document.write( "Question 153913: Please help me solve this problems :
\n" ); document.write( "1. On Dave's birthday, his brother Harry is 17 yrs younger than 3 times his age. The boy's father , Tom is 12 yrs. older than twice Harry's age. If Dave is 7 years younger than his brother , how many candles are on Dave's cake?\r
\n" ); document.write( "\n" ); document.write( "2. When Alvin divides each of 5 consecutive integers by his age, the sum of the 5 remainders he gets is 32. When Susan , several yrs. older, divides each of a different set of 5 consecutive integers by her age, the sum of the five remainders she gets is also 32. What is the sum of the ages of Alvin and Susan? \n" ); document.write( "

Algebra.Com's Answer #113351 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
Edwin's complete solution:\r
\n" ); document.write( "\n" ); document.write( "Please help me solve this problems :
\n" ); document.write( "1. On Dave's birthday, his brother Harry is 17 yrs younger than 3 times his age. The boy's father , Tom is 12 yrs. older than twice Harry's age. If Dave is 7 years younger than his brother , how many candles are on Dave's cake? \r
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( " Please help me solve this problems :
\n" ); document.write( "
\n" ); document.write( ">>...his brother Harry is 17 yrs younger than 3 times his age...<<
\n" ); document.write( "

 \r\n" );
document.write( "H = 3D - 17\r\n" );
document.write( "

\n" ); document.write( ">>...The boy's father , Tom is 12 yrs. older than twice Harry's age...<<
\n" ); document.write( "

 \r\n" );
document.write( "T = 2H + 12\r\n" );
document.write( "

\n" ); document.write( ">>...Dave is 7 years younger than his brother...<<
\n" ); document.write( "

 \r\n" );
document.write( "D = H - 7\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "-3D +  H     = -17\r\n" );
document.write( "    - 2H + T =  12\r\n" );
document.write( "  D -  H     =  -7\r\n" );
document.write( " \r\n" );
document.write( "Solve that and get D=12, H=19, T=50\r\n" );
document.write( "\r\n" );
document.write( "----------------------\r\n" );
document.write( "

\n" ); document.write( "2. When Alvin divides each of 5 consecutive integers by his age, the sum of the 5 remainders he gets is 32. When Susan , several yrs. older, divides each of a different set of 5 consecutive integers by her age, the sum of the five remainders she gets is also 32. What is the sum of the ages of Alvin and Susan?
\n" ); document.write( "

\r\n" );
document.write( "Let either one of their ages be A.\r\n" );
document.write( "Let the 5 consecutive integers be I1,I2,I3,I4,I5\r\n" );
document.write( "Let the remainders when these integers are divided\r\n" );
document.write( "by A be R1,R2,R3,R4,R5\r\n" );
document.write( "\r\n" );
document.write( "If none of the consecutive integers is A or\r\n" );
document.write( "or a multiple of A, then the 5 remainders will \r\n" );
document.write( "themselves be 5 consecutive integers.\r\n" );
document.write( "\r\n" );
document.write( "Let's see if there exist 5 consecutive integers \r\n" );
document.write( "which the remainders could be, so as to have the sum 32.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "No, that is not possible, for n doesn't come out\r\n" );
document.write( "to an integer.\r\n" );
document.write( "\r\n" );
document.write( "That means one of 5 consectutive integers must\r\n" );
document.write( "be a multiple of A, say kA\r\n" );
document.write( "\r\n" );
document.write( "Whichever one of the 5 is kA, The remainder left when\r\n" );
document.write( "dividing by A will of course be 0.  \r\n" );
document.write( "\r\n" );
document.write( "Any of the 5 remainders of members of the 5 that are \r\n" );
document.write( "less than kA, will be consecutive integers, the largest \r\n" );
document.write( "of which must be A-1, then the one below that will be\r\n" );
document.write( "A-2, and so on.\r\n" );
document.write( "\r\n" );
document.write( "Any of the 5 remainders that are greater than the multiple\r\n" );
document.write( "of A will be consecutive integers 1,2,3, and so on\r\n" );
document.write( "\r\n" );
document.write( "Then we have 5 cases to consider, that is, when kA comes\r\n" );
document.write( "1st, 2nd, 3rd, 4th, and 5th.\r\n" );
document.write( "\r\n" );
document.write( "         I1    I2    I3    I4    I5     R1   R2   R3   R4  R5   Sum  \r\n" );
document.write( "case 1.  kA   kA+1  kA+2  kA+3  kA+4     0    1    2    3   4    10\r\n" );
document.write( "case 2. kA-1   kA   kA+1  kA+2  kA+3    A-1   0    1    2   3   A+5\r\n" );
document.write( "case 3. kA-2  kA-1   kA   kA+1  kA+2    A-2  A-1   0    1   2    2A\r\n" );
document.write( "case 4. kA-3  kA-2  kA-1   kA   kA+1    A-3  A-2  A-1   0   1  3A-5    \r\n" );
document.write( "case 5. kA-4  kA-3  kA-2  kA-1   kA     A-4  A-3  A-2  A-1  0  4A-10\r\n" );
document.write( " \r\n" );
document.write( "We set each of those sums on the far right equal to 32 and see which,\r\n" );
document.write( "if any, have an integer solution:\r\n" );
document.write( "\r\n" );
document.write( "Case 1. This is out because 10 does not equal 32.\r\n" );
document.write( "\r\n" );
document.write( "Case 2.  A+5 = 32\r\n" );
document.write( "           A = 27\r\n" );
document.write( "\r\n" );
document.write( "That's a whole number, so one of their ages can be 27.\r\n" );
document.write( "\r\n" );
document.write( "Case 3.  2A = 32 \r\n" );
document.write( "          A = 16\r\n" );
document.write( "\r\n" );
document.write( "That's a whole number, so one of their ages can be 16.\r\n" );
document.write( "\r\n" );
document.write( "Case 4. 3A-5 = 32\r\n" );
document.write( "        2A   = 37\r\n" );
document.write( "         A   = 27/2\r\n" );
document.write( "\r\n" );
document.write( "That is not a whole number, so Case 4 is out.\r\n" );
document.write( "\r\n" );
document.write( "Case 5. 4A-10= 32\r\n" );
document.write( "        4A   = 42  \r\n" );
document.write( "         A   = 21/2\r\n" );
document.write( "\r\n" );
document.write( "That is not a whole number, so Case 5 is out.\r\n" );
document.write( "\r\n" );
document.write( "So Alvin is 16 and Susan is 27, making the sum of their ages 43.\r\n" );
document.write( "\r\n" );
document.write( "Now there is no way to tell whether Susan's 5 consecutive\r\n" );
document.write( "integers were \r\n" );
document.write( "\r\n" );
document.write( "26,27,28,29,30, or 53,54,55,56,57, or 80,81,82,83,84 or etc.\r\n" );
document.write( "\r\n" );
document.write( "There is also no way to tell whether Alvin's 5 consecutive\r\n" );
document.write( "integers were \r\n" );
document.write( "\r\n" );
document.write( "14,15,16,17,18, or 30,31,32,33,34, or 46,47,48,49,50 or etc.\r\n" );
document.write( " \r\n" );
document.write( "However he must be 16 and she must be 27.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );