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Math - Puzzle

by Emone Vinson
(Houston, Tx)










































Phillip read 7 books over the summer. Below is a list of the total pages in 6 books he read.

245 120 189 200 205 189

If the mean number of pages in the books that Phillip read was 192, how many pages long was the seventh book?

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Dec 29, 2011
Arithmetic Mean
by: Staff


Question:

by Emone Vinson
(Houston, Tx)


Phillip read 7 books over the summer. Below is a list of the total pages in 6 books he read.

245 120 189 200 205 189

If the mean number of pages in the books that Phillip read was 192, how many pages long was the seventh book?


Answer:

The arithmetic mean of a set of numbers is the central point (or central tendency) of a set (or group) of numbers. The arithmetic mean is also commonly called the “Mean”, or “Average”. These terms are often used interchangeably.

I am deliberately being somewhat formal by using the term arithmetic mean, rather than mean. This is to distinguish the arithmetic mean other computations such as the geometric mean, or harmonic mean.

The arithmetic mean is a concept which is familiar to everyone. For example, “on average” how many hours do you sleep each night? . . . or “on average” what are your grades this year? . . . or “on average” how fast can you drive on the freeway during rush hour traffic? . . . and so on.


The arithmetic mean is a ratio of two amounts: the grand total of all the numbers in the set, and the numerical count of how many different numbers were added to get the grand total.

For this particular problem, the arithmetic mean is computed by adding up the number of pages in all the books, and then dividing that total by 7 (since there are 7 books).

Note that you do not know the number of pages in the last book. This value is represented by the variable x.


Arithmetic Mean (average)

(245 + 120 + 189 + 200 + 205+ 189 + x)/7 = 192

(1148 + x)/7 = 192


Solve for x


Multiply each side of the equation by 7. This will remove the 7 from the denominator of the fraction on the left side of the equation.

[(1148 + x)/7]*7 = 192*7

(1148 + x)*(7/7) = 192*7

(1148 + x)*(1) = 192*7

(1148 + x) = 192*7

1148 + x = 192*7

1148 + x = 1344


Subtract 1148 from each side of the equation. This will remove the 1148 from the left side of the equation.

1148 + x - 1148 = 1344 - 1148

1148 - 1148 + x = 1344 - 1148

0 + x = 1344 - 1148

x = 1344 - 1148

x = 196



the final answer is: the seventh book is 196 pages long


check the answer by substituting 196 for x in the original equation
Mean (average)

(245 + 120 + 189 + 200 + 205+ 189 + 196)/7 = 192

(1148 + 196)/7 = 192

192 = 192, OK → x = 196 is a VALID solution



Thanks for writing.


Staff
www.solving-math-problems.com


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