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When a word problem involves a known formula, the general seven-step problem-solving strategy is adapted so that the **Translate** step uses the formula directly rather than constructing an equation from scratch. The revised steps are:

1. **Read** the problem and make sure all the words and ideas are understood.
2. **Identify** what you are looking for.
3. **Name** the unknown by choosing a variable to represent it.
4. **Translate** by writing the appropriate formula for the situation and substituting in the given information.
5. **Solve** the resulting equation using standard algebra techniques.
6. **Check** the answer in the original problem to make sure it is reasonable.
7. **Answer** the question with a complete sentence.

Organizing the known and unknown quantities into a small chart before substituting can help keep the information clear. For instance, when working with $$d = rt$$, listing which of $$d$$, $$r$$, and $$t$$ are given and which is unknown makes it straightforward to substitute and solve.

Strategy for Solving Formula-Based Applications

The **simple interest formula** relates four quantities associated with borrowing or investing money:

$$I = Prt$$

where

- $$I$$ = the amount of interest (earned on an investment or owed on a loan),
- $$P$$ = the principal (the original amount of money invested or borrowed),
- $$r$$ = the annual interest rate expressed as a decimal (not as a percent),
- $$t$$ = the time the money is invested or borrowed, measured in years.

Because the rate must be in decimal form, a percent such as $$5\%$$ is first converted to $$0.05$$ before being substituted into the formula. To apply this formula in a word problem, identify which three of the four quantities are given, substitute them into $$I = Prt$$, and solve for the remaining unknown using basic algebra. For example, to find the interest earned on a $$\$3{,}000$$ investment at $$4.5\%$$ annual interest for $$2$$ years, substitute $$P = 3000$$, $$r = 0.045$$, and $$t = 2$$ to obtain $$I = 3000 \times 0.045 \times 2 = 270$$, so the interest is $$\$270$$. This formula models situations in which the interest is computed only on the original principal and does not compound.

Simple Interest Formula

When solving geometry word problems, the general formula-based strategy is enhanced by incorporating a visual element. The adapted seven-step process is:

1. **Read** the problem and make sure all words and ideas are understood. **Draw the figure** and label it with the given information.
2. **Identify** what you are looking for.
3. **Label** the unknown by choosing a variable to represent it.
4. **Translate** into an equation by writing the appropriate geometry formula and substituting the given values.
5. **Solve** the equation using standard algebra techniques.
6. **Check** the answer by substituting it back into the equation and confirming it makes sense in the context of the problem.
7. **Answer** the question with a complete sentence.

The key adaptation for geometry is in Step 1: because these problems involve shapes, sketching a diagram and labeling it with known measurements and variables helps organize information visually. This makes it easier to select the correct formula and to see where each piece of given information fits before translating the situation into an equation.

Problem-Solving Strategy for Geometry Applications

A **triangle** is a polygon with three sides and three interior angles. Each side is conventionally labeled with a lowercase letter matching the uppercase letter of its opposite vertex — in $$\triangle ABC$$, side $$a$$ faces vertex $$A$$, side $$b$$ faces vertex $$B$$, and side $$c$$ faces vertex $$C$$. The plural of *vertex* is *vertices*, and a triangle is named by listing its three vertices (e.g., $$\triangle ABC$$).

The three interior angles of every triangle satisfy:

$$m\angle A + m\angle B + m\angle C = 180°$$

where $$m\angle A$$ is read as "the measure of angle $$A$$." This property holds for all triangles regardless of shape or size, so whenever two angles are known, the third can be found by subtracting their sum from $$180°$$.

Triangle Angle Sum Property

The **perimeter** of a figure is the total length of its boundary. For a triangle with sides of lengths $$a$$, $$b$$, and $$c$$, the perimeter is the sum of all three side lengths:

$$P = a + b + c$$

To find a triangle's perimeter, add together the lengths of its three sides. If the perimeter and two side lengths are known, the third side can be found by subtracting the two known sides from $$P$$.

Perimeter of a Triangle

The **area** of a triangle measures the amount of surface it encloses. Computing it requires knowing a **base** ($$b$$) and the corresponding **height** ($$h$$). The height is a perpendicular segment drawn from the base to the opposite vertex, forming a $$90°$$ angle with the base. The formula is:
$$A = \frac{1}{2}bh$$
In words, the area of a triangle equals one-half the product of its base and its height.

Area of a Triangle

Apply the geometry problem-solving strategy to find an unknown angle when two angles of a triangle are given.

