Algebraic Word Problems

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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.)

• $a_{\mathrm{g}}$: the time that it takes Peter to complete harvesting the garden
• $b_{\mathrm{g}}$: the time that it takes Pepper to complete harvesting the garden
• $a_{\mathrm{o}}$: the time that it takes Peter to complete harvesting the orchard
• $b_{\mathrm{o}}$: 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:

$\left(\frac{1}{a_{\mathrm{g}}}+\frac{1}{b_{\mathrm{g}}}\right)=\frac{1}{3}$

"Whereas Pepper, working alone, requires 12 hours." translates to:

$\frac{1}{b_{\mathrm{g}}}=\frac{1}{\mathrm{12}}$

Here we can clearly see that $b_{\mathrm{g}}=\mathrm{12}$

Next: "together they can harvest the orchard in 2 hours"

$\left(\frac{1}{a_{\mathrm{o}}}+\frac{1}{b_{\mathrm{o}}}\right)=\frac{1}{2}$

"whereas Peter, alone, takes 10 hours":

$\frac{1}{a_{\mathrm{o}}}=\frac{1}{\mathrm{10}}⟶<!--\; ⟶\; -->a_{\mathrm{o}}=\mathrm{10}$

Now we can solve for $a_{\mathrm{g}}$ and $b_{\mathrm{o}}$.

$\left(\frac{1}{a_{\mathrm{g}}}+\frac{1}{\mathrm{12}}\right)=\frac{1}{3}⟶<!--\; ⟶\; -->a_{\mathrm{g}}=4$

$\left(\frac{1}{\mathrm{10}}+\frac{1}{b_{\mathrm{o}}}\right)=\frac{1}{2}⟶<!--\; ⟶\; -->b_{\mathrm{o}}=\frac{5}{2}$

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 $t$ amount of time working together they will have completed harvesting. We represent that with the following:

$\left(\frac{1}{b_{\mathrm{o}}}\right)\left(\frac{5}{2}\right)+\left(\frac{1}{a_{\mathrm{g}}}+\frac{1}{b_{\mathrm{g}}}\right)t=1⟶<!--\; ⟶\; -->t=\frac{9}{8}$

So the total time spent is $\frac{5}{2}+\frac{9}{8}=\mathrm{3.62500}$. And the difference is $5-\mathrm{3.62500}=\mathrm{1.37500}$ hours. And that is the answer.