document.write( "Question 893076: In a first race, A gives b a Headstart of 100 feet and overtakes him in four minutes. In the second race he gains 750 feet on B when B runs 9000 feet. Find the rate at which each runs. \n" ); document.write( "

Algebra.Com's Answer #541042 by josmiceli(19441) $\"\"$ $\"About$

You can put this solution on YOUR website!
Call the runners A and B
\n" ); document.write( "Let their speeds be $\"+a+\"$ and $\"+b+\"$ in feet / minute
\n" ); document.write( "-----------------------
\n" ); document.write( "Start a stopwatch when A leaves
\n" ); document.write( "In the 1st race:
\n" ); document.write( "Let $\"+d+\"$ = the distance A runs when he overtakes B
\n" ); document.write( "A's equation:
\n" ); document.write( "(1) $\"+d+=+a%2A4+\"$
\n" ); document.write( "B's equation:
\n" ); document.write( "(2) $\"+d+-+100+=+b%2A4+\"$
\n" ); document.write( "================
\n" ); document.write( "In the 2nd race:
\n" ); document.write( "I think they both leave at the same time, so
\n" ); document.write( "Let $\"+t+\"$ = time in minutes until A is $\"+750+\"$ ft
\n" ); document.write( "ahead of B
\n" ); document.write( "------------------
\n" ); document.write( "A's equation:
\n" ); document.write( "(3) $\"+9000+%2B+750+=+a%2At+\"$
\n" ); document.write( "B's equation:
\n" ); document.write( "(4) $\"+9000=+b%2At+\"$
\n" ); document.write( "-------------------
\n" ); document.write( "This is actually 4 equations with 4 unknowns,
\n" ); document.write( "so it is solvable
\n" ); document.write( "Substitute (1) into (2)
\n" ); document.write( "(2) $\"+a%2A4+-+100+=+b%2A4+\"$
\n" ); document.write( "(2) $\"+4a+-+4b+=+100+\"$
\n" ); document.write( "(2) $\"+a+-+b+=+25+\"$
\n" ); document.write( "(2) $\"+a+=+b+%2B+25+\"$
\n" ); document.write( "-----------------------
\n" ); document.write( "Substitute (4) into (3)
\n" ); document.write( "(4) $\"+t+=+9000%2Fb+\"$
\n" ); document.write( "(3) $\"+9000+%2B+750+=+a%2A%28+9000%2Fb+%29+\"$
\n" ); document.write( "(3) $\"+9750+=++a%2A%28+9000%2Fb+%29+\"$
\n" ); document.write( "(3) $\"+9750b+=+9000a+\"$
\n" ); document.write( "Divide both sides by $\"+750+\"$
\n" ); document.write( "(3) $\"+13b+=+12a+\"$
\n" ); document.write( "Substitute (2) int (3)
\n" ); document.write( "(3) $\"+13b+=+12%2A%28+b+%2B+25+%29+\"$
\n" ); document.write( "(3) $\"+13b+=+12b+%2B+300+\"$
\n" ); document.write( "(3) $\"+b+=+300+\"$
\n" ); document.write( "and, since
\n" ); document.write( "(2) $\"+a+=+b+%2B+25+\"$
\n" ); document.write( "(2) $\"+a+=+325+\"$
\n" ); document.write( "-----------------
\n" ); document.write( "A's speed is 325 ft/min
\n" ); document.write( "B's speed is 300 ft/min
\n" ); document.write( "---------------------
\n" ); document.write( "check:
\n" ); document.write( "(1) $\"+d+=+a%2A4+\"$
\n" ); document.write( "(1) $\"+d+=+325%2A4+\"$
\n" ); document.write( "(1) $\"+d+=+1300+\"$ ft
\n" ); document.write( "and
\n" ); document.write( "(2) $\"+d+-+100+=+b%2A4+\"$
\n" ); document.write( "(2) $\"+d+-+100+=+300%2A4+\"$
\n" ); document.write( "(2) $\"+d+=+1200+%2B+100+\"$
\n" ); document.write( "(2) $\"+d+=+1300+\"$ ft
\n" ); document.write( "OK
\n" ); document.write( "You can check (3) and (4)\r
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