**Problem:** The measures of two angles of a triangle are $$55°$$ and $$82°$$. Find the measure of the third angle.

1. **Read and draw:** Sketch a triangle and label the known angles $$55°$$ and $$82°$$; label the unknown angle $$x$$.
2. **Identify:** The measure of the third angle.
3. **Label:** Let $$x$$ = the measure of the unknown angle.
4. **Translate:** Apply the angle sum property:

$$55 + 82 + x = 180$$

5. **Solve:** Combine the known values: $$137 + x = 180$$. Subtract $$137$$ from both sides: $$x = 43$$.
6. **Check:** $$55 + 82 + 43 \stackrel{?}{=} 180$$ → $$180 = 180$$ $$\checkmark$$
7. **Answer:** The measure of the third angle is $$43°$$.

This example illustrates how the triangle angle sum property converts a geometry question into a simple one-step linear equation: add the two known angles, then subtract from $$180$$.

Finding the Third Angle of a Triangle Given Two Angles

The Pythagorean Theorem defines the relationship between the three side lengths of any right triangle. It states that the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse: $$a^2 + b^2 = c^2$$. In this formula, $$a$$ and $$b$$ represent the lengths of the triangle's legs, and $$c$$ represents the length of the hypotenuse.

The Pythagorean Theorem

A **rectangle** is a four-sided figure with four right ($$90°$$) angles. Its opposite sides are always equal in length. By convention, one side is called the **length**, denoted $$L$$, and the adjacent side is called the **width**, denoted $$W$$. These two measurements serve as the basis for computing a rectangle's perimeter and area.

Key properties:

- Four sides and four right ($$90°$$) angles
- Opposite sides have equal lengths
- The perimeter is the sum of twice the length and twice the width: $$P = 2L + 2W$$
- The area is the product of the length and the width: $$A = L \cdot W$$

Properties of Rectangles

The **perimeter** of a rectangle is the total distance around its boundary. Because a rectangle has two pairs of equal opposite sides, traveling around the figure covers each length and each width twice. The formula is:

$$P = 2L + 2W$$

where $$L$$ is the length and $$W$$ is the width. Equivalently, the perimeter can be expressed as $$L + W + L + W$$, since traversing the boundary means passing along all four sides once. When any two of the three quantities ($$P$$, $$L$$, $$W$$) are known, the third can be found by substituting the known values into the formula and solving the resulting equation.

Perimeter of a Rectangle

The **area** of a rectangle measures the amount of surface it encloses, expressed in square units. It equals the product of the length and the width:

$$A = L \cdot W$$

For example, a rectangular rug that is $$2$$ feet long and $$3$$ feet wide covers $$2 \cdot 3 = 6$$ square feet — exactly the number of unit squares that tile the surface. When the area and one dimension are known, the missing dimension can be found by dividing the area by the known side length.

Area of a Rectangle

When a uniform motion problem involves two travelers or vehicles, the general formula-based strategy is enhanced with visual and organizational tools. The adapted seven-step process is:

1. **Read** the problem and make sure all the words and ideas are understood.
   - **Draw a diagram** to illustrate the situation (e.g., sketch the route with labeled arrows for each traveler).
   - **Create a table** with columns for Rate, Time, and Distance, and one row for each scenario.
   - Fill in the information that is given.
2. **Identify** what you are looking for.
3. **Name** the unknown by choosing a variable, then complete the table. Use variable expressions to represent unknown quantities in each row, and multiply rate by time to fill in the distance column (since $$d = rt$$).
4. **Translate** into an equation by restating the relationship between the two distances in one sentence, then converting that sentence into an equation.
5. **Solve** the equation using standard algebra techniques.
6. **Check** the answer in the original problem to make sure it is reasonable.
7. **Answer** the question with a complete sentence.

The key adaptations for distance, rate, and time problems are the diagram and the rate–time–distance table, which organize the two-scenario information visually and make it straightforward to identify the equation connecting the distances.

Problem-Solving Strategy for Distance, Rate, and Time Applications

Two angles are **complementary** if the sum of their measures equals $$90°$$. When two angles are complementary, each angle is called the **complement** of the other. For example, angles measuring $$58°$$ and $$32°$$ are complementary because $$58 + 32 = 90$$. This property provides a useful relationship: if $$x$$ and $$y$$ represent the measures of two complementary angles, then:

$$x + y = 90$$

This equation can be used alongside a second relationship between the angles to form a system of equations for solving angle problems.

Complementary Angles

Two angles are **supplementary** if the sum of their measures equals $$180°$$. When two angles are supplementary, each angle is called the **supplement** of the other. For example, angles measuring $$148°$$ and $$32°$$ are supplementary because $$148 + 32 = 180$$. This property provides a useful relationship: if $$x$$ and $$y$$ represent the measures of two supplementary angles, then:

$$x + y = 180$$

This equation can be paired with a second relationship between the angles to create a system of equations for solving angle problems.

