A smartphone originally costs $800 but is now sold for $ 680 \$680 . Determine the percent decrease in price.

Percent decrease = 15 % 15\% 15%

Percent decrease = 0.15 % 0.15\% 0.15%

Percent decrease = 0.18 % 0.18\% 0.18%

Percent decrease = 17.6 % 17.6\% 17.6%

A square garden has sides of 8 ft ⁡ 8\operatorname{ft} . The gardener reduces each side by 1 ft ⁡ 1\operatorname{ft} to make space for a pathway. What is the percent decrease in the perimeter of the garden?

Percent decrease = 12.5 % 12.5\% 12.5%

Percent decrease = 0.125 % 0.125\% 0.125%

Percent decrease = 62.5 % 62.5\% 62.5%

Percent decrease = 6.25 % 6.25\% 6.25%

3. Solving Word Problems

Percent Problem Solving

3. Solving Word Problems

Percent Problem Solving: Videos & Practice Problems

Video Lessons

Topic summary

Understanding percents involves converting percentages to decimals and solving equations like $x = 0.1 \times 50$ . Percent change measures the difference between values as a percent of the original, calculated by $PercentChange = \frac{New - Original}{Original} × 100%$ . This concept helps quantify increases or decreases, such as population shifts or price changes, providing context beyond absolute differences. Mastery of these calculations enhances problem-solving skills in real-world applications involving ratios, proportions, and percent-based reasoning.

concept

Translating Percent Problems

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Translating Percent Problems Video Summary

Understanding how to solve word problems involving percents is essential for everyday calculations, from determining earnings to interpreting exam scores. A percent represents a part per 100, which can be expressed as a fraction or a decimal. For example, 10% means 10 parts out of 100, which as a decimal is 0.1, obtained by dividing 10 by 100 and moving the decimal point two places to the left.

When translating percent problems into equations, it is important to recognize that the word "is" translates to "equals," and "of" translates to multiplication. For instance, the statement "a number is 10% of 50" can be written as $x = 10\% \times 50$. Since multiplying directly by a percent is not straightforward, convert the percent to decimal form first: \$10\% = 0.1$. Thus, the equation becomes $x = 0.1 \times 50$, which simplifies to $x = 5\(. This means 5 is 10% of 50.

Most percent problems follow the general formula:

\[\text{part} = \text{percent (in decimal)} \times \text{whole}\]

The goal is to identify which value is missing—part, percent, or whole—and solve accordingly. For example, if given "4 is 5% of what number?", translate it to \)4 = 0.05 \times x$. Solving for $x\( involves dividing both sides by 0.05:

\[x = \frac{4}{0.05} = 80\]

This means 4 is 5% of 80.

In another case, "35 is what percent of 28?" translates to:

\[35 = x \times 28\]

Solving for \)x\( gives:

\[x = \frac{35}{28} = 1.25\]

Since \)x$ is in decimal form, convert it back to a percent by moving the decimal point two places to the right:

\[x = 1.25 \times 100\% = 125\%\]

This indicates that 35 is 125% of 28, meaning it is greater than the whole amount. Percents can exceed 100% when the part is larger than the whole, such as scoring extra credit on a test.

Visualizing percents helps clarify their meaning: 0% means none of the whole, 100% means the entire whole, and values between 0% and 100% represent partial amounts. Values above 100% indicate more than the whole. Regardless of the percent value, the problem-solving approach remains consistent: translate the problem into an equation using the relationships between part, percent, and whole, then solve for the unknown variable.

example

Translating Percent Problems Example 1

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Translating Percent Problems Example 1 Video Summary

When solving problems involving percentages, it's important to translate the words into mathematical expressions accurately. For example, if a problem states that a number is 50% of 18, we represent the unknown number as x. The word "is" translates to an equals sign, so the equation becomes x = 50% of 18. Since percentages must be converted to decimals for calculations, 50% becomes 0.5. Therefore, the equation is x = 0.5 × 18, which simplifies to x = 9.

Similarly, if a number is 18% of 50, we again let the number be x. Converting 18% to decimal form gives 0.18, so the equation is x = 0.18 × 50, which also simplifies to x = 9.

