Apply the problem-solving strategy to find the rate of simple interest given the principal, interest earned, and time.

**Problem:** Hang borrowed $$\$7{,}500$$ from her parents to pay her tuition. In $$5$$ years, she paid them $$\$1{,}500$$ interest in addition to the $$\$7{,}500$$ she borrowed. What was the rate of simple interest?

1. **Read** the problem carefully.
2. **Identify** what to find: the rate of simple interest.
3. **Name** the unknown: Let $$r$$ = rate of interest.
4. **Translate** into an equation using the simple interest formula $$I = Prt$$. Given $$I = 1{,}500$$, $$P = 7{,}500$$, and $$t = 5$$:
$$1{,}500 = (7{,}500)r(5)$$
5. **Solve** the equation:
Multiply $$7{,}500$$ and $$5$$:
$$1{,}500 = 37{,}500r$$
Divide both sides by $$37{,}500$$:
$$0.04 = r$$
Change the decimal to percent form:
$$4\% = r$$
6. **Check** the result:
$$I = Prt$$
$$1{,}500 \stackrel{?}{=} (7{,}500)(0.04)(5)$$
$$1{,}500 = 1{,}500 \checkmark$$
7. **Answer**: The rate of interest was $$4\%$$.

Solving a Word Problem for the Rate of Simple Interest on Hang's Loan

Apply the problem-solving strategy to find the rate of simple interest given the principal, interest earned, and time.

**Problem:** Jim lent his sister $$\$5{,}000$$ to help her buy a house. In $$3$$ years, she paid him the $$\$5{,}000$$, plus $$\$900$$ interest. What was the rate of simple interest?

1. **Read** the problem carefully.
2. **Identify** what to find: the rate of simple interest.
3. **Name** the unknown: Let $$r$$ = rate of interest.
4. **Translate** into an equation using the simple interest formula $$I = Prt$$. Given $$I = 900$$, $$P = 5{,}000$$, and $$t = 3$$:
$$900 = (5{,}000)r(3)$$
5. **Solve** the equation:
Multiply $$5{,}000$$ and $$3$$:
$$900 = 15{,}000r$$
Divide both sides by $$15{,}000$$:
$$0.06 = r$$
Change the decimal to percent form:
$$6\% = r$$
6. **Check** the result:
$$I = Prt$$
$$900 \stackrel{?}{=} (5{,}000)(0.06)(3)$$
$$900 = 900 \checkmark$$
7. **Answer**: The rate of interest was $$6\%$$.

Solving a Word Problem for the Rate of Simple Interest on Jim's Loan

Apply the problem-solving strategy to find the rate of simple interest given the principal, interest earned, and time.

**Problem:** Loren lent his brother $$\$3{,}000$$ to help him buy a car. In $$4$$ years, his brother paid him back the $$\$3{,}000$$ plus $$\$660$$ in interest. What was the rate of simple interest?

1. **Read** the problem carefully.
2. **Identify** what to find: the rate of simple interest.
3. **Name** the unknown: Let $$r$$ = rate of interest.
4. **Translate** into an equation using the simple interest formula $$I = Prt$$. Given $$I = 660$$, $$P = 3{,}000$$, and $$t = 4$$:
$$660 = (3{,}000)r(4)$$
5. **Solve** the equation:
Multiply $$3{,}000$$ and $$4$$:
$$660 = 12{,}000r$$
Divide both sides by $$12{,}000$$:
$$0.055 = r$$
Change the decimal to percent form:
$$5.5\% = r$$
6. **Check** the result:
$$I = Prt$$
$$660 \stackrel{?}{=} (3{,}000)(0.055)(4)$$
$$660 = 660 \checkmark$$
7. **Answer**: The rate of interest was $$5.5\%$$.

Solving a Word Problem for the Rate of Simple Interest on Loren's Loan

The rate of interest, denoted by the variable $$r$$, is the percentage of the principal that is paid or earned as interest over a certain period, usually per year. When used in algebraic equations like the simple interest formula, this annual percentage rate must always be converted into its decimal equivalent to properly compute the interest.

