document.write( "Question 845926: I have these two problems. Can anyone help please?\r
\n" ); document.write( "\n" ); document.write( "6 5/9 - 9 3/4 + 7 1/3 - 1 5/6\r
\n" ); document.write( "\n" ); document.write( "3 3/5 x 1 5/9
\n" ); document.write( " 2/10\r
\n" ); document.write( "\n" ); document.write( "Thank you \n" ); document.write( "
Algebra.Com's Answer #509497 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
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Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
\n" ); document.write( " \"+system%28+%0D%0A++++1%5Cx+%2B+1%5Cy+=+2%2C%0D%0A++++1%5Cx+%2B+-1%5Cy+=+0+%29%0D%0A++\"We'll use substitution. After moving 1*y to the right, we get:
\n" ); document.write( " \"1%2Ax+=+2+-+1%2Ay\", or \"x+=+2%2F1+-+1%2Ay%2F1\". Substitute that
\n" ); document.write( " into another equation:
\n" ); document.write( " \"1%2A%282%2F1+-+1%2Ay%2F1%29+%2B+-1%5Cy+=+0\" and simplify: So, we know that y=1. Since \"x+=+2%2F1+-+1%2Ay%2F1\", x=1.
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\n" ); document.write( " Answer: \"system%28+x=1%2C+y=1+%29\".
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\n" ); document.write( "I am not sure that your problems are displayed correctly, but I will make my best guess at what you meant.
\n" ); document.write( "Not knowing your age, your grade, or how much math you have already learned, I do not know how much explanation you need.
\n" ); document.write( "Just in case, I will explain a lot, so it may get slow and boring.
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\n" ); document.write( "To calculate \"6%265%2F9-9%263%2F4%2B7%261%2F3-1%265%2F6\" ,
\n" ); document.write( "the first step would be to convert those mixed numbers into improper fractions.
\n" ); document.write( "\"6%265%2F9=6%2B5%2F9=54%2F9%2B5%2F9=59%2F9\" so the mixed number \"6%265%2F9\" can be written as the improper fraction \"59%2F9\" .
\n" ); document.write( "Doing the same with the other numbers,we get
\n" ); document.write( "\"6%265%2F9-9%263%2F4%2B7%261%2F3-1%265%2F6=59%2F9-39%2F4%2B22%2F3-11%2F6\"
\n" ); document.write( "Now the problem is that all the fractions to be added have different denominators.
\n" ); document.write( "We need a common denominator, one that is a multiple of all the denominators in those fractions.
\n" ); document.write( "We need a number that is a multiple of \"9\" , \"4\" , \"3\" and \"6\" .
\n" ); document.write( "The smallest one we can find is \"36=4%2A9\".
\n" ); document.write( "The multiples of 9 are 9, 18, 27, 36, 45, 54, and so on.
\n" ); document.write( "Of the multiples of \"9\" ,
\n" ); document.write( "\"9\" is a multiple of \"3\" , but not a multiple of the other numbers;
\n" ); document.write( "\"18\" is a multiple of \"3\" and \"6\" , but not a multiple of \"4\" ,
\n" ); document.write( "and \"27\" is not a multiple of \"4\" or \"6\" either.
\n" ); document.write( "The next smallest multiple of \"9\" is \"36\" , and it is a multiple of \"9\" , \"4\" , \"3\" and \"6\" .
\n" ); document.write( "That is the least common multiple of \"9\" , \"4\" , \"3\" and \"6\" .
\n" ); document.write( "So we want to convert all those fractions to fractions with \"36\" for a denominator.
\n" ); document.write( "To convert a fraction into an equivalent fraction, we have to multiply the numerator and denominator times the same number.
\n" ); document.write( "Multiplying the numerator and denominator of \"59%2F9\" times \"4\" we get
\n" ); document.write( "\"59%2F9=59%2A4%2F%289%2A4%29=236%2F36\" .
