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How Do You Solve a Word Problem with a Rational Equation?

The local high school is holding a car wash to raise funds for varsity sports. Bill can wash and wax one car in 3 hours. Jason can wash and wax one car in 4 hours. If Bill and Jason work together, how long will it take them to wash and wax one car?

Summary

  1. If time is 't', then 't/3' is the number of cars Bill can wash and wax in 't' hours, and 't/4' is the number of cars Jason can wash and wax in 't' hours
  2. Since we're combining their efforts to finish 1 car, we add 't/3' and 't/4', setting them equal to 1
  3. 't/3+t/4=1' is a rational equation
  4. The smallest number that the denominators '3' and '4' are both factors of is '12'
  5. Multiplying by the LCD will help us to get 't' by itself
  6. The time it takes Bill and Jason to wash and wax one car is '12/7' hours, but if we convert this to the mixed fraction '1 and 5/7', it's easier to understand

Notes

    1. We know the time it takes for each to wash and wax one car on their own
    2. Bill can wash and wax a car in 3 hours
    3. Jason can wash and wax a car in 4 hours
    1. We're looking for the time it takes Bill and Jason to wash and wax one car together
    1. So, 't' is the variable we're trying to solve for, and represents the 'time' it takes the two boys to wash and wax a car together
    1. Remember, 'time' will be represented by the variable 't'
    1. Remember, 't' is the variable we're trying to solve for, and represents the 'time' it takes the two boys to wash and wax a car together
    1. For Bill alone, "1 car = 3 hrs"
    1. For Bill alone, "1 car = 3 hrs"
    2. If we want to see what Bill can accomplish in one hour, divide by 3 to get: "1/3 car = 1 hr"
    1. For Bill alone, "1 car = 3 hrs"
    2. If we want to see what Bill can accomplish in one hour, divide by 3 to get: "1/3 car = 1 hr"
    3. So for any amount of time, 't', 't/3' will represent the number of cars Bill can wash and wax in 't' hours
    1. For Jason alone, "1 car = 4 hrs"
    1. For Jason alone, "1 car = 4 hrs"
    2. If we want to see what Jason can accomplish in one hour, divide by 4 to get: "1/4 car = 1 hr"
    1. For Jason alone, "1 car = 4 hrs"
    2. If we want to see what Jason can accomplish in one hour, divide by 4 to get: "1/4 car = 1 hr"
    3. So for any amount of time, 't', 't/4' will represent the number of cars Jason can wash and wax in 't' hours
    1. Since we don't know 't', we'll use 't/3' for Bill and 't/4' for Jason
    2. Combining them just means adding them together
    1. Since we don't know 't', we'll use 't/3' for Bill and 't/4' for Jason
    2. Combining them just means adding them together
    3. t/3 + t/4 = 1
    1. t/3 + t/4 = 1
    2. This is a rational equation
    1. Our equation is: 't/3 + t/4 = 1'
    2. This is a rational equation
    3. Remember, 't' is the variable we're trying to solve for, and represents the 'time' it takes the two boys to wash and wax a car together
    1. Our equation is: 't/3 + t/4 = 1'
    2. This is a rational equation
    1. Our equation is: 't/3 + t/4 = 1'
    2. This is a rational equation
    3. Our two fractions are 't/3' and 't/4'
    4. '3' and '4' are the denominators of our fractions
    5. The smallest number that '3' and '4' are both factors of is '12'
    1. Our equation is: 't/3 + t/4 = 1'
    2. This is a rational equation
    1. Up to this point, our equation looks like: '12(t/3 + t/4) = 12(1)'
    2. 12(t/3 + t/4) = 12(t/3) + 12(t/4) = 4t + 3t
    3. 12(1) = 12
    1. Up to this point, our equation looks like: '4t + 3t = 12'
    2. Remember, 't' is the variable we're trying to solve for, and represents the 'time' it takes the two boys to wash and wax a car together
    1. Up to this point, our equation looks like: '4t + 3t = 12'
    2. '4t' and '3t' are like terms!
    3. Combining them just means adding them together: 4t+3t=7t
    1. Up to this point, our equation looks like: '7t = 12'
    2. If we divide one side by '7', we must divide the other side by '7'
    3. 7t/7 = t
    1. Remember, 't' is the variable we're trying to solve for, and represents the 'time' it takes the two boys to wash and wax a car together
    2. We found 't' to equal '12/7 hours'
    1. '12/7' is not very easy to understand
    2. In this case, a mixed fraction form of '12/7' makes more sense
    1. '12/7' can be expressed as '1 and 5/7'
    2. Since we're dealing with hours, "one and five-sevenths hours" sounds better than "twelve-sevenths hours"