**Rewriting Formulas to Help Solve Problems**

**Objective:** When done with this lesson, you will have demonstrated how to rewrite formulas by solving for a specified variable.

**Washington State GLE:** 1.5.6 - Rearrange formulas to solve for a particular variable.

**Approximate completion time:** 3 hours

Formulas can be very helpful. Sometimes a formula is used so much that it is rewritten to make life even easier. Consider this situation.

Suppose you want to install a sprinkler system for your lawn. How many sprinklers would you need? How many different zones would you need? These questions and many more are often answered in a manufacturer's how-to guide. Hunter industries provides details steps to help in their guide. Here are a few of their recommended steps:

- "Plot Plan And Design". This gives the layout of the sprinkler system in a map on a coordinate plane.
- "Determine System Design Capacity". This determines the volume and pressure of water available to irrigate the lawn.
- "Select Sprinkler Heads". You have to choose wisely to make the plan work well.
- "Draw Sprinkler Head Locations". Each sprinkler must have its own location.
- "Divide Sprinklers Into Zones". The sprinklers likely will need to be turned on in small groups called "zones".

In their "Divide Sprinklers Into Zones" step, the following formula is used:

In algebra we would write this as

T÷D=Z. (Click on the image if you want.)Where

T= the total GPM of all heads in one area,

andD= the design capacity in GPM,

andZ= the number of zones in that area.

This formula is quite helpful if one were wanting to know how many zones were needed. Yet what if you bought an 8 zone timer and wanted to find out how many total sprinkler heads could go into one zone? You would have to solve for

Tin that equation. A simple algebra step and the equation becomes

T=Z·DDid you see that multiplying both sides of the equation

T÷D=ZbyDwill undo the divide byD? Look again closely to be sure you have that. Click on the image below to watch.

Now look at this example where rewriting a formula could be helpful.

To determine how much fertilizer a lawn needs, one could use this formula:

where

Pis the number of pounds of fertilizer,Ais the area of the lawn in square feet, andNis the nitrogen number on the fertilizer bag formula (as in the 28 on a bag labeled "28-3-6 fertilizer").If someone had 5,000 square feet of lawn and wanted to know how many pounds of 28-3-6 fertilizer to buy, they would simply use the formula and substitute giving

P= (0.15*5000)/28 which equals 26.7857 or about 27 pounds.What if they wanted to solve for

A, the area of lawn that could be fertilized with a certain nitrogen rating?

EquationReasonsGiven. NP= 0.15AMultiplying both sides by Nto undo the divide byN.Dividing both sides by 0.15 to undo the multiply by 0.15. Rewriting. So the formula is now written as .

Remember that the letters in the equation are unknown and simply represent some number. So when you divide by a number like 0.15, then you must write the equation with that represented. We all would prefer to have had numbers which could have been divided by 0.15 and get a result of something "clean" like A = 4500. Yet when we rewrite a formula, the variables are unknown so we don't know how much

NPdivided by 0.15 really is.

To many people, rewriting formulas is quite simple. To others the idea seems strange. With practice and time, you'll be sure to get it down.

Can you see how these formulas are the same?

FormulaAn Other Common FormDescriptiond=r·t distance = rate · timeF=m·a force = mass · accelerationV=I·R voltage = current · resistancea^{2}+b^{2}=c^{2} the pythagorean theorem the "sine ratio"Perhaps you recognize many of these formulas. They are only a few of the most popular and extremely important equations that we use in mathematics and science. Can you guess how to rewrite each of the formulas to get the other form?

For your practice, download this assignment.