document.write( "Question 1036364: Find three consecutive even integers such that four times the first equals 16 times the sum of the third and the number two.\r
\n" ); document.write( "\n" ); document.write( "Thank you! I simply don't know where to start or what to do :) \n" ); document.write( "
Algebra.Com's Answer #651016 by KMST(5328)\"\" \"About 
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THE ALGEBRA WAY:
\n" ); document.write( "Start by calling the three consecutive even integers
\n" ); document.write( "\"x\" , \"x%2B2\" , and \"x%2B4\" , or
\n" ); document.write( "\"x-2\" , \"x\" , and \"x%2B2\" .
\n" ); document.write( "Then, find expressions for:
\n" ); document.write( "\"four times the first\" =\"%22___%22\" ,
\n" ); document.write( "\"the sum of the third and the number two\" =\"%22___+%2B+2%22\" ,
\n" ); document.write( "and \"16 times\" that sum = \"16%2A%28%22___+%2B+2%22%29\" .
\n" ); document.write( "Finally write an equation that says that
\n" ); document.write( "the expression for \"four times the first\"
\n" ); document.write( "equals the expression for
\n" ); document.write( "\"16 times the sum of the third and the number two\".
\n" ); document.write( "When you solve that equation,
\n" ); document.write( "you should find an \"x\" that is an even integer.
\n" ); document.write( "If the \"x\" you find is not an even integer,
\n" ); document.write( "either you made a mistake in your calculations,
\n" ); document.write( "or there is no solution to the problem.
\n" ); document.write( "
\n" ); document.write( "THE FIFTH GRADER ANSWER:
\n" ); document.write( "The problem says \"integers\", meaning \"whole numbers\",
\n" ); document.write( "so \"0\" ,
\n" ); document.write( "and negative numbers, such as \"-8\" , could be involved.
\n" ); document.write( "Let us use some educated guess and check,
\n" ); document.write( "picking a first set of \"three consecutive even integers,
\n" ); document.write( "and then a second set of \"three consecutive even integer\";
\n" ); document.write( "seeing how far away \"four times the first\" is from
\n" ); document.write( "\"16 times the sum of the third and the number two\",
\n" ); document.write( "and deciding which way to move to choose the next set.
\n" ); document.write( "First try: 2, 4, and 6. With that choice,
\n" ); document.write( "\"four times the first\" = \"4%2A2=8\" ,
\n" ); document.write( "\"16 times the sum of the third and the number two\" =\"16%2A%286%2B2%29=16%2A8=128\" .
\n" ); document.write( "The two expressions are \"128-8=120\" apart.
\n" ); document.write( "Second try: 8, 10, and 12. With that choice,
\n" ); document.write( "\"four times the first\" = \"4%2A8=32\" ,
\n" ); document.write( "\"16 times the sum of the third and the number two\" =\"6%2A%2812%2B2%29=16%2A14=224\" .
\n" ); document.write( "The two expressions are \"224-32=192\" apart.
\n" ); document.write( "Since choosing the even integers so that they were more than 2, 4, and 6 made the difference greater (worse),
\n" ); document.write( "let us pick integers that are less than 2, 4, and 6.
\n" ); document.write( "How about 0, 2,and 4? With that choice,
\n" ); document.write( "\"four times the first\" = \"4%2A0=0\" ,
\n" ); document.write( "\"16 times the sum of the third and the number two\" =\"16%2A%284%2B2%29=16%2A6=96\" .
\n" ); document.write( "The two expressions are \"96-0=96\" apart.
\n" ); document.write( "The different got smaller, so we are getting warmer
\n" ); document.write( "How about -4, -2,and 0? With that choice,
\n" ); document.write( "\"four times the first\" = \"4%2A%28-4%29=-16\" ,
\n" ); document.write( "\"16 times the sum of the third and the number two\" =\"16%2A%280%2B2%29=16%2A2=32\" .
\n" ); document.write( "The two expressions are \"32-%28-16%29=48\" apart, and
\n" ); document.write( "\"16 times the sum of the third and the number two\"
\n" ); document.write( "is still more that \"four times the first\".
\n" ); document.write( "How about -8, -6,and -4? With that choice,
\n" ); document.write( "\"four times the first\" = \"4%2A%28-8%29=-32\" ,
\n" ); document.write( "\"16 times the sum of the third and the number two\" =\"16%2A%28-4%2B2%29=16%2A%28-2%29=-32\" ,
\n" ); document.write( "and we have hit the jackpot:
\n" ); document.write( "the solution is \"highlight%28-8%29\" , \"highlight%28-6%29\" , and \"highlight%28-4%29\" .
\n" ); document.write( "
\n" ); document.write( "NOTE 1:
\n" ); document.write( "When looking for consecutive even integers, or consecutive odd integers,
\n" ); document.write( "you do not need to get complicated with the expressions for each integer.
\n" ); document.write( "Just call one of those integers \"x\" or \"n\" ,
\n" ); document.write( "and the others have to be \"2\" apart, as in
\n" ); document.write( "\"n-2\" , \"n\" , and \"n%2B2\" .
\n" ); document.write( "You will either find the kind of integer (even or odd) that you need for the answer,
\n" ); document.write( "or you will find something unexpected, and will have to figure out if it is your mistake, or a tricky problem without a valid solution.
\n" ); document.write( "
\n" ); document.write( "NOTE 2:
\n" ); document.write( "The fifth grader probably does not know it, but the fifth grader's strategy works only because both expressions are simple linear expressions.
\n" ); document.write( "If there was a \"squared\" in one of those expressions, it may not work.
\n" ); document.write( "Anyway, the fifth grader got to the solution fairly easily, didn't she? \n" ); document.write( "
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