Hi guys, I was given a worksheet over the weekend from my Algebra 2 teacher, and I'm having trouble on solving the rational equations. They are all word problems, and I'm having trouble setting up the equation. Can you guys teach me how I'm supposed to start these? Thanks!
1. Mary takes twice as long as Carole to make the display case at school. Together it takes 4 hours to do the display case. How long does it take Mary to do it herself?
2.The numerator of a fraction is 4 more than twice the denominator. If 2 is subtracted from both the numerator and denominator, the result is 8. Find the original fraction.
Answers: 1. 12 hours 2. 10/3.
There are 12 questions in total, but im hoping that after I learn how to do these I can learn how to do the rest. Thanks alot guys!
1. Mary takes twice as long as Carole to make the display case at school. Together it takes 4 hours to do the display case. How long does it take Mary to do it herself?
2.The numerator of a fraction is 4 more than twice the denominator. If 2 is subtracted from both the numerator and denominator, the result is 8. Find the original fraction.
Answers: 1. 12 hours 2. 10/3.
There are 12 questions in total, but im hoping that after I learn how to do these I can learn how to do the rest. Thanks alot guys!
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1. Mary takes twice as long as Carole to make the display case at school. Together it takes 4 hours to do the display case. How long does it take Mary to do it herself?
Since it takes Mary twice as long as Carole, Mary's speed is half the speed of Carol. Or better yet, let's say that Carol's speed is twice the speed of Mary. Call "m" the speed of Mary and "c" the speed of Carol. Then:
4m + 4c = 1
But c = 2m, so:
4m + 4(2m) = 1 -----> 4m + 8m = 1 -----> 12m = 1 -----> m = 1 / 12
So Mary's speed is 1/12 the display case per hour. How long will it take her to make the display case?
1/12 --------1 hour
1 ----------- x hours
x = 1 / (1/12) = 12 hours. That's your answer. Ideally, to get started, the problem would say, for example, that someone takes a certain amount of hours to paint 8 rooms. The problem you presented is awkward for starters because it's about making a display case, so one tends to represent this amount of work as merely 1. It's harder to grasp that notion. Let me just add, for clarity, that in those 4 hours the amount of work done by Mary was 4*(1/12) = 1/3 of the total work, and since Carol's speed is twice the speed of Mary, then in those 4 hours the amount of work done by Carol is 4*(2/12) = 2/3 of the total work, so adding 1/3 + 2/3 we get 1 (complete job).
2.The numerator of a fraction is 4 more than twice the denominator. If 2 is subtracted from both the numerator and denominator, the result is 8. Find the original fraction.
We can say that the numerator is 2x + 4 and the denominator is x.
Then:
[(2x + 4) - 2] / [x - 2] = 8
(2x + 2) / (x - 2) = 8
2x + 2 = 8(x - 2)
2x + 2 = 8x - 16
6x = 18
x = 18/6
x = 3
Therefore, since the numerator is 2x + 4 and the denominator is x, the original fraction is:
(2x + 4) / x = (2*3 + 4) / 3 = (6 + 4) / 3 = 10 / 3
That's your answer.
I hope this helps...
Since it takes Mary twice as long as Carole, Mary's speed is half the speed of Carol. Or better yet, let's say that Carol's speed is twice the speed of Mary. Call "m" the speed of Mary and "c" the speed of Carol. Then:
4m + 4c = 1
But c = 2m, so:
4m + 4(2m) = 1 -----> 4m + 8m = 1 -----> 12m = 1 -----> m = 1 / 12
So Mary's speed is 1/12 the display case per hour. How long will it take her to make the display case?
1/12 --------1 hour
1 ----------- x hours
x = 1 / (1/12) = 12 hours. That's your answer. Ideally, to get started, the problem would say, for example, that someone takes a certain amount of hours to paint 8 rooms. The problem you presented is awkward for starters because it's about making a display case, so one tends to represent this amount of work as merely 1. It's harder to grasp that notion. Let me just add, for clarity, that in those 4 hours the amount of work done by Mary was 4*(1/12) = 1/3 of the total work, and since Carol's speed is twice the speed of Mary, then in those 4 hours the amount of work done by Carol is 4*(2/12) = 2/3 of the total work, so adding 1/3 + 2/3 we get 1 (complete job).
2.The numerator of a fraction is 4 more than twice the denominator. If 2 is subtracted from both the numerator and denominator, the result is 8. Find the original fraction.
We can say that the numerator is 2x + 4 and the denominator is x.
Then:
[(2x + 4) - 2] / [x - 2] = 8
(2x + 2) / (x - 2) = 8
2x + 2 = 8(x - 2)
2x + 2 = 8x - 16
6x = 18
x = 18/6
x = 3
Therefore, since the numerator is 2x + 4 and the denominator is x, the original fraction is:
(2x + 4) / x = (2*3 + 4) / 3 = (6 + 4) / 3 = 10 / 3
That's your answer.
I hope this helps...