document.write( "Question 564756: The camera has 10.1 million pixels. The product of the pixels across the screen times the pixels up the screen gives the number of pixels the camera has. The ratio of pixels across the screen to pixels up the screen is 4 : 3. How many pixels does this camera have across the screen? Explain what you are calculating as you solve the problem. \n" ); document.write( "

Algebra.Com's Answer #852271 by ikleyn(52879) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "The camera has 10.1 million pixels. The product of the pixels across the screen times the pixels up the screen
\n" ); document.write( "gives the number of pixels the camera has. The ratio of pixels across the screen to pixels up the screen is 4 : 3.
\n" ); document.write( "How many pixels does this camera have across the screen? Explain what you are calculating as you solve the problem.
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\n" ); document.write( "\n" ); document.write( " The \" solution \" by @Theo is FATALLY WRONG in 39 lines.\r
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\n" ); document.write( "\n" ); document.write( " I came to bring a correct normal solution to this simple elementary problem.\r
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document.write( "Let 4x be the number of pixels across the screen.\r\n" );
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document.write( "Then 3x is the number of pixels up the screen.\r\n" );
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document.write( "The total number of pixels in the screen is then  (4x)*(3x) = 12x^2.\r\n" );
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document.write( "So, we have this equation\r\n" );
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document.write( "    12x^2 = 10.1 million,  or  12x^2 = 10,100,000.\r\n" );
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document.write( "Find x from this equation\r\n" );
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document.write( "    x =  = 917.4239296.\r\n" );
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document.write( "The number of pixels across the screen is about  4x = 4*917.4239296 = 3669.696.\r\n" );
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document.write( "Round to the closest integer  3670.\r\n" );
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document.write( "Answer.  The number of pixel across the screen is about 3670. \r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "////////// Below is my comment to the Edwin solution //////////\r
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\n" ); document.write( "\n" ); document.write( "This morning, I saw Edwin's solution after my solution, which I submitted yesterday.
\n" ); document.write( "On the one hand, I was glad that Edwin's solution was close to mine.
\n" ); document.write( "From the other side, some aspects in Edwin' solution seem strange to me.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, I'd like to discuss Edwin' solution, in order for to create clarity in the reader's mind.\r
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\n" ); document.write( "\n" ); document.write( "First of all, throughout the post, Edwin mistakenly and consistently writes 1010000 for 10.1 million.
\n" ); document.write( "It is a typo - it should be read as 10100000 at every appearance.
\n" ); document.write( "Edwin should fix this typo, so as not to mislead a reader.\r
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\n" ); document.write( "\n" ); document.write( "Secondly, while solving the problem, Edwin uses an approach, which is not standard.\r
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\n" ); document.write( "\n" ); document.write( "He introduces the system of equation, which is correct and quite elementary, but then solves it using
\n" ); document.write( "online solver, bypassing standard Algebra methods.\r
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\n" ); document.write( "\n" ); document.write( "Yes, we live in a free country, and everybody is free to apply a method he wants.
\n" ); document.write( "But in teaching Algebra, for such standard elementary system of equations we usually use other simple
\n" ); document.write( "algebraic methods (solving by hands) that are standard in teaching.
\n" ); document.write( "Why I speak about it - because this site is for teaching Algebra - not for
\n" ); document.write( "tutors to demonstrate their superiority. So, traditional Algebra methods
\n" ); document.write( "are assumed and should be presented first, as I did in my solution.\r
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\n" ); document.write( "\n" ); document.write( "Thirdly, Edwin insists that the number 3669.7 should be rounded to 3669, largest whole number
\n" ); document.write( "that does not exceed 3669.\r
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\n" ); document.write( "\n" ); document.write( "This Edwin argument is incorrect, since the standard rounding rules directly dictate to round 3669.7 to 3670.\r
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\n" ); document.write( "\n" ); document.write( "In whole, I am very pleasant that Edwin practically confirmed my solution (3670 pixels
\n" ); document.write( "across the camera against incorrect millions pixels in the post by @Theo - it is why
\n" ); document.write( "I developed and presented my solution here).\r
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\n" ); document.write( "\n" ); document.write( "Thank you, Edwin, for it.\r
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\n" ); document.write( "\n" ); document.write( "But Edwin performed his intention somewhat awkwardly, so I hasten to correct Edwin's mistakes
\n" ); document.write( "to support the right concepts in the reader's mind.\r
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\n" ); document.write( "\n" ); document.write( "Actually, in this problem, it does not matter, which answer to accept, 3670 or 3669.\r
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\n" ); document.write( "\n" ); document.write( "Since the number $\"10100000%2F12\"$ is not a perfect square, we can not expect to get an exact integer value.
