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Chapter 1: Problem 17

a. A glass jar contains 17 blue marbles and 25 red marbles. What percentage of the marbles are blue? b. A different glass jar has 430 marbles, and \(63 \%\) of them are blue. How many blue marbles are in the jar? c. A different glass jar contains \(45 \%\) red marbles and has 90 red marbles in it. What is the total number of marbles in the jar?

Short Answer

Expert verified
a. The percentage of blue marbles is 40.47%. b. The number of blue marbles is 271. c. The total number of marbles is 200.

Step by step solution

01

Percent Calculation for Problem a

Here, the part is the number of blue marbles, which is 17 and the whole is the total number of marbles, which is obtained by adding the number of blue and red marbles, i.e., 17 + 25 = 42. Hence, the percentage of blue marbles is \(\frac{17}{42} * 100 = 40.47 \% \)
02

Calculating the Number of Blue Marbles for problem b

Here, the total number of marbles is given (430) and the percentage of blue marbles is also given (63%). To find the number of blue marbles, the formula part = \(\frac{Percentage * Whole}{100}\) is used. Hence, the number of blue marbles is \( \frac{63 * 430}{100} = 270.9 \). Since we cannot have part of a marble, this result has to be rounded to the nearest whole number, which is 271.
03

Total Marbles Calculation for problem c

Here, the number of red marbles (45) and their percentage (90%) is given. The total number of marbles can be found using the formula total = \(\frac{Part}{Percentage}\). Hence, the total number of marbles is \(\frac{90}{45} * 100 = 200\).

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Key Concepts

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

Percent Calculation
Percent calculation is a fundamental concept in mathematics and various real-world applications, particularly in statistics. It involves determining what fraction of a whole is represented by a certain number, then expressing that fraction as a percentage. A percentage represents a number out of 100, an easily digestible format for comparing quantities.

In problem a, percent calculation is utilized to find out how much of the jar is filled with blue marbles. By dividing the number of blue marbles (17) by the total number of marbles (42), and then multiplying the result by 100, we convert the fraction to a percentage. It's crucial to ensure the division is done prior to the multiplication to avoid a common mistake.

The formula for finding the percentage is: \[ \text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) * 100 \. \] Remember to express the result to two decimal places for accuracy, unless instructed otherwise鈥攁s percentages are all about precision.
Proportional Reasoning
Proportional reasoning is about understanding the relationship between parts and wholes. It allows one to solve problems by finding equivalent ratios or fractions that represent the same relationship.

When faced with problem b, we can apply proportional reasoning to calculate the quantity of blue marbles. Here, the problem gives us the whole (430 marbles) and the percent of the whole that are blue (63%), we seek the 'part'鈥攖he number of blue marbles. To find the number of blue marbles, we establish a proportional relationship between the percentage and the quantity:\[ \text{Part} = \frac{\text{Percentage} * \text{Whole}}{100} \]

By plugging in the given values into the formula, we compute the 'part' to be 270.9, which practically gets rounded to the nearest whole number, 271 marbles. The concept of proportional reasoning reinforces the importance of direct proportion鈥攖he higher the total number of marbles, the more blue marbles there will be, given the percentage.
Statistical Numeracy
Statistical numeracy is the ability to interpret, evaluate, and communicate statistical information. It enables individuals to make sense of data and use it in decision-making. For example, understanding the percentage of red marbles like in problem c requires statistical numeracy.

The data given is a certain number of red marbles (90) which represent 45% of the total. To find the total number of marbles in the jar, we use the formula:\[ \text{Total} = \frac{\text{Part}}{\text{Percentage}} * 100 \]

This calculation reveals that there are 200 marbles in total, which includes both red and any other colored marbles in the jar. In everyday life, statistical numeracy is key for making well-informed decisions on matters such as budgeting, interpreting survey results, and understanding news reports that use percentage data.

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Most popular questions from this chapter

The accompanying table gives the population of the six U.S. states with the largest populations in 2008 and the area of these states. (Source: www.infoplease.com) $$\begin{array}{|l|l|l|} \hline \text { State } & \text { Population } & \text { Area (square miles) } \\\ \hline \text { Pennsylvania } & 12,448,279 & 44,817 \\ \hline \text { Illinois } & 12,901,563 & 55,584 \\ \hline \text { Florida } & 18,328,340 & 53,927 \\ \hline \text { New York } & 19,490,297 & 47,214 \\ \hline \text { Texas } & 24,326,974 & 261,797 \\ \hline \text { California } & 36,756,666 & 155,959\\\ \hline \end{array}$$ a. Determine and report the rankings of the population density by dividing each population by the number of square miles to get the population density (in people per square mile). Use rank 1 for the highest density. b. If you wanted to live in the state (of these \(\operatorname{six}\) ) with the lowest population density, which would you choose? c. It you wanted to live in the state (of these six) with the highest population density, which would you choose?

A study by Pezzoli and Cereda was reported in Neurology, May \(28,2013 .\) The report said that the use of pesticides is associated with the development of Parkinson's disease, which is a neurological disease that causes people to shake. The study reported that exposure to bug killers and weed killers is "associated with" an increase of \(33 \%\) to \(80 \%\) in the chances of getting Parkinson's. Does this study show that pesticides cause Parkinson's disease? Why or why not? (Source: Pezzoli and Cereda, Exposure to pesticides or solvents and risk of Parkinson disease, Neurology, vol. 80, no. 22: 2035-2041, May 28,2013 )

Philip, an employee at an IT firm in Mexico, kept track of the duration of his daily meetings (in minutes) for one week. He also took note of whether the meetings took place in the morning or in the afternoon. Morning meetings: \(45,90,75,35,60,80\) Afternoon meetings: \(50,65,45,70\) Write these data as they might appear in (a) stacked format with codes and (b) unstacked format.

In the fall of 2004 , there was a shortage in flu vaccine in the United States after it was discovered that vaccines from one of the manufacturers were contaminated. The New England Journal of Medicine reported on a study that was done to see whether a smaller dose of the vaccine could be used successfully. If that were the case, then a small amount of vaccine could be divided into more flu shots. In this study, the usual amount of vaccine was injected into half the patients, and the other half of the patients had only a small amount of vaccine injected. The response was measured by looking at the production of antibodies (more antibodies generally result in less risk of getting the flu). In the end, the lower dose of vaccine was just as effective as a higher dose for those under 65 years old. What more do we need to know to be able to conclude that the lower dose of vaccine was equally effective at preventing the flu for those under \(65 ?\) (Source: Beishe et al., Serum antibody responses after intradermal vaccination against influenza, New England Journal of Medicine, 2004 )

Effects of Light Exposure (Example 9) A study carried out by Baturin and colleagues looked at the effects of light on female mice. Fifty mice were randomly assigned to a regimen of 12 hours of light and 12 hours of dark (LD), while another fifty mice were assigned to 24 hours of light (LL). Researchers observed the mice for two years, beginning when the mice were two months old. Four of the LD mice and 14 of the LL mice developed tumors. The accompanying table summarizes the data. (Source: Baturin et al., The effect of light regimen and melatonin on the development of spontaneous mammary tumors in mice, Neuroendocrinology Letters, 2001) $$\begin{array}{lcc} & \text { LD } & \text { LL } \\ \text { Tumors } & 4 & 14 \\ \hline \text { No tumors } & 46 & 36 \end{array}$$ a. Determine the percentage of mice that developed tumors from each group (LL and LD). Compare them and comment. b. Was this a controlled experiment or an observational study? How do you know? c. Can we conclude that light for 24 hours a day causes an increase in tumors in mice? Why or why not?
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