Claude

Basic percent problems can be solved by translating English sentences directly into algebraic equations, rather than setting up proportions. Every basic percent sentence involves three quantities — the **percent**, the **base** (the whole amount), and the **amount** (the part of the whole) — and a typical problem supplies two of these while asking for the third. The same translation keywords used in general equation translation apply in percent contexts: the word "is" translates to an equals sign ($$=$$), the word "of" translates to multiplication ($$\times$$), and phrases like "what number" or "what percent" become a variable. It is helpful to choose a variable letter that serves as a reminder of the unknown (e.g., $$n$$ for "number" or $$p$$ for "percent"). Before placing the percent into the equation, it **must** be converted to its decimal form. This approach produces three standard equation types:

- **Finding the amount:** "What number is $$35\%$$ of $$90$$?" translates to $$n = 0.35 \times 90$$.
- **Finding the base:** "$$6.5\%$$ of what number is $$1.17$$?" translates to $$0.065 \times n = 1.17$$.
- **Finding the percent:** "$$144$$ is what percent of $$96$$?" translates to $$144 = p \times 96$$.

Once the equation is written, it is solved using the same properties of equality (multiplication, division) that apply to any linear equation with decimal coefficients. Many everyday situations — such as calculating tips, determining recommended daily nutrient amounts, or finding what fraction of calories comes from fat — reduce to one of these three equation types once the problem is translated into a single percent sentence.

Translating Basic Percent Sentences to Equations

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

To translate and solve "What number is $$35\%$$ of $$90$$?", apply the percent-sentence-to-equation process:

1. **Assign a variable:** Let $$n$$ = the unknown number.
2. **Translate keyword by keyword:** "What number" becomes $$n$$; "is" becomes $$=$$; "$$35\%$$" converts to the decimal $$0.35$$; "of" becomes multiplication; "$$90$$" remains $$90$$.
3. **Write the equation:** $$n = 0.35 \times 90$$
4. **Multiply:** $$n = 31.5$$

Therefore, $$31.5$$ is $$35\%$$ of $$90$$.

This type of percent equation — where the **amount** is unknown — produces a one-step multiplication problem because the variable is already isolated on one side of the equals sign. Converting $$35\%$$ to its decimal equivalent $$0.35$$ before multiplying is the essential prerequisite step.

Solving 'What number is 35% of 90?'

To translate and solve "$$6.5\%$$ of what number is \$1.17?", apply the percent-sentence-to-equation process:

1. **Assign a variable:** Let $$n$$ = the unknown number.
2. **Translate keyword by keyword:** "$$6.5\%$$" converts to the decimal $$0.065$$; "of" becomes multiplication; "what number" becomes $$n$$; "is" becomes $$=$$; "\$1.17" becomes $$1.17$$.
3. **Write the equation:** $$0.065n = 1.17$$
4. **Divide both sides by $$0.065$$:** $$\frac{0.065n}{0.065} = \frac{1.17}{0.065}$$
5. **Simplify:** $$n = 18$$

Therefore, $$6.5\%$$ of \$18 is \$1.17.

In this type of percent equation, the **base** (the whole amount) is unknown, so the variable is not isolated after translation — it appears as part of the product $$0.065n$$. Dividing both sides by the decimal coefficient isolates $$n$$ and reveals the answer.

Solving '6.5% of what number is \$1.17?'

To translate and solve "$$144$$ is what percent of $$96$$?", apply the percent-sentence-to-equation process:

1. **Assign a variable:** Let $$p$$ = the unknown percent (which will initially be found in decimal form).
2. **Translate keyword by keyword:** "$$144$$" stays as $$144$$; "is" becomes $$=$$; "what percent" becomes $$p$$; "of" becomes multiplication; "$$96$$" stays as $$96$$.
3. **Write the equation:** $$144 = p \times 96$$, i.e., $$144 = 96p$$.
4. **Divide both sides by $$96$$:** $$\frac{144}{96} = \frac{96p}{96}$$
5. **Simplify:** $$p = 1.5$$
6. **Convert to percent form:** $$p = 150\%$$

Therefore, $$144$$ is $$150\%$$ of $$96$$.

