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+---
+title: Algebraic Word Problems
+author: James R. Bracy
+layout: post
+---
+
+The next section in the book *Problem Solving Through Recreation
+Mathematics* that I am doing is about taking a word problem, representing it
+in algebraic form and then solving it.
+
+While reading the chapter I found what the term *Algebra* actually means and
+where it came from. From the page 70 of the book:
+
+> The word algebra is derived from the title of a book on the subject (as it
+> existed at the time), *Hisâb al-jabr w'al-muqâbalah*, which
+> was written by the Persian mathematician Al-Khowarizmi (c. 825). The term
+> al-jabr apparently meant "restoring," as in restoring the balance between
+> the two sides of an equation by subtracting from one side what has been
+> subtracted from another.
+
+So all of these years of simply accepting the term *algebra* and now I know
+what it means. It actually makes a lot of sense.
+
+## The Problem
+
+On with the problem:
+
+> **3.51.** Peter Piper and his wife Pepper have a vegetable garden and a
+> fruit orchard. Working together they can collect the harvest from the garden
+> in 3 hours, whereas Pepper, working alone, requires 12 hours. Furthermore,
+> together they can harvest the orchard in 2 hours, whereas Peter, alone,
+> takes 10 hours.
+>
+> It would seem that to harvest both the garden and the orchard, they should
+> first spend 3 hours in the garden and then 2 hours in the orchard -- a total
+> of five hours in all. However, Peter is much more skillful at picking
+> vegetables and Pepper is better at picking fruit, so that they can time by
+> having Peter work in the garden and Pepper work in the orchard until one of
+> them finishes. That person would then help the other.
+>
+> How much time can they save in this manner?
+
+First lets understand what the problem is asking for. Sometimes I go through
+the whole process of solving the equations, but don't do the last step. We
+want to know the difference between the time it would take both Peter and
+Pepper to harvest both the garden and orchard together one right after
+another, 5 hours, and the time it would take if they worked separately at what
+each did best then join together on whatever remains when one finishes.
+
+## Work it Out
+
+First we extract the variables and formulas from the text. (Note that you must
+have a MathML enabled browser to view the equations.)
+
+- <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>g</mn></msub></mrow></math>:
+ the time that it takes Peter to complete harvesting the garden
+- <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>b</mi><mn>g</mn></msub></mrow></math>:
+ the time that it takes Pepper to complete harvesting the garden
+- <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>o</mn></msub></mrow></math>:
+ the time that it takes Peter to complete harvesting the orchard
+- <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>b</mi><mn>o</mn></msub></mrow></math>:
+ the time that it takes Pepper to complete harvesting the orchard
+
+"Working together they can collect the harvest from the garden in 3 hours"
+translates to:
+
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfenced>
+<mrow>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>a</mi><mn>g</mn></msub></mrow>
+</mfrac>
+<mo>+</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>b</mi><mn>g</mn></msub></mrow>
+</mfrac>
+</mrow>
+</mfenced>
+</mrow>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>3</mi></mrow>
+</mfrac>
+</math>
+
+"Whereas Pepper, working alone, requires 12 hours." translates to:
+
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>b</mi><mn>g</mn></msub></mrow>
+</mfrac>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>12</mi></mrow>
+</mfrac>
+</mrow>
+</math>
+
+Here we can clearly see that
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<msub><mi>b</mi><mn>g</mn></msub>
+<mo>=</mo>
+<mi>12</mi>
+</math>
+
+Next: "together they can harvest the orchard in 2 hours"
+
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfenced>
+<mrow>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>a</mi><mn>o</mn></msub></mrow>
+</mfrac>
+<mo>+</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>b</mi><mn>o</mn></msub></mrow>
+</mfrac>
+</mrow>
+</mfenced>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>2</mi></mrow>
+</mfrac>
+</mrow>
+</math>
+
+"whereas Peter, alone, takes 10 hours":
+
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>a</mi><mn>o</mn></msub></mrow>
+</mfrac>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>10</mi></mrow>
+</mfrac>
+<mo>&#x27F6;<!-- &longrightarrow; --></mo>
+<msub><mi>a</mi><mn>o</mn></msub>
+<mo>=</mo>
+<mi>10</mi>
+</mrow>
+</math>
+
+Now we can solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>g</mn></msub></mrow></math>
+and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>b</mi><mn>o</mn></msub></mrow></math>.
+
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfenced>
+<mrow>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>a</mi><mn>g</mn></msub></mrow>
+</mfrac>
+<mo>+</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>12</mi></mrow>
+</mfrac>
+</mrow>
+</mfenced>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>3</mi></mrow>
+</mfrac>
+<mo>&#x27F6;<!-- &longrightarrow; --></mo>
+<msub><mi>a</mi><mn>g</mn></msub>
+<mo>=</mo>
+<mi>4</mi>
+</mrow>
+</math>
+
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfenced>
+<mrow>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>10</mi></mrow>
+</mfrac>
+<mo>+</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>b</mi><mn>o</mn></msub></mrow>
+</mfrac>
+</mrow>
+</mfenced>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><mi>2</mi></mrow>
+</mfrac>
+<mo>&#x27F6;<!-- &longrightarrow; --></mo>
+<msub><mi>b</mi><mn>o</mn></msub>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>5</mi></mrow>
+<mrow><mi>2</mi></mrow>
+</mfrac>
+</mrow>
+</math>
+
+So it takes 2.5 hours for Pepper to finish the orchard and 4 hours for Peter
+to finish the garden. After 2.5 hours Pepper will then help out in the garden
+because she will be done with the orchard. After spending
+<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi></mrow></math>
+amount of time working together they will have completed harvesting. We
+represent that with the following:
+
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfenced>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>b</mi><mn>o</mn></msub></mrow>
+</mfrac>
+</mfenced>
+<mfenced>
+<mfrac>
+<mrow><mi>5</mi></mrow>
+<mrow><mi>2</mi></mrow>
+</mfrac>
+</mfenced>
+<mo>+</mo>
+<mfenced>
+<mrow>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>a</mi><mn>g</mn></msub></mrow>
+</mfrac>
+<mo>+</mo>
+<mfrac>
+<mrow><mi>1</mi></mrow>
+<mrow><msub><mi>b</mi><mn>g</mn></msub></mrow>
+</mfrac>
+</mrow>
+</mfenced>
+<mi>t</mi>
+<mo>=</mo>
+<mi>1</mi>
+<mo>&#x27F6;<!-- &longrightarrow; --></mo>
+<mi>t</mi>
+<mo>=</mo>
+<mfrac>
+<mrow><mi>9</mi></mrow>
+<mrow><mi>8</mi></mrow>
+</mfrac>
+</mrow>
+</math>
+
+So the total time spent is
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mfrac>
+<mrow><mi>5</mi></mrow>
+<mrow><mi>2</mi></mrow>
+</mfrac>
+<mo>+</mo>
+<mfrac>
+<mrow><mi>9</mi></mrow>
+<mrow><mi>8</mi></mrow>
+</mfrac>
+<mo>=</mo>
+<mi>3.62500</mi></mrow></math>. And the difference is <math xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow><mi>5</mi>
+<mo>-</mo>
+<mi>3.62500</mi>
+<mo>=</mo>
+<mi>1.37500</mi></mrow></math> hours. And that is the answer.
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