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Solving Rational Equations
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Solving Rational Equations
On a tour of the Caribbean, Grandpa Lindbergh had to make an emergency landing, leaving his tour group on a deserted island.
Being a resourceful group, they know they need a few things to survive the first night: shelter, a fire and food plus the knowledge of Solving rational equations to distribute the work efficiently. Luckily, they crashed in the morning, so they have a little while to work. However, time’s-a-tickin’ and they’re not sure if they’ll be able to finish all the tasks before sundown. Mike volunteers to find food on the island, General Good collects leaves for the roof, leaving Jasmine and Grandpa to collect wood for the fire.
Setting up a rational equation
After a while, Jasmine comes back with 10 pieces of wood, Grandpa returns 10 minutes later with 15 pieces, but they’re not quite sure how long they were off collecting wood. We call the time it took for Jasmine to collect 10 pieces of wood ‘x’. Since Grandpa Lindbergh came back 10 minutes later than Jasmine, we can say that Grandpa took ‘x’ + 10 minutes to gather 15 pieces of wood.
Since they collected wood at the same rate, they can set up a rational equation to help them determine the time it took them to complete the task.
Solving a rational equation
To solve this problem, we can use a method called cross-multiplication.
We simply have to multiply the numerators on one side of the equal sign by the denominators on the other side of the equal sign.
Like this.Then we use PEMDAS and opposite operations to isolate the variable and solve. Distribute the 10 giving us 10x + 100, substract 10x from both sides of the equation and finally divide both sides by 5 to give us our answer.
So, Jasmin collected 10 pieces of wood in 20 minutes which averages out to one piece every two minutes. Grandpa Lindbergh collected wood at the same rate. He gathered 15 pieces in 30 minutes which is also one piece every two minutes
To make sure they have enough for the night, Jasmine goes off to collect more wood.
Setting up the rational equation
In the meantime, General Good comes back from gathering leaves, which Grandpa and General Good weave together for a roof so the group has some shelter to protect them from the elements. In order to finish the shelter before sundown, Grandpa and General Good need to weave together 80 leaves in 3 hours, which is the same as180 min.
Solving the rational expression
Although they can weave leaves at the same unknown rate of 1 over ‘x’, Grandpa needs a bit of a breather after every leaf, changing his rate to 1 over x plus 6.
How long do they have to weave each leaf?
To solve this rational equation, we have to find a common denominator first because we have to sum the numbers on the right side of the equation.
To find the common denominator, we multiply the numerator and denominator of the first fraction by the denomiator of the second fraction, (x+6), and vice versa.
Next, we have to use the Distributive Property, giving us ‘x’ plus 6 divided by the quantitiy x squared plus 6x plus ‘x’ divded by quantity x squared plus 6x.
Now that we have common denominators, we can simply add the numerators.
From here, we can cross-multiply, as we did in the previous example. Just use the Distributive Property again. All terms are divisible by 40, so we divide both sides by 40.
Next, we bring the equation into standard form and simplify by combining like terms.
For our last step, we can factor the equation, leaving us with (2x+9) times (x-3).
To solve for ‘x’, we use the Zero Product Property and set both factors equal to zero. Finally, we can solve for ‘x’ using opposite operations.
Negative time?!? Remember, when solving word problems with rational expressions, you not only have to plug your answers to check your work but you also have to think about the solution and whether or not it makes sense.
So, General Good has to weave at least 1 leaf every 3 minutes, whereas Grandpa Lindbergh has to weave 1 leaf every 9 minutes in order to finish the roof on time.
Together, they can weave 4 leaves every 9 minutes.
Everyone’s glad that they were all able to finish the tasks before sundown, but they’re famished!
By the way – Where’s Mike? More importantly, where’s the food?
Wait, where did Mike get fast food? Maybe they should have looked around the island a bit before trying to play survivor.
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Rational Expressions (4 Videos)
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