Question

In Exercises \(50-52,\) use the data from a survey about T-shirts. The survey found that 93 out of \(100,\) or \(0.93,\) of the adults responding own at least one T-shirt. The bar graph shows where those adults who own T-shirts got their oldest T-shirt. Suppose the survey takers counted 186 people who own at least one T-shirt. Choose the equation you could use to find the number of people in the survey. About how many people were surveyed? A. \(x+0.93=186\) B. \(0.93 \div x=186\) C. \(0.93 x=186\)

Short answer

The correct equation to identify the total number of people surveyed is \(0.93 \times x = 186\). With this equation, it can be estimated that about 200 individuals were surveyed.

Step-by-step solution

Identify the correct equation

First, identify the correct equation from the alternatives given. In this case, the correct equation is directly related to the proportion of adults who own at least one T-Shirt which is 0.93. As a proportion of the total number of adults surveyed (x), this would equal to the number of people who own at least one T-Shirt (186). Therefore, the correct equation should be \(0.93 \times x = 186\). So, the correct choice is C.

Calculate the total number of people surveyed

Knowing that \(0.93 \times x = 186\), solve for 'x'. This can be achieved by dividing both sides by 0.93. This results in \(x = 186 \div 0.93\). Performing this calculation gives \(x \approx 200\), indicating that about 200 people were surveyed.

Key concepts

Proportion Equations

Understanding proportion equations is critical when you're dealing with parts of a whole or ratios. A proportion equation sets two ratios equal to each other. In the format \( a:b = c:d \) or frequently written as \( \frac{a}{b} = \frac{c}{d} \) they represent the same relationship between quantities. For instance, the statement '93 out of 100 adults own at least one T-shirt' can be written as \( \frac{93}{100} \) or 0.93 in decimal form. This ratio is then applied to find out the proportion in a larger or smaller group, like the survey problem in the exercise.

When turning survey findings into proportion equations, ensure that you keep the components consistent. Here, 0.93 represents the adults who own a T-shirt in relation to the total number surveyed, which is a yet unknown quantity 'x'. Writing down the proportion as an equation allows us to solve for the missing part—the total number of people in the survey.

Solving Proportions

To solve proportions like those we encounter in survey problems, we use algebraic techniques. The equation \( 0.93x = 186 \) allows us to find 'x'—the total number of people surveyed—by performing simple algebraic manipulations. Dividing both sides of the equation by the known proportion, 0.93, we isolate 'x'. This operation gives us \( x = \frac{186}{0.93} \) which can be calculated to get the total number of individuals surveyed.

It's important to remember that solving proportions relies heavily on equivalent fractions or ratios. By keeping cross-products equal or by ensuring that one part of the proportion is equal to another, we maintain the integrity of the original ratio even when we manipulate the equation to find our unknowns.

Bar Graph Analysis

Bar graphs are an excellent tool for visual analysis of categorical data, such as survey responses. With bars of proportional length representing different quantities, they quickly communicate key elements of the data. When analyzing bar graphs, pay attention to the scale and titles on each axis, the length of each bar, and any provided labels or legends.

For example, in the context of the T-shirt survey, such a bar graph could illustrate the various sources from which adults acquired their oldest T-shirts. Data from these graphs can then be translated into proportions, percentages, or even raw numbers for further analysis or to reinforce conclusions drawn from the survey data.

Survey Data Interpretation

Interpreting survey data hinges on understanding the context and method of data collection as well as the manner in which results are presented. The given problem tells us that a survey found that 93 out of 100 adults own at least one T-shirt. Beyond simply noting this proportion, data interpretation involves extrapolating these results to larger populations or subsets of the survey group.

Assuming that the same proportion applies, we can predict the number of T-shirt owners in a different group size, such as the 186 T-shirt owners mentioned. Through critical thinking and algebraic application in solving proportions, we infer about 200 people were part of the survey, providing a glimpse into the broader community's T-shirt ownership.