Day: June 3, 2018

Week 16 – Application of Rational Expressions

keishan2015

This week in pre-calculus 11, our rational expressions unit came to an end. One thing I learned from that unit was the application of rational expressions in real life situations.

Things to know:

  • Word problems are filled with a lot of valuable information that is easy to look past one you read it onece.t is very important to read a word problem more than once, making sure to read slowly and carefully. It helps when you highlight or underline the important information in the question. Be sure to take note of key words such as less than, greater than, equal to, the difference of, the sum of, the product of, etc.
  • Due to the fact that it is a word problem, make sure that you write your final answer in a sentence always making sure to include the right units if necessary.
  • You should always check to make sure that your solution ‘makes sense’.

Example 1:

Q: Jake rows a boat 6km upstream in the same time that it takes the boat to travel 12km downstream. The average speed of the current is 5km/hr. What is the average speed of the boat in still water?

Based on the word problem we know the following information:

*Let x = the speed of the boat in still water

  Downstream Upstream
Distance 12 km 6 km
Speed X + 5 X – 5
Time \frac{12}{x+5} \frac{6}{x-5}

*we know that because the boat is moving up stream the speed of the boat will be a bit slow considering it is working against the current, hence the -5. We know that the speed of the boat going down stream will be a bit faster considering that it is moving with the current, hence the +5. We also know that time is equal to when distance is divided by speed.

After reading the word problem over again, we know that the two times are equal. Due to this information we can now write an equation.

  • \frac{12}{x+5}=\frac{6}{x-5} –> cross multiply (fraction = fraction)
  • 12(x-5) = 6(x+5) –> distribute the coefficient in front of the brackets
  • 12x-60 = 6x + 30 –> Re-arrange so that the variable is on one side
  • 6x = 90 –> isolate x by dividing both sides by 6
  • \frac{6x}{6}=\frac{90}{6} –>Divide
  • x = 15 –>Rewrite as a sentence
  • The average speed of the boat in still water is 15km/hr.

Example 2:

Q: How much sugar should be added to 6L of water to make a solution that is 20% sugar?

We know:

  • \frac{part}{total}
  • Total = sugar + water
  • Part = sugar
  • 20% sugar –> \frac{20}{100} –> \frac{1}{5}
  • x = amount of sugar needed

Equation based on information:

  • \frac{part}{total} = \frac{1}{5}
  • \frac{1}{5}=\frac{x}{6+x} –> cross multiply
  • 6+x = 5x –> Move variable to one side
  • 6 = 4x –> Divide each side by 4 to isolate the variable
  • x = 1.5 –> Write answer in a sentence
  • 1.5L of sugar is needed to make the solution 20% sugar.