a Use the d=rt formula. How many miles did Lucita travel at the faster pace?
B
b Use the d=rt formula. At what pace did Lucita travel the second part of the distance?
C
c Use the expressions from Part A and Part B.
A
a 50/x
B
b 50/x-15
C
c 100x-750/x(x-15)
Practice makes perfect
a Let x represent the faster pace in miles per hour. We know that Lucita traveled half the distance at this rate. Since the distance is 100 miles, she traveled 50 miles at rate x. Let's recall the formula for the distance d depending on the rate r and the time t.
d=rtWe want to find the time spent at the faster pace. Therefore, we will solve this formula for t.
Lucita traveled the distance of 50 miles at the faster rate, x. Let's substitute these values into the formula above to find the time t_1 spent at the faster pace.
t_1=50/x
b We will use the formula from Part A for the time t depending on the distance d and the rate r.
t=d/r
Since Lucita traveled 50 miles at the faster rate, she must have traveled 50 miles at the slower rate as well. We know that the slower rate is 15 less than the faster rate x. Therefore, the slower rate is x-15. We can substitute these values into the formula to find the time t_2 spent at the slower pace.
t_2=50/x-15
c We want to write an expression for the amount of time Lucita needs to complete the trip. In Part A we have found the time t_1 she spent at the faster pace. In Part B we have found the time t_2 she spent at the slower pace.
t_1 = 50/x t_2 = 50/x-15
The total time t of the trip is the time spent at the faster pace plus the time spent at the slower pace.
t=t_1+t_2=50/x+50/x-15
To add these two rational expressions, they must have a common denominator. Let's find their least common denominator (LCD)! Since both of the denominators are binomials, they are in their simplest form. Therefore, the LCD is a product of these two binomials.
Least Common Denominator: x(x-15)
Now we can expand the fractions so that their denominators are equal to the LCD.
