Formulas

 

Learning Objective(s)

·         Evaluate a formula using substitution.

·         Rearrange formulas to isolate specific variables.

 

Introduction

 

Many real-world problems have well-known equations that describe relationships among different quantities. These equations that state a rule for a relationship are called formulas. You may have used formulas to figure out things like the area of a rectangle (area = length • width), the speed of a moving object (speed = distance ÷ time) or to convert from one system of measurement to another. Most formulas include more than one variable. Although they have a special name, formulas are written and solved like any other equations.

 

Evaluating Formulas Using Substitution

 

There are many formulas that we know that relate to the activities of our daily lives. For example, the formula distance = rate • time expresses a frequently used relationship in algebra. Formulas are also commonly used for measurement in geometry. For example, the formula for the area of a parallelogram, a four-sided figure with two pairs of parallel sides, is base times height, or.

 

Formulas are written so that one variable is already isolated. You need to only evaluate the expression on the other side for the given value(s) for the variable.

 

 

Example

Problem

Juan drove his car for 7 hours at 50 miles per hour. Use the formula d = r • t where d = distance, r = rate, and

t = time to find the distance Juan drove.

 

Substitute in the values given for the rate (or speed) and the time traveled. Then compute.

Answer

Juan drove his car a distance of 350 miles.

 

 

The next example illustrates a geometry formula.

 

 

Example

Problem

Using the formula A = bh, find the area (A) of a parallelogram with a base (b) of 3.5 inches and a height (h) of 7.2 inches.

 

   

   

   

Substitute the given lengths into the formula for area.

Multiply.

Answer

The area of the parallelogram is 25.2 square inches.

 

 

You can also solve for any of the variables in a formula using the algebra techniques you have learned.

 

 

Example

Problem

Find the base (b) of a triangle with an area (A) of 20 square feet and a height (h) of 8 feet. Use the formula for the area of a triangle, .

 

  

 

     

 

     

    

     

       

 

Substitute the given lengths into the formula.

 

 

 

 

 

Solve for b.

Answer

The base of the triangle measures 5 feet.

 

 

Advanced Example

Problem

In order to entice customers to invest their money, many banks will offer interest-bearing accounts. They work like this: a customer deposits a certain amount of money (called the Principal, or P), which then grows slowly according to the interest rate (R, measured in percent) and the length of time (T, usually measured in months) that the money stays in the account. The amount earned over time is called the interest (I), which is then given to the customer.

 

The simplest way to calculate interest earned on an account is through the formula .

 

If a customer deposits a principal of $2000 at a monthly rate of 0.7%, what is the total amount that she has after 24 months?

 

 

 

Substitute in the values given for the Principal, Rate, and Time.

 

 

Rewrite 0.7% as the decimal 0.007, then multiply.

Add the interest and the original principal amount to get the total amount in her account.

Answer

She has $2336 after 24 months.

 

 

Solving a Formula for a Specified Variable

 

If you were going to use the formula to find numerous values of b, given A and h, it would be more efficient to solve the formula for b first. Since formulas are equations, you can solve for a different variable just as you solve an equation. This is called solving the formula for b.

 

In the example below, we solve the formula d = r • t  for t. This would be useful if we wanted to calculate the amount of time a number of car trips will take to complete.

 

 

Example

Problem

Solve the equation d = r • t for t.

 

 

 

 

To solve this formula for t, isolate this variable on one side of the equation.

 

You can do this by using the Multiplication Property of Equality to multiply both sides by .

 

 

 

 

 

             

Multiplying by  is the equivalent of dividing by r as long as r ≠ 0.

 

The quantity , so .

 

 

Answer

              

 

 

 

Note the answer  is a correct answer. However, we traditionally write the variable on the left. So with equality, the expressions can be switched to .

 

 

Solve the formula for the volume of a rectangular solid, , for width (w).

 

A)

B)

C)

D)

 

Show/Hide Answer

A)

Incorrect. To isolate the variable w, you should have divided both sides of the formula by lh (or hl). The correct answer is .

 

B)

Correct. To isolate the variable w, you just need to divide both sides of the formula by l and h. The order in which these variables are written doesn’t affect their product. So, .

 

C)

Incorrect. To isolate the variable w, you should have divided both sides of the formula by lh (or hl), not just exchanged V and w in the formula. The correct answer is .

 

D)

Incorrect. This equation is true, but it does not solve for w. To isolate the w, you must divide both sides of the formula by lh (or hl), not by w. The correct answer is .

 

 

 

Advanced Question

Solve the formula for the area of a trapezoid,  for .

 

A)

B)

C)

D)

 

Show/Hide Answer

A)

Incorrect. To isolate the variable , begin by multiplying both sides of the equation by 2. Then divide by h. The new equation looks like this: . The correct solution is: .

 

B)

Incorrect. It looks like you multiplied and divided incorrectly. To remove the fraction from the right side, multiply both sides of the equation by 2. This gives you the new equation: . Now divide by h in order to isolate . The correct solution is: .

 

C)

Correct. To isolate the variable , multiply both sides of the equation by 2, divide by h, and then subtract . Written in terms of , the formula is .

 

D)

Incorrect. To isolate the variable , begin by multiplying both sides of the equation by 2. Then divide by h. The new equation looks like this: . The correct solution is: .

 

 

 

Solving Complex Formulas for a Specific Variable

 

Some formulas can be solved for a different variable with just a single step. Many others are multi-step formulas, and these will require a series of steps. It may sound like a big task, but it can be done, one step at a time.

 

For example, let’s look at the formula for the perimeter of a rectangle, . This formula is useful if we are given the length and width. But what if you are given the perimeter and lengths of the sides instead? If you are asked to calculate the width of several rectangles, it might be more efficient to solve the formula for w, the width, first.

 

The steps for rearranging a more complex formula are shown in the example.

 

 

Example

Problem

Express the formula for the perimeter of

a rectangle, , in terms of the width, w.

 

 

Isolate the term containing the variable, w, by subtracting the other term in its expression,, from both sides.

 

 

Next, clear the coefficient of w by dividing both sides of the equation by 2.

 

 

 

 

 

You can rewrite the equation so the isolated variable is on the left side.

Answer

 

 

 

Let’s look at a more complex formula that includes parentheses and fractions, the formula for converting from the Fahrenheit temperature scale to the Celsius scale.

 

You may or may not recall the formula for converting from the Celsius scale to the Fahrenheit scale. However, you can solve the formula that you do know (or are given) for the other to find that formula.

 

 

Example

Problem

Solve the formula shown below for converting from the Fahrenheit scale to the Celsius scale for F.

 

 

To isolate the variable F, it would be best to clear the fraction involving F first.

 

Multiply both sides of the equation by .

 

 

 

Add 32 to both sides.

Answer

         

 

 

 

Advanced Example

Problem

Express the formula for the surface area of a cylinder, , in terms of the height, h.

 

Isolate the term containing the variable, h, by subtracting  from both sides.

 

 

 

Next, isolate the variable h by dividing both sides of the equation by .

 

You can rewrite the equation so the isolated variable is on the left side.

Answer

 

 

 

Summary

 

Formulas are a type of equation. They usually contain multiple variables, describe important relationships, and provide a quick way to calculate frequently needed quantities. Although they are written to find the value for a particular variable, formulas can be solved for other variables in the formula following standard algebraic rules.