The reason both parts yield the same result lies in the commutative property of multiplication, which states that the order of factors does not affect the product. Since a percent means "per 100," converting percentages to decimals involves dividing by 100. Thus, the expressions can be rewritten as:

\[x = \frac{50}{100} \times 18\]

and

\[x = \frac{18}{100} \times 50\]

Both simplify to:

\[x = \frac{50 \times 18}{100}\]

This demonstrates that multiplying 50% by 18 or 18% by 50 results in the same value. Understanding this property is useful for simplifying percentage calculations, especially when performing mental math without a calculator. For instance, calculating 18% of 80 might seem challenging, but recognizing the commutative property allows you to find an easier equivalent calculation.

example

Translating Percent Problems Example 2

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Translating Percent Problems Example 2 Video Summary

Caroline earns \$3,452 per month and pays \$1,142 in rent each month. To determine what percentage of her monthly paycheck goes towards rent, we use the concept of percent as a ratio between a part and the whole. By dividing the rent amount by her total monthly income, we calculate the fraction of her paycheck allocated to rent:

\[\frac{1,142}{3,452} \approx 0.33\]

Converting this decimal to a percentage involves multiplying by 100, resulting in approximately 33%. This means that about one-third of Caroline's monthly income is spent on rent, illustrating how percentages can effectively represent portions of a total amount in real-life financial contexts.

Problem

Saphia's lunch bill was $dollar sign 17.65$ . She wants to leave an $18 percent sign$ tip. How much should the tip be?

$dollar sign 31.80$

$dollar sign 14.47$

$dollar sign 3.18$

$dollar sign 14.18$

Problem

Carson bought some new sneaker on sale for $dollar sign 250$ . The sale price was $65 percent sign$ of the original price. What was the original price?

$dollar sign 412.50$

$dollar sign 384.62$

$dollar sign 162.50$

$dollar sign 337.50$

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Translating Percent Problems Example 3

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Translating Percent Problems Example 3 Video Summary

A circle graph, or pie chart, visually represents the distribution of the world population by continent. To determine the percentage of people who do not live in Asia, subtract Asia's population percentage from the total 100%. Since Asia accounts for 59%, the remaining percentage is 100% - 59% = 41%. Thus, 41% of the global population resides outside Asia.

Given the total world population of 8,142,000,000, calculating the number of people living in Africa involves multiplying this total by Africa's population percentage, 19%, expressed as a decimal (0.19). The calculation is:

\[8,142,000,000 \times 0.19 = 1,546,980,000\]

This means approximately 1.547 billion people live in Africa.

To find the percentage of the world population living in Oceania, use the fact that all continental percentages sum to 100%. The known percentages are 19% (Africa), 7%, 5x% (South America), 9%, x% (Oceania), and 59% (Asia). Setting up the equation:

\[100 = 19 + 7 + 5x + 9 + x + 59\]

Combine like terms:

\[100 = 94 + 6x\]

Subtract 94 from both sides:

\[6 = 6x\]

Divide both sides by 6:

\[x = 1\]

Therefore, Oceania represents 1% of the world population.

Since South America is represented as 5 times x, and x equals 1%, South America's population percentage is:

\[5 \times 1\% = 5\%\]

Hence, 5% of the world population lives in South America.

Problem

If sales tax on a \$36 pair of shoes is \$1.26, find the sales tax on a $dollar sign 250$ pair of shoes.

$dollar sign 8.75$

$dollar sign 3.15$

$dollar sign 1.26$

$dollar sign 7.49$

Problem

Suppose the local sales tax rate is $6.5 percent sign$ and you buy a car for $dollar sign 12 comma 500$ . What is the car's total cost?
Hint: Find how much tax is due first.

