Math Help! Test Tomorrow!?
Please help me with these questions:
1. Marta and Laird hiked the Snowy Mt. Trail, hiking at 2.4 mph on the way up and at 3.6 mph on the same trial on the way down. The entire hike took 5 hours. What was the distance that they hiked?
2.It takes Mia twice the time that it takes Kerry to trim some hedges. If together they can trim the hedges in 3 hours, how long would it take Kerry to trim the hedges by herself?
3.Stewart and Felicia rode their ATVs on her uncle's land. Felicia started out 15 minutes before Stewart, at and average speed of 24 miles per hour. Stewart averaged 30 miles per hour. How long did it take Stewart to catch up to Felicia?
Please show me step by step and not just say the answer. I need to know how you get the answer. I would really appreciate it. Thank You!
- jsjsLv 510 years agoBest Answer
1. Let d be the distance of the hike in miles (one direction), so that 2d is the answer. Then on the way up, they hiked for d/2.4 hours. On the way down, they hiked for d/3.6 hours. Since the entire hike took 5 hours, we have 5 = d/2.4 + d/3.6. Now put the fractions over a common denominator and add:
5 = 10d/24 + 10d/36
5 = 30d/72 + 20d/72 = 50d/72
d = 5 * 72/50 = 7.2.
Therefore, the hike is 7.2 miles in one direction, so they hiked 14.4 miles total.
Alternate solution: Since their speed was always greater than or equal to 2.4 mph, and they walked 5 hours total, they must have walked at least 2.4 * 5 = 12 miles. 14.4 miles is the only answer that is greater than 12 miles, so it must be the right one.
2. Since Mia takes twice the time that Kerry does, we know that in 3 hours, Mia trims half as much of the hedge as Kerry. Since together they trim the whole hedge, Mia must have trimmed 1/3 of it, and Kerry trimmed 2/3. (More formally, this comes from solving the system of equations M+K=1, K=2M.) Since Kerry trims at a rate of
(2/3 of the hedges) / (3 hours) = 2/9 of the hedges per hour, it would take her 9/2 = 4.5 hours to trim all the hedges.
Alternate solution: If it takes her 3 hours with help, it will obviously take her more than 3 hours without help, so you can rule out answers a and b. (Kerry+Mia) is less than fast than (Kerry+another copy of Kerry), since Mia is not as fast as Kerry. Therefore, Kerry+Mia take more than time than 2 Kerry's, so they take more than half as much time as 1 Kerry. Put another way, the time Kerry spends by herself, must be less than twice the time spent by Kerry+Mia. Since Kerry+Mia spend 3 hours, Kerry spends less than 6 hours by herself, which rules out answer d. The only remaining answer is c. (Sorry I couldn't find a simpler way to explain that last part, but it's worth making sure you understand why it works -- once you get it, it's not nearly as complicated as I made it sound.)
3. Since Felicia got a quarter-hour head start at 24 mph, she got a 6 mile head start. After that, Stewart was going 6 mph faster than her, so you might as well imagine that Felicia was staying still and Stewart was going 6 mph. (Their actual speeds don't affect when they meet, only their relative speeds. Make sure you understand this, too.) If Stewart were going at 6 mph, it would take him one hour to make up Felicia's 6 mile head start, so the answer is 1 hour.
Alternate solution: If you don't like my "relative speed" thing, you can do it this way. Let t denote the time after Stewart starts, in hours. Then the total distance traveled by Stewart is 30t, and the total distance traveled by Felicia is 6 + 24t. Now solve
6 + 24t = 30t. You get t=1 hour.