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Chapter 7: Problem 59

In a recent year, children and adolescents comprised 6.3 million hospital stays. If this represents \(18 \%\) of all hospital stays, what was the total number of hospital stays?

Short Answer

Expert verified
The total number of hospital stays was 35 million.

Step by step solution

01

Understand the Problem

We know that 6.3 million hospital stays represent 18% of the total number. We need to find the total number of hospital stays.
02

Set Up the Equation

Let the total number of hospital stays be denoted as \(T\). According to the problem, 18% of \(T\) is equal to 6.3 million. This can be written as: \(0.18T = 6.3 \)
03

Isolate the Variable

Divide both sides of the equation by 0.18 to isolate \(T\): \(T = \frac{6.3}{0.18} \)
04

Perform the Division

Calculate the value of \(T\): \(T = 35 million \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

percentage calculation
Understanding percentage calculation is essential for solving many real-life problems, like the one in the exercise about hospital stays. A percentage represents a part of a whole divided into 100 equal parts. When we say 18% of hospital stays involve children and adolescents, it means that 18 out of every 100 hospital stays are for children and adolescents. To find out the total number of hospital stays based on a given percentage, you set up an equation where the percentage (as a decimal) multiplied by the total equals the given part. For example, in this problem, 18% as a decimal is 0.18, and our given part is 6.3 million. This helps us set up the equation: \(0.18T = 6.3\).
algebraic equations
Algebraic equations help us solve for unknown values by setting up a mathematical statement indicating the equality between two expressions. In our exercise, we needed to determine the total number of hospital stays, so we used an algebraic equation. First, we defined the total number of hospital stays as \(T\). Then, we represented 18% of \(T\) as \(0.18T\). Given that this equals 6.3 million hospital stays, we wrote the equation: \(0.18T = 6.3\). Our next step was to solve for \(T\) by isolating the variable. This involves performing operations to get \(T\) on one side of the equation by itself.
division
Division is a key mathematical operation used to break a number into equal parts. In our example, to isolate our variable \(T\) in the equation \(0.18T = 6.3\), we needed to perform a division. Specifically, we divided both sides of the equation by 0.18, which removes the 0.18 on the left side, leaving us with \(T = \frac{6.3}{0.18}\). Performing this division gives us the solution: \(T = 35\). This means the total number of hospital stays is 35 million. Understanding and correctly performing division is critical for solving equations and finding accurate solutions.

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