How to Solve Combined Labor Problems

Step-by-Step Guide

1. 1

   Step 1: Read the problem carefully.

   Use this method if the problem represents two or more people working together to complete a job.
   
   The problem should also give you the amount of time it would take each person to complete the job alone.
   
   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?
2. 2

   Step 2: Determine the hourly rate of each individual.

   The hourly rate is represented by creating a fraction, where the total number of hours it takes to complete the job is the denominator (bottom number), and 1 is the numerator (top number).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}}}. , This will be 1t{\displaystyle {\frac {1}{t}}}, where t{\displaystyle t} equals the amount of time it takes them to complete the job together., Because they are working together, their combined hourly rate will equal the sum of their individual hourly rates.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}}}. , You will need to find the least common denominator.
   
   For complete instructions on how to add fractions, you can read the article Add Fractions.
   
   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}}} , To do this, cross multiply.In this instance, you can also simply take the inverse of the fraction.For example:712=1t{\displaystyle {\frac {7}{12}}={\frac {1}{t}}}7t=12{\displaystyle 7t=12}t=127{\displaystyle t={\frac {12}{7}}} , This will give you the number of hours it takes for the workers to complete the job together.
   
   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).
3. 3

   Step 3: Create a ratio for their combined hourly rate.

4. 4

   Step 4: Set up the equation.

5. 5

   Step 5: Add the fractions together.

6. 6

   Step 6: Solve for t{\displaystyle t}.

7. 7

   Step 7: Simplify the fraction

8. 8

   Step 8: if necessary.

Detailed Guide

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