For example, the problem might ask, “If Tommy can paint a room in 3 hours, and Winnie can paint the same room in 4 hours, how long will it take them to paint the room together?
For example, if Tommy can paint a room in 3 hours, his hourly rate is 13{\displaystyle {\frac {1}{3}}}; that is, each hour he completes 13{\displaystyle {\frac {1}{3}}} of a room. If Winnie takes 4 hours to paint a room, her hourly rate is 14{\displaystyle {\frac {1}{4}}}.
For example, if Tommy paints 13{\displaystyle {\frac {1}{3}}} of a room in 1 hour, Winnie paints 14{\displaystyle {\frac {1}{4}}} of a room in 1 hour, and together they complete 1t{\displaystyle {\frac {1}{t}}} of a room in 1 hour, the equation will be:13+14=1t{\displaystyle {\frac {1}{3}}+{\frac {1}{4}}={\frac {1}{t}}}.
For example, 12 is the least common denominator of 13{\displaystyle {\frac {1}{3}}} and 14{\displaystyle {\frac {1}{4}}}, thus:13+14=1t{\displaystyle {\frac {1}{3}}+{\frac {1}{4}}={\frac {1}{t}}}412+312=1t{\displaystyle {\frac {4}{12}}+{\frac {3}{12}}={\frac {1}{t}}}712=1t{\displaystyle {\frac {7}{12}}={\frac {1}{t}}}
For example:712=1t{\displaystyle {\frac {7}{12}}={\frac {1}{t}}}7t=12{\displaystyle 7t=12}t=127{\displaystyle t={\frac {12}{7}}}
For example, if Tommy takes 3 hours to paint a room, and Winnie takes 4 hours to complete a room, together they can complete a room in 127{\displaystyle {\frac {12}{7}}}, or 157{\displaystyle 1{\frac {5}{7}}} of an hour. This equals almost two hours (about 1 hour, 43 minutes).
For example, the problem might ask, “If a hose can fill a pool 6 hours, and an open drain can empty it in 2 hours, how long will it take the open drain to empty the pool with the hose on?”
For example, if a drain can empty a pool in 2 hours, and you need to calculate how long it takes to empty the pool, then the drain is completing the job. Its hourly rate is 12{\displaystyle {\frac {1}{2}}}; that is, each hour it empties12{\displaystyle {\frac {1}{2}}} of the pool.
For example, if the hose can fill a pool in 3 hours, but the goal is to empty the pool, then the hose is undoing the job. If the hose fills the pool in 6 hours, Its hourly rate is 16{\displaystyle {\frac {1}{6}}}; that is, each hour it fills 16{\displaystyle {\frac {1}{6}}} of the pool.
For example, if a drain empties 12{\displaystyle {\frac {1}{2}}} of a pool in 1 hour, a hose fills 16{\displaystyle {\frac {1}{6}}} of a pool in 1 hour, and together they empty 1t{\displaystyle {\frac {1}{t}}} of a pool in 1 hour, the equation will be:12−16=1t{\displaystyle {\frac {1}{2}}-{\frac {1}{6}}={\frac {1}{t}}}.
For example, 6 is the least common denominator of 12{\displaystyle {\frac {1}{2}}} and 16{\displaystyle {\frac {1}{6}}}, thus:12−16=1t{\displaystyle {\frac {1}{2}}-{\frac {1}{6}}={\frac {1}{t}}}36−16=1t{\displaystyle {\frac {3}{6}}-{\frac {1}{6}}={\frac {1}{t}}}26=1t{\displaystyle {\frac {2}{6}}={\frac {1}{t}}}
For example:26=1t{\displaystyle {\frac {2}{6}}={\frac {1}{t}}}2t=6{\displaystyle 2t=6}2t=6{\displaystyle 2t=6}t=62{\displaystyle t={\frac {6}{2}}}
For example, if a hose fills a pool in 6 hours, and a drain empties the pool in 2 hours to, working against each other, the pool will drain in 62{\displaystyle {\frac {6}{2}}} hours, or 3{\displaystyle 3} hours.
For example, the problem might be: “Damarion can clean the cat shelter in 8 hours, and Cassandra can clean the shelter in 4 hours. They work together for 2 hours, but then Cassandra leaves to take some cats to the vet. How long will it take for Damarion to finish cleaning the shelter on his own?”
For example, if Damarion can clean the cat shelter in 8 hours, his hourly rate is 18{\displaystyle {\frac {1}{8}}}; that is, each hour he completes 18{\displaystyle {\frac {1}{8}}} of a room. If Cassandra takes 4 hours to clean the shelter, her hourly rate is 14{\displaystyle {\frac {1}{4}}}.
For example, if Damarion cleans 18{\displaystyle {\frac {1}{8}}} of the room in an hour, and Cassandra completes 14{\displaystyle {\frac {1}{4}}} of the room an hour, together they will complete 18+14{\displaystyle {\frac {1}{8}}+{\frac {1}{4}}} of the room in an hour:18+14{\displaystyle {\frac {1}{8}}+{\frac {1}{4}}}=216+416{\displaystyle ={\frac {2}{16}}+{\frac {4}{16}}}=616{\displaystyle ={\frac {6}{16}}}
For example, if Damarion and Cassandra together clean 616{\displaystyle {\frac {6}{16}}} of the shelter in 1 hour, in two hours they complete twice that much:616×2{\displaystyle {\frac {6}{16}}\times 2}=1216{\displaystyle ={\frac {12}{16}}}=34{\displaystyle ={\frac {3}{4}}} of the shelter
For example, if Damarion and Cassandra cleaned 34{\displaystyle {\frac {3}{4}}} of the shelter in 2 hours, then after Cassandra leaves, Damarion has to clean 14{\displaystyle {\frac {1}{4}}} of the shelter on his own.
For example, if Damarion cleans the shelter at a rate of 18{\displaystyle {\frac {1}{8}}} per hour, and he needs to complete 14{\displaystyle {\frac {1}{4}}} of the job on his own, you equation will be 18h=14{\displaystyle {\frac {1}{8}}h={\frac {1}{4}}}, or, more simply, h8=14{\displaystyle {\frac {h}{8}}={\frac {1}{4}}}
For example:h8=14{\displaystyle {\frac {h}{8}}={\frac {1}{4}}}4h=8{\displaystyle 4h=8}h=2{\displaystyle h=2}So, it will take Damarion 2 hours to complete the job on his own.