Because the question asks for "what percent," the final answer must be expressed in percent form, not as a decimal. The decimal $$1.5$$ is converted to $$150\%$$ by moving the decimal point two places to the right and attaching the percent sign. Notice that a percent can exceed $$100\%$$ whenever the amount is greater than the base.

Solving '144 is what percent of 96?'

Apply the seven-step problem-solving strategy to a percent application in which the **amount** (the tip) is the unknown.

**Problem:** Dezohn's restaurant bill was \$68.50, and he wants to leave an $$18\%$$ tip. How much should the tip be?

1. **Identify** what to find: the dollar amount of the tip.
2. **Name** the unknown: Let $$t$$ = the tip amount.
3. **Translate** into a percent sentence and equation: "The tip is $$18\%$$ of the total bill" becomes

$$t = 0.18 \times 68.50$$

4. **Solve** by multiplying:

$$t = 12.33$$

5. **Check** — a quick reasonableness test: $$20\%$$ of \$70 would be \$14, so \$12.33 for $$18\%$$ of \$68.50 is sensible.
6. **Answer:** Dezohn should leave a tip of \$12.33.

This problem is a "find the amount" percent equation — the percent and the base are both known, so the variable is already isolated after translation and only a single multiplication step is needed. Choosing the letter $$t$$ (for "tip") makes the equation easier to read and interpret in context.

Solving a Tip Calculation Using a Percent Equation

Apply the seven-step problem-solving strategy to a percent application in which the **base** (the whole) is the unknown.

**Problem:** One serving of breakfast cereal provides $$85$$ mg of potassium, which is $$2\%$$ of the recommended daily amount. What is the total recommended daily amount of potassium?

1. **Identify** what to find: the total recommended daily amount of potassium.
2. **Name** the unknown: Let $$a$$ = the total amount of potassium recommended daily.
3. **Translate** into a percent sentence and equation: "$$85$$ mg is $$2\%$$ of the total amount" becomes

$$85 = 0.02 \times a$$

4. **Solve** by dividing both sides by $$0.02$$:

$$\frac{85}{0.02} = a$$

$$a = 4{,}250$$

5. **Check** — $$2\%$$ is a small percent, and $$85$$ is indeed a small portion of $$4{,}250$$, so the answer is reasonable.
6. **Answer:** The total recommended daily amount of potassium is $$4{,}250$$ mg.

This problem is a "find the base" percent equation. After translation the variable $$a$$ is multiplied by the decimal coefficient $$0.02$$, so dividing both sides by that coefficient — using the Division Property of Equality — isolates the unknown and reveals the whole amount.

Finding the Total Recommended Daily Amount of Potassium

Apply the seven-step problem-solving strategy to a percent application in which the **percent** is the unknown.

**Problem:** Each gourmet brownie contains $$480$$ total calories, of which $$240$$ calories come from fat. What percent of the total calories is from fat?

1. **Identify** what to find: the percent of total calories that come from fat.
2. **Name** the unknown: Let $$p$$ = the percent (to be found first in decimal form).
3. **Translate** into a percent sentence and equation: "What percent of $$480$$ is $$240$$?" becomes

$$p \times 480 = 240$$

4. **Solve** by dividing both sides by $$480$$:

$$\frac{480p}{480} = \frac{240}{480}$$

$$p = 0.5$$

5. **Convert to percent form:** $$p = 50\%$$
6. **Check** — $$240$$ is exactly half of $$480$$, so $$50\%$$ is clearly correct.
7. **Answer:** Of the total calories in each brownie, $$50\%$$ come from fat.