$dollar sign 14562.5$

$dollar sign 12506.5$

$dollar sign 13312.5$

$dollar sign 14000$

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Translating Percent Problems Example 4

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Translating Percent Problems Example 4 Video Summary

To calculate the total cost of purchasing three packs of laundry detergent priced at \$6.49 each with a \$1 discount per pack when buying three, and including a 7.5% sales tax, we follow a two-step process. First, determine the discounted price for the three packs. Since each pack costs \$6.49 and the coupon provides a \$1 discount per pack, the price per pack after the discount is \$6.49 − \$1 = \$5.49. Multiplying this by three packs gives the subtotal before tax: \$5.49 × 3 = \$16.47.

Next, calculate the sales tax on the subtotal. The sales tax rate of 7.5% is converted to decimal form as 0.075. Multiplying the subtotal by the tax rate gives the tax amount: \$16.47 × 0.075 = \$1.24.

Finally, add the sales tax to the subtotal to find the total amount charged: \$16.47 + \$1.24 = \$17.71. This total reflects the cost of three packs of laundry detergent after applying the coupon discount and including the sales tax.

concept

Percent Change

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Percent Change Video Summary

Percent change is a valuable way to describe how values increase or decrease relative to their original amounts, expressed as a percentage. It quantifies the difference between two values by comparing the change to the initial value, making it easier to understand the significance of that change in various contexts, such as price fluctuations or population shifts.

To calculate percent change, start by finding the difference between the new value and the original value using the formula:

\[\text{Change} = \text{New Value} - \text{Original Value}\]

This difference can be positive, indicating a percent increase, or negative, indicating a percent decrease. Next, divide this change by the original value to find the ratio of change relative to the starting point:

\[\text{Ratio of Change} = \frac{\text{Change}}{\text{Original Value}}\]

Finally, convert this ratio to a percentage by multiplying by 100 or moving the decimal point two places to the right:

\[\text{Percent Change} = \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100\%\]

For example, if the price of an item rises from \$50 to \$60, the change is \$10. Dividing \$10 by the original \$50 gives 0.2, which converts to a 20% increase. Conversely, if a rabbit population decreases from 12,000 to 8,000, the change is -4,000. Dividing -4,000 by 12,000 yields -0.33, or a 33% decrease. When expressing decreases, the negative sign is often omitted because the term "decrease" already implies a reduction.

Using percent change rather than absolute change allows for meaningful comparisons across different scales. For instance, a loss of 4,000 rabbits is significant if the population is 12,000 but less so if the population were 120,000. Percent change standardizes this measure, providing a clearer understanding of the magnitude of change relative to the original value.

Problem

A city's population grew from $120 comma 000$ to $138 comma 000$ people in one year. What is the percent increase in population?

$0.15\%$

$13\%$

$0.13\%$

Problem

A smartphone originally costs \$800 but is now sold for $dollar sign 680$ . Determine the percent decrease in price.

Percent decrease = $15\%$

Percent decrease = $0.15\%$

Percent decrease = $0.18\%$

Percent decrease = $17.6\%$

Problem

Last season, a farmer harvested $of 500 kg$ of rice. Due to improved techniques, the harvest is estimated to increase by $20 percent sign$ this season. How much rice is the farmer looking to harvest this season?

$400\operatorname{kg}$

$500\operatorname{kg}$

$600\operatorname{kg}$

$520\operatorname{kg}$

Problem

A reservoir's water level decreased by $20 percent sign$ over the summer due to evaporation. If the water level is currently at $120$ million liters, how much water was there initially?

$24$ million liters

$150$ million liters.

$96$ million liters.

$30$ million liters.

Problem

A square garden has sides of $8 ft$ . The gardener reduces each side by $1 ft$ to make space for a pathway. What is the percent decrease in the perimeter of the garden?

Percent decrease = $0.125\%$

Percent decrease = $62.5\%$

Percent decrease = $6.25\%$

Percent decrease = $12.5\%$

Here’s what students ask on this topic:

To convert a percent to a decimal, you divide the percent value by 100. This is equivalent to moving the decimal point two places to the left. For example, 10% becomes 0.10 as a decimal. This conversion is essential because when solving percent problems, you multiply the decimal form of the percent by the whole amount to find the part or the unknown value. For instance, if you want to find 10% of 50, you convert 10% to 0.10 and then multiply: $x = 0.10 \times 50$ , which equals 5. This method simplifies calculations and helps in translating word problems into solvable equations.

