A student writes,
Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better?
This is a surprisingly subtle question. It turns out that the answer is partly “Do what’s mathematically sound,” but mostly “Do what’s useful.” It’ll take me 6 posts to show what that means:
- Manipulating Algebraic Equations and Expressions with Fractions 1: A Quick Quiz
- Manipulating Algebraic Equations and Expressions with Fractions 2: Expressions
- Manipulating Algebraic Equations and Expressions with Fractions 3: Another Example
- Manipulating Algebraic Equations and Expressions with Fractions 4: Equations
- Manipulating Algebraic Equations and Expressions with Fractions 5: A Word Problem
- Manipulating Algebraic Equations and Expressions with Fractions 6: Systems of Equations
All the problems we’ve so far considered have presented as algebra. However, quite a few word problems translate into algebraic equations with fractions. In particular, some of the hardest work-rate problems become very easy if you can correctly translate them into algebraic equations with fractions, and then correctly solve those equations.
Consider this problem:
Little Texas Drilling Company has three wells, each producing oil at a constant rate. Well A produces one barrel every two minutes. Well B produces one barrel every three minutes. Well C produces one barrel every four minutes. How many hours does it take Little Texas Drilling Company to produce 195 barrels of oil?
A combined rate problem that asks about simultaneous action allows a nice shortcut. If every worker starts and stops at the same time or is working continuously for the duration of the story, we can simply add the various work rates together to determine the combined work rate. Be sure to express work rates as work/time, and be careful to use the same units throughout the problem.
In this case, the rate for Well A is 1/2 (one barrel/two minutes), for Well B 1/3, and for well C 1/4. The combined rate is 195 barrels in x hours, which we can express as 60x minutes.
All that remains is to solve for x.
Multiply through by 60x to clear the fractions.
30x + 20x + 15x = 195
Combine like terms.
65x = 195
And divide each side of the equation by 65.
x = 3