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Chapter 15: Problem 59

Mike's average on the first five exams in Econ \(1 \mathrm{~A}\) is 88 What must he earn on the next exam to raise his overall average to \(90 ?\)

Short Answer

Expert verified
Mike must score 100 on his next exam to raise his overall average to 90.

Step by step solution

01

Find the total sum of scores

First, we need to find the total sum of Mike's scores on the first five exams. Since his average is 88 and he's taken 5 exams, we calculate the total sum of scores by multiplying his average score by the number of exams. So, the total sum of scores is \(88 * 5 = 440\).
02

Set up the equation

Now, we set up an equation to find the score Mike needs to get on his 6th exam. We know his desired average is 90 and this average will be based on 6 exams - the 5 he's already taken and the one he's yet to take. The equation is thus: \( \frac{440 + x}{6} = 90\). In this equation, 'x' stands for the score Mike should get to earn his desired average.
03

Solve the equation

Multiplying both sides of the equation by 6, we get \(440 + x = 540\). If we solve this for 'x', by subtracting 440 from both sides, we have \(x = 540 - 440\). This results in \(x = 100\).

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Key Concepts

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

Understanding Average Calculations
The concept of averages is a fundamental part of mathematics, especially useful in assessments like exams. An average provides a single value that summarizes or represents a set of numbers. To calculate the average, sum up all the numbers in your data set, then divide by the number of items in that set.

For instance, Mike's average score for his first five exams is calculated by totaling his scores and dividing by 5, the number of exams. This process simplifies comparison and helps to identify targets, just like Mike needs an average of 90 after six exams.

  • Step 1: Total Scores = Average 脳 Number of Items
  • Step 2: Average = Total Scores / Number of Items

These formulas help us understand the distribution of values and determine what must be achieved to reach a particular average.
Solving the Equation
Equations help us find unknown values by balancing two sides of a statement, using arithmetic operations to maintain equality.

In the exercise, we set up the equation \( \frac{440 + x}{6} = 90 \), where 440 represents the total of known scores, 6 is the total exams, and \'x\' is the unknown score for the 6th exam. The goal is to solve for \'x\'.

  • First, eliminate the fraction by multiplying both sides by 6, getting \( 440 + x = 540 \).
  • Next, isolate \'x\' by subtracting 440 from both sides, leaving \( x = 100 \).

These steps ensure that the mathematical operations lead us directly to the correct and desired outcome.
Mastering Arithmetic Operations
Arithmetic operations are the basic building blocks of all mathematical calculations. These include addition, subtraction, multiplication, and division. Each has its unique purpose and can be used in various scenarios, like solving equations or calculating averages.

In Mike's case, finding out what score he needs involves:

  • Multiplication: Used to find the total current scores by calculating 88 脳 5.
  • Addition and Subtraction: Determine the needed score by lining up known totals against the desired overall outcome.
  • Division: Essential in average calculation, showing how individual components contribute to the whole.

Understanding these operations is crucial. They enable the manipulation of numbers to solve problems efficiently and accurately. This foundation supports not just calculations but logical reasoning needed to deduce solutions in broader contexts.

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Most popular questions from this chapter

You should use the following definition of an outlier: An outlier is any data value that is above the third quartile by more than 1.5 times the IQR [Outlier \(\left.>Q_{3}+1.5(I Q R)\right]\) or below the first quartile by more than 1.5 times the IQR [Outlier \(\left.

Table \(15-17\) shows the percentage of U.S. workers who are members of unions \((2000-2011)\). (a) Draw a pictogram for the data in Table 15-17. Assume you are trying to convince your audience that unions are holding their own and that the percentage of union members in the workforce is steady. (b) Draw a different pictogram for the data in Table \(15-17\) where you are trying to convince your audience that there is a steep decline in union membership in the U.S. workforce. $$ \begin{array}{l|r|r|r|r|r|r} \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Percent } & 13.4 & 13.3 & 13.3 & 12.9 & 12.5 & 12.5 \\ \hline \text { Year } & 2006 & 2007 & 2008 & 2009 & 2010 & 2011 \\ \hline \text { Percent } & 12.0 & 12.1 & 12.4 & 12.3 & 11.9 & 11.8 \end{array} $$

Refer to the data set shown in Table \(15-12 .\) The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each. $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } & \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } \\ \hline 1362 & 50 & 4315 & 70 \\ \hline 1486 & 70 & 4719 & 70 \\ \hline 1721 & 80 & 4951 & 60 \\ \hline 1932 & 60 & 5321 & 60 \\ \hline 2489 & 70 & 5872 & 100 \\ \hline 2766 & 10 & 6433 & 50 \\ \hline 2877 & 80 & 6921 & 50 \\ \hline 2964 & 60 & 8317 & 70 \\ \hline 3217 & 70 & 8854 & 100 \\ \hline 3588 & 80 & 8964 & 80 \\ \hline 3780 & 80 & 9158 & 60 \\ \hline 3921 & 60 & 9347 & 60 \\ \hline 4107 & 40 & & \end{array} $$ (a) Make a frequency table for the Chem 103 test scores. (b) Draw a bar graph for the data in Table \(15-12\).

Refer to histograms with unequal class intervals. When sketching such histograms, the columns must be drawn so that the frequencies or percentages are proportional to the area of the column. Figure \(15-21\) illustrates this. If the column over class interval 1 represents \(10 \%\) of the population, then the column over class interval 2 , also representing \(10 \%\) of the population, must be one-third as high, because the class interval is three times as large (Fig. \(15-21\) ). Two hundred senior citizens are tested for fitness and rated on their times on a one-mile walk. These ratings and associated frequencies are given in Table 15-23. Draw a histogram for these data based on the categories defined by the ratings in the table. $$ \begin{array}{l|c|c} \text { Time } & \text { Rating } & \text { Frequency } \\ \hline 6^{+} \text {to } 10 \text { minutes } & \text { Fast } & 10 \\ \hline 10^{+} \text {to } 16 \text { minutes } & \text { Fit } & 90 \\ \hline 16^{+} \text {to } 24 \text { minutes } & \text { Average } & 80 \\ \hline 24^{+} \text {to } 40 \text { minutes } & \text { Slow } & 20 \end{array} $$

A realty company has sold \(N=341\) homes in the last year. The five-number summary for the sale prices is \(\operatorname{Min}=\$ 97,000, \quad Q_{1}=\$ 115,000, \quad M=\$ 143,000, \quad Q_{3}=\) \(\$ 156,000,\) and \(\operatorname{Max}=\$ 249,000 .\) (a) Find the interquartile range of the home sale prices. (b) How many homes sold for a price between \(\$ 115,000\) and \(\$ 156,000\) (inclusive)? (Note: If you don't believe that you have enough information to give an exact answer, you should give the answer in the form of "at
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