The formula for percent change is $PercentChange = \frac{New - Original}{Original} \times 100%$ . To apply this, first find the difference between the new value and the original value. Then, divide this difference by the original value to get a decimal. Finally, multiply by 100 to convert it to a percent. For example, if a price changes from \$50 to \$60, the change is \$10. Dividing 10 by 50 gives 0.2, and multiplying by 100 gives a 20% increase. This formula helps quantify how much a value has increased or decreased relative to its original amount.

When you know the part (partial amount) and the percent, but need to find the whole, you set up the equation as $Part = Percent \times Whole$ . Convert the percent to decimal form first. For example, if 4 is 5% of a number, write $4 = 0.05 \times x$ . To find $x$ , divide both sides by 0.05: $x = \frac{4}{0.05} = 80$ . So, 4 is 5% of 80. This approach is useful for finding the total when given a part and its percentage.

A percent greater than 100% means the part is larger than the whole. For example, if 35 is what percent of 28, you set up $35 = x \times 28$ . Solving for $x$ gives $x = \frac{35}{28} = 1.25$ , which is 125% when converted to percent form. This indicates that 35 is 125% of 28, meaning it is 25% more than the whole. In real-world terms, this could represent extra credit or an increase beyond the original amount.

A negative percent change indicates a decrease from the original value. For example, if a rabbit population decreases from 12,000 to 8,000, the change is $8,000 - 12,000 = - 4,000$ . Dividing by the original population gives $\frac{-}{4,000} = - 0.33$ , or a 33% decrease. This means the population shrank by one-third. When stating the result, you can say the population decreased by 33%, dropping the negative sign because 'decrease' already implies a reduction. This helps quantify the extent of decline relative to the original amount.

Percent change is important because it provides context to the size of the change relative to the original value. An absolute change, like a decrease of 4,000 rabbits, might seem large or small depending on the total population. For example, losing 4,000 rabbits from 12,000 is significant (33% decrease), but the same loss from 120,000 is minor (about 3.3%). Percent change standardizes the comparison, allowing you to understand the impact proportionally, which is especially useful in economics, biology, and other fields where relative changes matter more than absolute numbers.

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Patrick Ford

Physics and Math Lead Instructor

Percent Problem Solving: Videos & Practice Problems

Translating Percent Problems

Translating Percent Problems Video Summary

Translating Percent Problems Example 1

Translating Percent Problems Example 1 Video Summary

Translating Percent Problems Example 2

Translating Percent Problems Example 2 Video Summary

Saphia's lunch bill was $dollar sign 17.65$ . She wants to leave an $18 percent sign$ tip. How much should the tip be?

Carson bought some new sneaker on sale for $dollar sign 250$ . The sale price was $65 percent sign$ of the original price. What was the original price?

Translating Percent Problems Example 3

Translating Percent Problems Example 3 Video Summary

If sales tax on a \$36 pair of shoes is \$1.26, find the sales tax on a $dollar sign 250$ pair of shoes.

Suppose the local sales tax rate is $6.5 percent sign$ and you buy a car for $dollar sign 12 comma 500$ . What is the car's total cost?
Hint: Find how much tax is due first.

Translating Percent Problems Example 4

Translating Percent Problems Example 4 Video Summary

Percent Change

Percent Change Video Summary

A city's population grew from $120 comma 000$ to $138 comma 000$ people in one year. What is the percent increase in population?

A smartphone originally costs \$800 but is now sold for $dollar sign 680$ . Determine the percent decrease in price.

Last season, a farmer harvested $of 500 kg$ of rice. Due to improved techniques, the harvest is estimated to increase by $20 percent sign$ this season. How much rice is the farmer looking to harvest this season?

A reservoir's water level decreased by $20 percent sign$ over the summer due to evaporation. If the water level is currently at $120$ million liters, how much water was there initially?

A square garden has sides of $8 ft$ . The gardener reduces each side by $1 ft$ to make space for a pathway. What is the percent decrease in the perimeter of the garden?

Here’s what students ask on this topic:

How do you convert a percent to a decimal for solving percent problems?