Supplementary Angles

A **square** is a rectangle in which the length and width are equal. If the side length is denoted by $$s$$, then the area of the square is:

$$A = s^2$$

When the area is known and the side length is needed, the formula is solved for $$s$$ by taking the square root of both sides:

$$A = s^2$$

$$\sqrt{A} = \sqrt{s^2}$$

$$\sqrt{A} = s$$

Thus the side length formula is:

$$s = \sqrt{A}$$

The first formula gives the area when the side length is known; the second gives the side length when the area is known. Because the area must be positive, the principal (positive) square root is used.

Area of a Square

On Earth, if an object is dropped from a height of $$h$$ feet, the time $$t$$ in seconds it takes to reach the ground is given by:

$$t = \frac{\sqrt{h}}{4}$$

This formula relates the drop height directly to the fall time through a square root. To use it, substitute the known height into the formula and simplify. For example, if an object is dropped from a height of 64 feet, substitute $$h = 64$$:

$$t = \frac{\sqrt{64}}{4} = \frac{8}{4} = 2$$

The object takes 2 seconds to reach the ground. Because the formula involves a square root, the result will be a whole number only when the height is a perfect square multiplied by 16; otherwise, the answer will be an irrational number that must be approximated.

Falling Objects Formula

Police officers investigating car accidents measure the length of the skid marks left on the pavement and use a square root formula to determine the vehicle's speed before the brakes were applied. If the length of the skid marks is $$d$$ feet, then the speed $$s$$ of the car (in miles per hour) before braking can be found using:

$$s = \sqrt{24d}$$

To apply this formula, substitute the measured skid mark length for $$d$$, multiply by $$24$$, and then take the square root of the product. When the product $$24d$$ is not a perfect square, the result will be an irrational number that should be approximated with a calculator and rounded as directed by the problem.

Skid Marks and Speed of a Car Formula

The **projectile motion formula** gives the height, in feet, of an object launched straight upward from ground level with an initial velocity of $$v_0$$ feet per second. After $$t$$ seconds, the height $$h$$ is:

$$h = -16t^2 + v_0 t$$

The term $$-16t^2$$ accounts for the downward pull of gravity (the coefficient $$-16$$ comes from half the acceleration due to gravity in feet per second squared), and the term $$v_0 t$$ represents the upward distance the object would travel at constant speed if gravity were absent. Initially the upward velocity dominates, causing the object to rise, but gravity steadily decelerates it. Eventually the object reaches a peak height and then falls back down. Setting $$h$$ equal to a specific height and solving the resulting quadratic equation determines the time(s) at which the object is at that height.

Projectile Motion Formula

A logistics coordinator is using the formula d = rt (distance = rate x time) to calculate the travel time for a delivery. Arrange the steps of the formula-based problem-solving strategy in the correct order from start to finish.

A logistics manager is using the formula d = rt (distance = rate x time) to determine the required speed for a delivery truck to arrive on time. Match each step of the formula-based problem-solving strategy to the specific action the manager takes.

A facilities coordinator is using a standard power distribution formula to calculate the total wattage required for a new office floor. According to the seven-step problem-solving strategy for formula-based applications, which step is the coordinator performing when they write out the required formula and plug the known electrical values into it?

A shipping coordinator is calculating the total distance of a delivery route using the formula $d = rt$. According to the problem-solving strategy for formula-based applications, the coordinator is performing the ____ step when they write out the formula and substitute the known speed and time values into it.

A production supervisor is using a formula to calculate the total capacity of a storage tank. According to the seven-step problem-solving strategy for formula-based applications, what organizational tool is recommended to manage the known and unknown quantities, and what is one primary benefit of using this tool?

Organizing Quantities in Formula-Based Problems

A warehouse manager is using the formula $V = LWH$ to calculate the total storage capacity of a shipping container. True or False: According to the seven-step strategy for formula-based applications, the manager performs the 'Translate' step by substituting the known dimensions into the formula directly rather than constructing a new equation from the problem's description.

To ensure the audit documentation is compliant, list the seven steps of the formula-based problem-solving strategy in the correct order. In your description of the 'Translate' step, specifically explain how the formula and the given information are utilized.

Quality Control: Storage Tank Audit

Imagine you are a supervisor training a new team member on how to efficiently solve technical problems that involve known industry formulas. Outline the seven-step problem-solving strategy for formula-based applications. In your response, list the steps in their correct sequence, describe the specific actions taken during the 'Translate' step, and explain why organizing known and unknown quantities into a chart is a recommended practice before solving.

Standard Procedure for Formula-Based Applications

A financial analyst is using the simple interest formula ($I = Prt$) to calculate the interest on a small business expansion loan. According to the seven-step problem-solving strategy for formula-based applications, what specific step is the analyst performing when they choose the variable 't' to represent the number of years for the loan?

A production supervisor is using a formula to calculate the total output of a factory line for the day. Before choosing a variable or substituting any values into the formula, the supervisor clearly writes at the top of their report: "Objective: Find the total number of units produced during the morning shift." According to the seven-step strategy for formula-based applications, which step is the supervisor performing?

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