\n" ); document.write( "With necessity, we will get some irrational number between two consecutive integer numbers.\r
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\n" ); document.write( "\n" ); document.write( "Having two possibilities, 3670 and 3669, the better what we can say is that the answer is close to 3670 pixels
\n" ); document.write( "across the camera - precisely as I did in my solution.\r
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\n" ); document.write( "\n" ); document.write( "What is really important, is to apply an adequate approach/method/technique,
\n" ); document.write( "be accurate and provide a solution in the frame of a school curriculum,
\n" ); document.write( "or as close as possible, in order for the solution be educative.\r
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\n" ); document.write( "\n" ); document.write( "I would never start debating for such insignificant issue as the difference
\n" ); document.write( "between 3670 and 3669 pixels, but Edwin first begun to question my solution.
\n" ); document.write( "Thus, I have an obligation to clarify.\r
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\n" ); document.write( "\n" ); document.write( "\\\\\\\\\ Edwin, it is my response to your response as mccravyedwin /////////\r
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\n" ); document.write( "\n" ); document.write( "In his response, Edwin writes (I copy-paste)\r
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document.write( "Ikleyn ignores the fact that there are exceptions to the 'rounding off' rule,:\r\n" );
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document.write( "\"Round up if first digit you're dropping is 5 or more\" rule, otherwise round\r\n" );
document.write( "down\". \r\n" );
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document.write( "One exception would be that if the question of a problem asks the number of\r\n" );
document.write( "people that can fit into a room, and your calculation gives 11.9, the answer\r\n" );
document.write( "would be 11 rather than 12.\r\n" );
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document.write( "The exceptions to that rule are in word problems dealing with things that we\r\n" );
document.write( "cannot have fractions of, such as people or pixels.  We must have whole \r\n" );
document.write( "numbers of people and pixels.\r\n" );
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document.write( "Therefore, in this problem we must be conservative and round down the number of\r\n" );
document.write( "pixels, for there is not enough room on the camera to squeeze in one whole extra\r\n" );
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document.write( "Answer: 3669, not 3670.\r\n" );
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document.write( "Also, Ikleyn frowns on the use of online technology. I see nothing wrong with it\r\n" );
document.write( "in solving word problems, for the skill in a word problem is translating English\r\n" );
document.write( "sentences to algebra sentences (aka equations). That is, setting up the\r\n" );
document.write( "equations, not solving them.\r\n" );
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document.write( "Edwin

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\n" ); document.write( "\n" ); document.write( "- - - - - - - - Below is my (iKleyn) reasoning - - - - - - - -\r
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\n" ); document.write( "\n" ); document.write( "It is really interesting point, deserving a discussion.\r
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document.write( "So, in my solution I got x = 917.4239296, 4x = 4*917.4239296 = 3669.696, and I rounded it to 3670,\r\n" );
document.write( "                                               following standard rounding rules.\r\n" );
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document.write( "                                          3x = 3*917.4239296 = 2752.272, and I rounded it to 2752,\r\n" );
document.write( "                                               following standard rounding rules.\r\n" );
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document.write( "Now the product (4x)*(3x) is  3670*2752 = 10099840, and it is less than 10.1 million.\r\n" );
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document.write( "So,  prevents me from claiming that 3670 is the  number of pixels across the camera.\r\n" );
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document.write( "On contrary, if I take the number 3669, then the product  (4x)*(3x)  will be 3669*2752 = 10097088,  \r\n" );
document.write( "which means that I loose many (2752) pixels for nothing (due to using wrong conception).\r\n" );
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\n" ); document.write( "\n" ); document.write( "This analysis shows that 3670 is $\"highlight%28highlight%28BETTER%29%29\"$ answer than 3669.\r
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\n" ); document.write( "\n" ); document.write( "Then we account for greater number of pixels, still being within 10.1 million of pixels.\r
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\n" ); document.write( "\n" ); document.write( "Thus, you see that the pair (x,y) = (3660,2752) is BETTER than the pair (x,y) = (3669,2752).\r
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\n" ); document.write( "\n" ); document.write( "For completeness, let's consider the pair (3659,2753).   It produces the product\r
\n" ); document.write( "\n" ); document.write( "3659*2753 = 10100757, which is greater than 10.1 million of pixels, and therefore does not work in this problem.\r
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\n" ); document.write( "\n" ); document.write( "So, this analysis shows that the pair (3670,2752) provides the BEST possible answer in this problem.\r
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\n" ); document.write( "\n" ); document.write( "ANSWER.   3670 is THE $\"highlight%28highlight%28UNIQUE%29%29\"$ possible number of pixels across the camera in this problem,\r
\n" ); document.write( "\n" ); document.write( "                   on contrary to Edwin's INCORRECT answer 3669.\r
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