Question

Translate to an equation and solve. According to the Bureau of Justice Statistics, in 2009 \(93 \%\) of America's inmates were men. If there were about 1.6 million inmates in America, how many were men? (Source: Bureau of Justice Statistics.)

Short answer

Answer: In 2009, there were 1,488,000 male inmates in America.

Step-by-step solution

Determine the percentage of male inmates

We know that 93% of inmates are men.

Convert the percentage to a decimal

To convert the percentage to a decimal, divide by 100: \(93\% = \frac{93}{100} = 0.93\)

Represent the total number of inmates

We know there are 1.6 million inmates in total: \(1,600,000\)

Create an equation representing the number of male inmates

Use the decimal value for the percentage (0.93) and the total number of inmates (1.6 million) to create an equation: \(0.93 \cdot (1,600,000) = x\), where x is the number of male inmates.

Solve for x (the number of male inmates)

Multiply the values to find the number of male inmates: \(0.93 \cdot (1,600,000) = 1,488,000\). Hence, in 2009, there were 1,488,000 male inmates in America.

Key concepts

Percentage to Decimal Conversion

Understanding how to convert percentages to decimals is foundational in handling many real-life calculations, including finance, data analysis, and yes, solving textbook problems. The concept is quite straightforward—since 'percent' literally means 'per hundred', you are essentially translating a number out of 100 into its decimal form.

To convert a percentage to a decimal, simply divide the percentage by 100. For instance, when you're working with 93%, you divide 93 by 100, which gives you 0.93. This method works for any percentage, providing a quick and simple way to transition between these two formats of expressing proportions.

Equation Creation

Crafting equations is a significant skill in mathematical problem-solving. It's how you translate words into numbers and operations. When faced with a word problem, the first step is to identify the key variables and the relationships between them.

In our example regarding inmates, the percentage of inmates that are men and the total inmate population are the two key variables. The equation is set up by multiplying the decimal equivalent of the percentage by the total number of inmates. In essence: Equation for Male Inmates:
Proportion of Men (as a decimal) x Total Number of Inmates = Number of Male Inmates

This simple equation allows you to solve for the unknown value, which in this case is the number of male inmates.

Mathematical Problem Solving

Problem solving in mathematics requires a systematic approach that includes understanding the problem, devising a plan, carrying out the plan, and then reflecting on the solution. Our exercise is an excellent example of this process in action.

• Firstly, you decipher the text to understand that you're seeking the number of male inmates.
• Next, you develop a plan which involves converting percentages to decimals and creating an equation.
• Then, you execute the plan by plugging the known values into the equation and performing the arithmetic.
• Finally, you reflect on the result and consider whether it makes sense within the context of the problem.

By practicing these steps regularly, you'll sharpen your problem-solving skills and gain confidence in tackling a variety of mathematical challenges.