step 1: Finding the Relationship Between Men and Women's Work

The first crucial step in tackling this problem is figuring out the relationship between how much work a man can do compared to a woman. We know that 12 men can finish a job in the same time as 18 women (30 days). This is our starting point, and it gives us a direct comparison of their work rates. To make things easier to compare, let's think about this in terms of individual work rates.

If 12 men and 18 women take the same amount of time to do the same job, it means the total work they accomplish is equal. We can express this mathematically. Let's say a single man's work rate is 'M' (meaning the amount of work he does in a day) and a single woman's work rate is 'W'. The equation we get from the given information is:

12M * 30 days = 18W * 30 days

Notice that both sides are multiplied by 30 days because they both take the same time to complete the work. Now, we can simplify this equation. Since both sides have a "* 30 days", we can divide both sides by 30, which gives us:

12M = 18W

This is a much simpler form! Now we want to find the ratio between M and W. To do this, we can divide both sides by 6, simplifying further to:

2M = 3W

This equation is super important. It tells us that 2 men do the same amount of work as 3 women. Or, we can say that one man does 3/2 (or 1.5) times the work of a woman. This ratio is the key to unlocking the rest of the problem. We can use this to convert our workforce into a single unit – either all men or all women – which will make the calculations much simpler.

Step 2: Calculating Total Work and New Workload

Now that we understand the relationship between men's and women's work rates, let's calculate the total amount of work involved in the original project. This will serve as our baseline and help us determine the workload for the new, more difficult project. We can use either the men's or women's work rate for this calculation, as they both represent the same amount of work.

Let's use the information about the women. We know 18 women can complete the job in 30 days. If we think about the total work as the number of 'woman-days' required, we can calculate it like this:

Total Work (in woman-days) = 18 women * 30 days = 540 woman-days

This means the original job requires 540 'woman-days' of effort. It's a way of quantifying the amount of work in terms of how many days a single woman would need to complete it.

Now, the problem states that the new job is three times as difficult. This means the new workload is three times the original workload. So, let's calculate that:

New Total Work (in woman-days) = 3 * 540 woman-days = 1620 woman-days

So, the new job requires a whopping 1620 'woman-days' of work! This is a significant increase from the original project, and it highlights why we need to adjust our workforce. We now have a clear target – we need to figure out how many women, along with the 8 men, are needed to complete this 1620 woman-day job in 36 days.

Step 3: Converting Men to Women and Setting up the Equation

We're getting closer! Now, we need to figure out how the 8 men we already have contribute to the workload. Since we've been working with 'woman-days', it makes sense to convert the men's work rate into an equivalent number of women. Remember from Step 1, we found that 2 men do the same work as 3 women. This is the key to our conversion.

We can set up a proportion to find out how many women are equivalent to 8 men:

(2 men) / (3 women) = (8 men) / (x women)

To solve for x, we can cross-multiply:

2 * x = 3 * 8
2x = 24
x = 12

So, 8 men are equivalent to 12 women in terms of their work rate. This means our existing team of 8 men provides the same amount of work as 12 women. Now, we can combine this with the additional women we need to hire to complete the project.

Let's say we need to add 'y' women to the team. The total number of women (in equivalent work rate) will be 12 (from the men) + y. We know this combined team needs to complete 1620 woman-days of work in 36 days. We can set up an equation to represent this:

(12 + y) women * 36 days = 1620 woman-days

This equation represents the core of the problem. We know the total work, the time available, and the contribution of the men. The only unknown is 'y', the number of additional women we need. Now, let's solve for 'y'!

Step 4: Solving for the Number of Additional Women

We've got our equation set up, now it's time to solve for 'y', which represents the number of additional women needed. Our equation from the previous step is:

(12 + y) * 36 = 1620

First, we need to isolate the term with 'y'. We can do this by dividing both sides of the equation by 36:

(12 + y) = 1620 / 36
(12 + y) = 45

Now, it's a simple matter of subtracting 12 from both sides to solve for 'y':

y = 45 - 12
y = 33

So, we need to add 33 women to the team to complete the job on time! This is our final answer. Let's recap the steps we took to get here.

Step 5: Final Answer and Recap

Okay, we've cracked the code! The final answer is that we need to add 33 women to the existing team of 8 men to complete the job, which is three times as difficult as the original, in 36 days.

Let's quickly recap the steps we took to solve this problem:

  1. Find the relationship between men and women's work: We determined that 2 men do the same amount of work as 3 women.
  2. Calculate total work: We found the original job required 540 woman-days of work, and the new job required 1620 woman-days.
  3. Convert men to women: We converted the 8 men into an equivalent of 12 women.
  4. Set up the equation: We created the equation (12 + y) * 36 = 1620, where 'y' is the number of additional women.
  5. Solve for 'y': We solved the equation and found that y = 33.

So, the correct answer is (b) 33. These types of work-rate problems might seem daunting, but by breaking them down into smaller, manageable steps, you can conquer them every time. Keep practicing, and you'll become a pro at these in no time!

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