This is a "find the percent" equation. The variable $$p$$ represents the unknown percent, which appears as a decimal coefficient multiplying the base. After dividing both sides by the base, the resulting decimal must be converted to percent form (move the decimal point two places to the right and attach $$\%$$) to answer the original question.

Finding the Percent of Calories from Fat

When a quantity grows from an original value to a larger new value, the **percent increase** expresses the size of that growth relative to the original value. The calculation follows two steps:

1. **Find the amount of increase** by subtracting the original amount from the new amount:

$$\text{new amount} - \text{original amount} = \text{increase}$$

2. **Determine what percent the increase is of the original amount** by translating the question "The increase is what percent of the original amount?" into an equation and solving it. Let $$p$$ represent the unknown percent. The equation takes the form:

$$\text{increase} = p \times \text{original amount}$$

Divide both sides by the original amount to isolate $$p$$, then convert the resulting decimal to percent form.

The key idea is that percent increase is always calculated relative to the **original** (starting) amount, not relative to the new amount. This ensures a consistent basis for comparison when describing how much a quantity has grown.

Procedure for Finding Percent Increase

When a quantity falls from an original value to a smaller new value, the **percent decrease** expresses the size of that decline relative to the original value. The calculation follows two steps:

1. **Find the amount of decrease** by subtracting the new amount from the original amount:

$$\text{original amount} - \text{new amount} = \text{decrease}$$

2. **Determine what percent the decrease is of the original amount** by translating the question "The decrease is what percent of the original amount?" into an equation and solving it. Let $$p$$ represent the unknown percent. The equation takes the form:

$$\text{decrease} = p \times \text{original amount}$$

Divide both sides by the original amount to isolate $$p$$, then convert the resulting decimal to percent form.

This procedure mirrors the one for percent increase, with one key difference: the subtraction in Step 1 reverses the order — the original amount comes first — so that the result is a positive number representing how much the quantity dropped. As with percent increase, the comparison is always made against the **original** amount.

Procedure for Finding Percent Decrease

A **discount** is a reduction in the original (regular) price of an item, typically expressed as a percent. Two key relationships govern discount problems:

1. **Amount of discount:**

$$\text{amount of discount} = \text{discount rate} \times \text{original price}$$

2. **Sale price:**

$$\text{sale price} = \text{original price} - \text{amount of discount}$$

The discount rate is a percent that must be converted to its decimal form before multiplying. For instance, if a jacket originally priced at $$\$80$$ is offered at a $$25\%$$ discount, the amount of discount is $$0.25 \times 80 = 20$$, so the sale price is $$80 - 20 = \$60$$. To solve discount word problems, identify which quantities are given (original price, discount rate, discount amount, or sale price), set up the appropriate equation using the relationships above, and solve for the unknown. These calculations are a direct application of the basic percent equation structure: the discount amount "is" a certain percent "of" the original price.

Discount Applications in Percent Problems

A **mark-up** is an increase added to the wholesale (or cost) price of an item to determine its retail selling price, typically expressed as a percent of the wholesale cost. Two key relationships govern mark-up problems:

1. **Amount of mark-up:**

$$\text{amount of mark-up} = \text{mark-up rate} \times \text{wholesale cost}$$

2. **List (selling) price:**

$$\text{list price} = \text{wholesale cost} + \text{amount of mark-up}$$

The mark-up rate is a percent that must be converted to decimal form before multiplying. For example, if a store purchases a lamp at a wholesale cost of $$\$40$$ and applies a $$55\%$$ mark-up, the mark-up amount is $$0.55 \times 40 = 22$$, so the list price is $$40 + 22 = \$62$$. To solve mark-up word problems, identify the given quantities (wholesale cost, mark-up rate, mark-up amount, or list price), set up the corresponding equation, and solve for the unknown. Like discount calculations, mark-up problems are a direct application of the basic percent equation structure: the mark-up amount "is" a certain percent "of" the wholesale cost.

