Question

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. The ratio of male to female clients in a nursing facility is 3 to \(5 .\) If there are 40 women in the facility, how many men are there?

Short answer

There are 24 men in the facility.

Step-by-step solution

Understand the Problem

We are given the ratio of males to females as 3:5, and the number of females is 40. We need to determine the number of males.

Set Up the Proportion

Use the ratio to set up a proportion: \( \frac{\text{males}}{\text{females}} = \frac{3}{5} \). Plug in the given number for females: \( \frac{\text{males}}{40} = \frac{3}{5} \).

Solve the Proportion

Cross-multiply to solve for the number of males: \( \text{males} \times 5 = 3 \times 40 \). This simplifies to \( 5 \times \text{males} = 120 \).

Solve for Males

Divide both sides of the equation by 5 to isolate males: \( \text{males} = \frac{120}{5} = 24 \).

Label the Answer

There are 24 men in the facility based on the calculated ratio.

Key concepts

Understanding Ratios

In real-world scenarios, we often deal with ratios, which are a way to compare two quantities. A ratio tells us how much of one thing we have in relation to another. For example, in this problem, the ratio of male to female clients is 3 to 5.
This means for every 3 males, there are 5 females. Ratios help us establish these relationships clearly and concisely without listing all the quantities involved. When working with ratios, it's key to maintain the order provided. Here, male numbers precede female ones, aligning with the stated relationship.

Analyzing Nursing Facility Clients

Let's apply ratios in the context of a nursing facility, where understanding client demographics could be crucial. The problem involves a nursing facility with a known ratio of male to female clients. One reason this may matter is in planning for client needs, where knowing the gender distribution can aid in resource allocation.
Ratios outline this demographic distribution clearly, as shown in the exercise. A set number of women (40 in this case) allows us to calculate how many clients of another gender (men) are present. This information could help administrators in decision-making regarding staffing, amenities, or healthcare provisions.

Solving Proportions

Solving proportions involves equating two ratios and using mathematical steps to solve for unknowns. A proportion can be set up using the ratio and the specific numbers from the problem. Let's look at the details:

• Firstly, establish a proportion using the data. Here, the given male-to-female ratio is \( \frac{3}{5} \), and females are set at 40.
• Insert the known quantity into the proportion: \( \frac{\text{males}}{40} = \frac{3}{5} \).

To solve, cross-multiply to find the number of males: \( \text{males} \times 5 = 3 \times 40 \), leading to \( 5 \times \text{males} = 120 \).
Isolate "males" by dividing both sides by 5, giving \( \text{males} = \frac{120}{5} = 24 \). Hence, the facility houses 24 men, maintaining the established proportion. Understanding these steps empowers students to analyze similar word problems confidently.