Math
Homework Problem- Kady can complete a homework assignment 3 minutes faster than Fran. When they work together they complete the job in 2 minutes. How long does it take Kady to do the assignment on her own?
Step 1: To begin this problem we have to identify the variables. We are trying to find out how many minutes Kady takes to finish an assignment and we can't know that without knowing how long it takes Fran, so we set Fran equal to x. Since Kady can complete the assignment 3 minutes FASTER than Fran, we know that she completes the job in less time and can set her equal to x-3.
To complete the first step, all we have to do is set each variable up as a ration, or in this case, fraction: for Fran, 1 job: x; for Kady, 1 job: x-3; and together, they would be 1 job: 2 minutes. Now all theres left to do is to set these ratios up as an equation. |
Step 2: For step two, we can begin adding the fractions together, but we cannot do that until we find what's called a common denominator, meaning that the bottom numbers of the fraction have to be the same. To do this, we can multiply the two denominators by each other (make sure that what ever you multiply the bottom of the fraction by you multiply by the top as well). We can start by multiplying both the top and bottom of 1/x by x-3 and then multiply the top and bottom of 1/(x-3) by x. This, in the end, gives us (x-3) / [x(x-3)] + x / [x(x-3)]= 1/2.
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Step 4: Now the outcome of step 3 left us with two fractions equal to each other. To solve for x, we have to make all the numbers in our equation whole numbers, meaning that the numbers are not fractions, decimals or percentages. To achieve only whole numbers we can use a method called cross multiplying, where we multiply the opposite denominators and numerators by each other. Fran likes us to say, "Float like a butterfly, sting like a bee" because of the butterfly shape you can make when encircling the numbers you're multiplying together.
The equation we're left with after cross multiplication is 2(2x-3) = x(x-3). |
Step 7: Now that we've got the equation in standard form, we can factor out numbers in the equation. To factor means you find two binomials (an expression with two variables such as (x+5) or (3x-8)) that when multiplied together, get you your equation. In this case we end up with 0 = (x-1) (x-6). We know these binomials work because when you foil them out (multiplying two binomials) you get x^2 - 6x - x + 6 or x^2 - 7x + 6.
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Step 8: Now that we have our binomials, we can finally solve for x. In this case, we will have two possible solutions for x: 1 or 6. To solve for x, we can set both of our binomials equal to zero so we get 0 = x-1 and 0 = x-6. Now we solve for x in both equations and get x = 1 and x = 6 which means that the amount of time it takes Fran to finish the assignment could be either 1 minute OR 6 minutes.
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Step 9: Finally, now that we have Fran's time, we can figure out how many minutes it takes Kady to finish the assignment. However, since we have two answers, we have to test both to be certain of Kady's time. For example, if it takes Fran 1 minute to finish the assignment, that means it takes Kady 1 - 3 minutes which equals -2 minutes and is NOT POSSIBLE. But, if it takes Fran 6 minutes then it takes Kady 6 - 3 minutes which equals 3 minutes.
So we can conclude that it takes Kady 3 MINUTES to complete the assignment. |
Reflection
The Struggle: The issues I had with this math problem did not include completion of the math involved, but more of why and how the numbers and methods connected to the words themselves. I mean, that’s a long process for something as simple as “3 minutes”. In the end, I learned that there can be more than one answer to a math problem, like when x can equal both 1 AND 6. It's really up to you using critical thinking and methods of deciphering to really understand the outcome of the problem.
The Success: The success is that despite my confusion, I actually managed to complete the problem and remember the process was supposed to work through and get the right answer in the process. So even though I didn’t know why I was doing what I was doing, I could still complete the process. And the process is only half the battle. Once I understood what I was supposed to do, it was easier to compare and connect the problem to why I was supposed to do it.
Real World Context: Well, instead of using Kady and Fran and math problems, I could use this problem to determine how long it would take to finish multiple jobs, such as painting a fence or typing a two person essay. Anything that involves team work. However, feel that what I should really be focusing on is not the equation itself, but rather how to apply numbers and equations to real world situations such as average times. The senior math teacher gave me a math packet the other day full of word problems that sometimes make my brain hurt. I found that I am actually able to sit there and find numbers that fit to words that I wasn't able to before.
The Success: The success is that despite my confusion, I actually managed to complete the problem and remember the process was supposed to work through and get the right answer in the process. So even though I didn’t know why I was doing what I was doing, I could still complete the process. And the process is only half the battle. Once I understood what I was supposed to do, it was easier to compare and connect the problem to why I was supposed to do it.
Real World Context: Well, instead of using Kady and Fran and math problems, I could use this problem to determine how long it would take to finish multiple jobs, such as painting a fence or typing a two person essay. Anything that involves team work. However, feel that what I should really be focusing on is not the equation itself, but rather how to apply numbers and equations to real world situations such as average times. The senior math teacher gave me a math packet the other day full of word problems that sometimes make my brain hurt. I found that I am actually able to sit there and find numbers that fit to words that I wasn't able to before.
Step Two: For step two we can simplify (to combined like terms and simplify the amount of variables in your problem. ex: a + 6 + 2a + 3 = 3a + 9) each expression, starting with finding common denominators (explained above) for the fractions that we are adding and subtracting and then expand on the numbers.
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Step Three: Now it's time to bring both expressions together again. Because the fractions are being divided, we can use the Keep Change Flip rule to get rid of the Super Fraction. (the Keep Change Flip rule indicates that we have to keep the first fraction as is, change the division sign into a multiplication sign, and then flip the second fraction into its reciprocal)
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Reflection
The Struggle: the difficulties I had with this problem was not that I couldn't understand how to solve it. On the contrary, I could understand it just fine. I was the length and amount of numbers that messed me up. The fact that it was more of a test of mental perseverance plus skill made it a tricky question, in my book. In the end, this problem taught me to be patient with my math so that I don't make the little mistakes I did, again.
The Strength: I found the strength to push through the problem for one. It was also a good test of basic math skills considering the large account of simplification required. Even though I messed up on one multiplication portion (initially setting up a domino effect of mistakes on the rest of the problem), when I realized my mistake, I quickly fixed it and solved for the correct answer.
Real World Connection: Just like my previous question this is a rational expression, meaning it can be applied to many real world situations, such as time. Say Mike, Mabel and Kaylee are all cleaning Fran's room. Kaylee can clean the room twice as fast as Mike can, mabel takes 16 minutes and together, all three finish the room in 12 minutes. How long does it take Kaylee to clean Fran's room by herself? By setting up the equation (1/16) + (1/x) + (1/2x) = (1/12) we can determine that Mikes time, x, is 48 minutes and because Kaylee's time is twice as fast, she finishes in 24 minutes.
The Strength: I found the strength to push through the problem for one. It was also a good test of basic math skills considering the large account of simplification required. Even though I messed up on one multiplication portion (initially setting up a domino effect of mistakes on the rest of the problem), when I realized my mistake, I quickly fixed it and solved for the correct answer.
Real World Connection: Just like my previous question this is a rational expression, meaning it can be applied to many real world situations, such as time. Say Mike, Mabel and Kaylee are all cleaning Fran's room. Kaylee can clean the room twice as fast as Mike can, mabel takes 16 minutes and together, all three finish the room in 12 minutes. How long does it take Kaylee to clean Fran's room by herself? By setting up the equation (1/16) + (1/x) + (1/2x) = (1/12) we can determine that Mikes time, x, is 48 minutes and because Kaylee's time is twice as fast, she finishes in 24 minutes.