5x + 4x = 20 9x = 20 x = 20/9Hints for solving the problems(8.4)
Rational Expressions
Problem No. 1 - Working along it would take Josie 4 hours to paint a room. It would take Curtis 5 hours to paint the same room by himself. How long would it take them to paint the room if they work together?
Remember work problems uses the idea that rate x time equals the work done.
Josie 4 hr. rate is ¼ per hour
Curtis 5 hr. rate is 1/5 per hour
together x hr. The rate is always the reciprocal of the time it takes to do the job.
(1/4)(x) + (1/5)(x) = 1 The time together to do the job (x) is multiplied times the rate of each person and added together, and the whole job is done (100% = 1)
5x + 4x = 20 9x = 20 x = 20/9
This is the answer, but it should be put into an improper fraction 2 2/9 hrs.
Problem No. 12 - In a certain cookie recipe, the ratio of cups of flour to cups of sugar is 3 to 1. If the recipe uses 2 1/4 cups of flour, how much sugar does it use?
This is a ratio problem and a proportion must be set up.
cross multiply 3x = 9/4
Divide both sides of the equation by 3 x = 9/12 x = ¾ So ¾ cup of sugar is the
answer.
Problem No. 17 - Leroy rows a boat as fast as Sasha Rows a boat. If Leroy rows for 30 minutes, he travels 1 mile farther than Sasha when she rows for 20 minutes. How far does each row?
This is a motion problem and a chart is helpful. (r x t = d)
The formula for these problems is rate x time equals the distance. (shown above)
r t d
Leroy x 30 30x distance Leroy rows
Sasha x 20 20x distance Sasha rows
The problem says Leroy travels 1 mile farther than Sasha, so to set up an equation you add 1 to Sasha’s distance.
30x = 20x + 1
10x = 1
x = 1/10
This shows the rate of each person is 1/10 mile per second.
But the problem asks for the distance each has traveled.
So you have to multiply 1/10 times each time.
Leroy (1/10)(30) = 3 miles
Sasha (1/10)(20) = 2 miles
Problem No. 19 - Ranji and Paula spend the same amount of time driving to work. Ranji averages 60 miles per hour and Paula averages 40 miles per hour. Ranji drives 15 miles farther than Paula. How far does each drive to work?
r t d
Ranji 60 x 60x
Paula 40 x 40x
Ranji drives 15 miles farther than Paula, so to set up the equation you have to add 15 to Paula’s distance to make the equation equal.
60x = 40x + 15
20x = 15
x = 15/20 = ¾
So the time driving to work for both is ¾ of an hour. The problem asks for the distance driven to work, so you have to multiply the rate times the time.
Ranji 60(3/4) = 45 miles
Paula 40(3/4) = 30 miles The answers to the problem.
The following problems are either direct or indirect variations.
direct – y varies directly as x:
indirect – y varies indirectly (inversely) as x:
Notice the difference, in a direct variation it is 1 to 2 = 1 to 2 and in an indirect variation it is 1 to 2 = 2 to 1.
Problem No. 23 - A person’s weight on the moon varies directly as the person’s weight on Earth. A person weighing 144 pounds on Earth weights only 24 pounds on the moon. How much does a person weigh on Earth who weighs 30 pounds on the moon?
This problem is a direct variation, so: 
m is the moon’s weight and e is the earth’s weight.
Substitute the numbers into the proportion: 
cross
multiply
(24)(x) =(30)(144) 24x = 4320 x = 180 lbs. the ans.
Problem No. 26 - The time it takes a car to travel a fixed distance varies inversely with the rate at which it travels. It takes the car 4 hours to travel a fixed distance when it travels at a rate of 50 mules per hour. How fast does the car have to travel to cover the same distance in 2 ½ hours?
The variation equation: 
t = time, r = rate
substitute number into the
equation
cross multiply
(4)(50) =
200 =
400 = 5x
80 = x 80 mph is the ans.
Don’t let these thought problems get you down. Try them and go on and review the topics for the next Exam. Try to understand a little about each of these types of problems.