What is the formula for calculating percent change and how is it applied?

How do you solve a problem where you know the part and the percent but need to find the whole?

What does it mean when a percent is greater than 100% in a problem?

How do you interpret a negative percent change in a real-world context?

Why is it important to use percent change instead of absolute change when comparing values?

Your Intermediate Algebra tutors

Percent Problem Solving: Videos & Practice Problems

Translating Percent Problems

Translating Percent Problems Video Summary

Translating Percent Problems Example 1

Translating Percent Problems Example 1 Video Summary

Translating Percent Problems Example 2

Translating Percent Problems Example 2 Video Summary

Saphia's lunch bill was $17.65\$17.65. She wants to leave an 18%18\% tip. How much should the tip be?

Carson bought some new sneaker on sale for $250\$250. The sale price was 65%65\% of the original price. What was the original price?

Translating Percent Problems Example 3

Translating Percent Problems Example 3 Video Summary

If sales tax on a \$36 pair of shoes is \$1.26, find the sales tax on a $250\$250 pair of shoes.

Suppose the local sales tax rate is 6.5%6.5\% and you buy a car for $12,500\$12,500. What is the car's total cost?Hint: Find how much tax is due first.

Translating Percent Problems Example 4

Translating Percent Problems Example 4 Video Summary

Percent Change

Percent Change Video Summary

A city's population grew from 120,000120,000 to 138,000138,000 people in one year. What is the percent increase in population?

A smartphone originally costs \$800 but is now sold for $680\$680. Determine the percent decrease in price.

Last season, a farmer harvested 500kg500\operatorname{kg} of rice. Due to improved techniques, the harvest is estimated to increase by 20%20\% this season. How much rice is the farmer looking to harvest this season?

A reservoir's water level decreased by 20%20\% over the summer due to evaporation. If the water level is currently at 120120 million liters, how much water was there initially?

A square garden has sides of 8ft8\operatorname{ft}. The gardener reduces each side by 1ft1\operatorname{ft} to make space for a pathway. What is the percent decrease in the perimeter of the garden?

Here’s what students ask on this topic:

How do you convert a percent to a decimal for solving percent problems?

How do you convert a percent to a decimal for solving percent problems?

What is the formula for calculating percent change and how is it applied?

What is the formula for calculating percent change and how is it applied?

How do you solve a problem where you know the part and the percent but need to find the whole?

How do you solve a problem where you know the part and the percent but need to find the whole?

What does it mean when a percent is greater than 100% in a problem?

What does it mean when a percent is greater than 100% in a problem?

How do you interpret a negative percent change in a real-world context?

How do you interpret a negative percent change in a real-world context?

Why is it important to use percent change instead of absolute change when comparing values?

Why is it important to use percent change instead of absolute change when comparing values?

Your Intermediate Algebra tutors

Saphia's lunch bill was $dollar sign 17.65$ . She wants to leave an $18 percent sign$ tip. How much should the tip be?

Carson bought some new sneaker on sale for $dollar sign 250$ . The sale price was $65 percent sign$ of the original price. What was the original price?

If sales tax on a \$36 pair of shoes is \$1.26, find the sales tax on a $dollar sign 250$ pair of shoes.

Suppose the local sales tax rate is $6.5 percent sign$ and you buy a car for $dollar sign 12 comma 500$ . What is the car's total cost?
Hint: Find how much tax is due first.

A city's population grew from $120 comma 000$ to $138 comma 000$ people in one year. What is the percent increase in population?

A smartphone originally costs \$800 but is now sold for $dollar sign 680$ . Determine the percent decrease in price.

Last season, a farmer harvested $of 500 kg$ of rice. Due to improved techniques, the harvest is estimated to increase by $20 percent sign$ this season. How much rice is the farmer looking to harvest this season?

A reservoir's water level decreased by $20 percent sign$ over the summer due to evaporation. If the water level is currently at $120$ million liters, how much water was there initially?

A square garden has sides of $8 ft$ . The gardener reduces each side by $1 ft$ to make space for a pathway. What is the percent decrease in the perimeter of the garden?