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Applications of Rational Equations
Shoveling Snow
After a heavy snowfall, Brian can shovel all of the driveway in 30 minutes. If his younger brother Allen helps, the job takes only 20 minutes. How long would it take Allen to do the job by himself?
To solve the problem, answer each of the questions:
If it takes Brian 30 minutes to shovel the entire driveway, which fraction of the driveway would be shoveled in 20 minutes?
Brian shovels EMBED Equation.3 of driveway in 20 minutes. (1)
Let x be the number of minutes it would take Allen to shovel the entire driveway by himself. Which fraction of the driveway (in terms of x) would Allen shovel in 20 minutes?
Allen shovels EMBED Equation.3 of driveway in 20 minutes. (2)
When Allen helps Brian to shovel all of the driveway, the sum of Brians fraction (1) and Allens fraction (2) equals 1. In other terms, we have:
EMBED Equation.3 + EMBED Equation.3 = 1 (3)
Solve equation (3) to find x
x = _______
Mixing water and antifreeze
How much pure antifreeze liquid should be added to 1 gallon of a 40% antifreeze solution to get a 60% antifreeze concentration?
Note. There is 0.4 gallons of pure antifreeze in 1 gallon of 40% antifreeze solution.
To solve the problem, answer each of the questions:
We add x gallons of pure antifreeze to 1 gallon of 40% antifreeze solution.
The total volume of pure antifreeze in the new solution is
______ + _____ = _____________ gallons (1)
The total volume of liquid in the new solution is
______ + _____ = _____________ gallons (2)
To satisfy the volume of 60% concentration, we divide the amount found in (1) by the amount found in (2) and we obtain the equation
EMBED Equation.3 (3)
Solve equation (3) to find x.
x = _________
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