Mark-up Applications in Percent Problems

A logistics coordinator is calculating a fuel surcharge for a shipping route. The coordinator uses the sentence: 'What number is 8% of $1,250?' Which equation represents the correct translation of this sentence into an algebraic equation?

A payroll specialist is translating a policy manual's instructions into mathematical equations to calculate employee benefits. Match each English word or phrase commonly found in these percent-based calculations with its correct algebraic representation.

When translating a professional sentence like 'The new hire's salary is 90% of the standard rate' into an algebraic equation, the word 'is' should be replaced with an equals sign (=).

A payroll supervisor is training a new clerk to translate policy instructions, such as '8% of the gross pay is the retirement contribution,' into mathematical equations. The supervisor explains that every basic percent sentence like this involves three specific types of quantities that must be identified. Based on the rules of translation, what are the technical names of these three quantities?

Identifying Quantities in Payroll Percent Calculations

A human resources specialist is translating a company benefits policy: 'The employer's retirement contribution is 7% of the employee's gross monthly pay.' To represent this relationship in an algebraic equation, the word 'of' must be translated into the mathematical operation of ________.

A project manager is translating a budget requirement into a mathematical formula: 'The licensing fee is 15% of the total project cost.' Arrange the following steps in the correct order to translate this sentence into a basic algebraic equation.

Based on the scenario provided, identify and list the four fundamental rules for translating a basic percent sentence into an algebraic equation. Your response should specify the mathematical representations for the keywords 'is' and 'of', how to represent unknown phrases like 'what number', and the required numerical conversion for any percent values.

Developing Professional Standards for Percent Equation Translation

A department manager is creating a 'Best Practices' guide for administrative assistants who handle monthly expense reports. You have been assigned to write the section on 'Translating Budget Sentences to Equations.' In a brief essay, describe the standard protocol for converting a sentence like 'The travel expense is 15% of the total monthly allowance' into a mathematical format. Your response must identify the specific mathematical symbols or operations that the keywords 'is' and 'of' translate to, explain how to represent unknown values (such as 'what amount'), and state the mandatory conversion step required for the percent value before it is placed in an equation.

Professional Standards for Translating Percent-Based Financial Sentences

A quality control inspector is translating the following professional sentence into an algebraic equation: 'The defect rate was 0.5% of the total production run.' According to the standard rules for translating percent sentences, what is the correct decimal form of 0.5% that must be written in the equation?

A warehouse supervisor is calculating total inventory levels based on a defect report. The supervisor uses the sentence: '3.5% of what number is 42?' Which of the following algebraic equations correctly translates this sentence?

To find the percent of calories from fat when a muffin has $$240$$ total calories and $$60$$ calories from fat, translate the question "What percent of $$240$$ is $$60$$?" into an equation. Let $$p$$ represent the unknown percent. The equation is $$p \times 240 = 60$$. Dividing both sides by $$240$$ gives $$p = 0.25$$. Converting the decimal to a percent yields $$25\%$$. Thus, $$25\%$$ of the total calories in each muffin come from fat.

Solving for the Percent of Calories from Fat in Muffins

To determine the percent of calories from fat in a brownie with $$190$$ total calories and $$76$$ calories from fat, translate the question "What percent of $$190$$ is $$76$$?" into an algebraic equation. Let $$p$$ be the unknown percent. The situation translates to $$p \times 190 = 76$$. Solving for $$p$$ by dividing both sides by $$190$$ results in $$p = 0.4$$. Converting this decimal to a percent gives $$40\%$$. This means $$40\%$$ of the total calories are from fat.

Solving for the Percent of Calories from Fat in Brownies

In algebraic finance problems, the initial amount of money deposited into a bank or borrowed as a loan is called the principal, denoted by the variable $$P$$. Whether earning interest on a savings account or paying interest on a loan, the amount of interest calculated is always based directly on this original principal sum.