\n" ); document.write( "Multiplying the numerator and denominator of \"39%2F4\" times \"9\" we get
\n" ); document.write( "\"39%2F4=39%2A9%2F%284%2A9%29=351%2F36\" .
\n" ); document.write( "Multiplying the numerator and denominator of \"22%2F3\" times \"12\" we get
\n" ); document.write( "\"22%2F3=22%2A12%2F%283%2A12%29=264%2F36\" .
\n" ); document.write( "Multiplying the numerator and denominator of \"11%2F6\" times 6 we get
\n" ); document.write( "\"11%2F6=11%2A6%2F%286%2A6%29=66%2F36\" .
\n" ); document.write( "So far we have
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\n" ); document.write( "Now it is just a question of adding/subtracting the numerators to get
\n" ); document.write( "\"%28236-351%2B264-66%29%2F36=83%2F36\"
\n" ); document.write( "The way I calculate \"236-351%2B264-66\" has 3 steps:
\n" ); document.write( "1) I add together all the numbers with a + sign in front, and add together all the numbers with a - sign in front,
\n" ); document.write( "2) I figure out which total wins, to use the sign of the winning total, and
\n" ); document.write( "3) I calculate the difference of the totals.
\n" ); document.write( "I could write that as
\n" ); document.write( "\"236-351%2B264-66=%28236%2B264%29-%28351%2B66%29=500-417=83\"
\n" ); document.write( "Putting all of the above together, I would just write
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\n" ); document.write( "(The rest of my calculations would be in my head, or on the margin of the paper, or on separate scrap paper).
\n" ); document.write( "\"83%2F26\" cannot be simplified into an equivalent fraction with a denominator smaller than \"36\",
\n" ); document.write( "so the last step I could do is converting that \"83%2F36\" improper fraction into the mixed number \"2%2611%2F36\" .
\n" ); document.write( "I would do that in my head (or pretend I did), but my calculation would be
\n" ); document.write( "that \"83\" divided by \"36\" is \"2\" with a remainder of \"11\" ,
\n" ); document.write( "because \"36\" times \"2\" is \"72\" ,
\n" ); document.write( "and \"83\" is \"83-72=11\" more than \"72\" ,
\n" ); document.write( "so \"83%2F36=%2872%2B11%29%2F36=72%2F36%2B11%2F36=2%2B11%2F36=2%2611%2F36\" .
\n" ); document.write( "On my paper, I would only write
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\n" ); document.write( "although a lot of calculating went into that.
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\n" ); document.write( "I believe your second problem is \"3%263%2F5\" x \"1%265%2F9\" divided by \"2%2F10\" .
\n" ); document.write( "I would write that as
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\n" ); document.write( "To begin with, \"2%2F10=1%2F5\" . It is strange that it was given as \"2%2F10\" ,
\n" ); document.write( "because \"2%2F10\" is the same as \"1%2F5\" and \"1%2F5\" is the simplest way to write that number.
\n" ); document.write( "You must also know that dividing by a fraction is the same as multiplying times its reciprocal (the fraction turned upside down), so dividing by \"1%2F5\" is the same as multiplying by the fraction \"5%2F1\" , which is a fancy way of writing \"5\".
\n" ); document.write( "The final \"18%2A14%2A5%2F%285%2A9%29=28\" calculation can be done with a calculator,
\n" ); document.write( "but it is easy to do in my head if I simplify (cancel out) factors that appear in numerator and denominator.
\n" ); document.write( "In my head I see it like this:
\n" ); document.write( " .
\n" ); document.write( "Some teachers do not like to see cross-out cancelling,
\n" ); document.write( "because sometimes students go cross-out crazy,
\n" ); document.write( "and they cross out numbers here and there without rhyme or reason,
\n" ); document.write( "ending with a result that is not equal to what they started with.
\n" ); document.write( "They think that it should be written as
\n" ); document.write( " ,
\n" ); document.write( "That shows that the two \"5\"s cancel out because they can be rearranged into a factor that equals \"1\" , and multiplying times \"1\" does not change anything. \n" ); document.write( "
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