Principal (Algebra)

Rate of Interest (Algebra)

Apply the seven-step problem-solving strategy to a percent application where the percent is unknown. **Problem:** Veronica is planning to make muffins from a mix. The package says each muffin will be $$240$$ calories and $$60$$ calories will be from fat. What percent of the total calories is from fat? 1. **Identify** what to find: What percent of the total calories is fat? 2. **Name** the unknown: Let $$p$$ = percent of fat. 3. **Translate** into a percent sentence and equation: "What percent of $$240$$ is $$60$$?" becomes $$p \times 240 = 60$$. 4. **Solve** by dividing both sides by $$240$$: $$p = \frac{60}{240}$$, so $$p = 0.25$$. 5. **Convert to percent form:** $$p = 25\%$$. 6. **Check** if this makes sense: Yes, $$25\%$$ is one-fourth, and $$60$$ is one-fourth of $$240$$, so $$25\%$$ makes sense. 7. **Answer:** Of the total calories in each muffin, $$25\%$$ is fat.

Example: Solving for the Percent of Calories from Fat in Muffins

The percent change describes how much a quantity has increased or decreased over a specified period. To calculate the percent change, follow a two-step procedure. First, calculate the absolute amount of change by finding the difference between the new amount and the original amount ($$\text{change} = \text{new amount} - \text{original amount}$$). Second, determine what percent the amount of change represents relative to the original amount by translating the conceptual question "change is what percent of the original amount?" into an algebraic percent equation.

Calculating Percent Change

To translate and solve the percent equation 'What number is $$45\%$$ of $$84$$?': 1. Assign a variable: Let $$n$$ = the number. 2. Translate: 'What number' becomes $$n$$, 'is' becomes $$=$$, '$$45\%$$' becomes $$0.45$$, and 'of $$84$$' becomes $$	imes 84$$. 3. Write the equation: $$n = 0.45 	imes 84$$. 4. Multiply: $$n = 37.8$$. Therefore, $$37.8$$ is $$45\%$$ of $$84$$.

Solving 'What number is 45% of 84?'

To translate and solve the percent equation '$$8.5\%$$ of what amount is \$4.76?': 1. Assign a variable: Let $$n$$ = the amount. 2. Translate: '$$8.5\%$$' becomes $$0.085$$, 'of what amount' becomes $$	imes n$$, 'is' becomes $$=$$, and '\$4.76' becomes $$4.76$$. 3. Write the equation: $$0.085 	imes n = 4.76$$. 4. Solve: Divide both sides by $$0.085$$, which gives $$n = 56$$. Therefore, $$8.5\%$$ of \$56 is \$4.76.

Solving '8.5% of what amount is \$4.76?'

To translate and solve the percent equation '$$168$$ is what percent of $$112$$?': 1. Assign a variable: Let $$p$$ = the percent. 2. Translate: '$$168$$' is $$168$$, 'is' becomes $$=$$, 'what percent' becomes $$p$$, and 'of $$112$$' becomes $$	imes 112$$. 3. Write the equation: $$168 = p 	imes 112$$. 4. Solve: Divide both sides by $$112$$, which yields $$1.5 = p$$. 5. Convert to percent form: Multiply by $$100$$ to get $$150\%$$. Therefore, $$168$$ is $$150\%$$ of $$112$$.

Solving '168 is what percent of 112?'

Apply the seven-step problem-solving strategy to a percent application involving finding the base amount. Problem: The label on Audrey's yogurt said that one serving provided $$12$$ grams of protein, which is $$24\%$$ of the recommended daily amount. What is the total recommended daily amount of protein? 1. Identify: Find the total recommended amount of protein. 2. Name: Let $$a$$ = total amount of protein. 3. Translate: '$$12$$ grams is $$24\%$$ of the total amount' translates to $$12 = 0.24 	imes a$$. 4. Solve: Divide both sides by $$0.24$$, which gives $$a = 50$$. 5. Check: $$24\%$$ is about one-fourth, and $$12$$ is about one-fourth of $$50$$, so the answer makes sense. 6. Answer: The amount of protein that is recommended is $$50$$ grams.

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