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%

How do you solve a percent problem when the percent is unknown?

When the percent is unknown, set up an equation where the partial amount equals the percent (as a decimal) times the whole amount. For example, if 35 is what percent of 28, write 35 = x × 28 , where x is the decimal form of the percent. Solve for x by dividing both sides by 28: x = 35 28 = 1.25 . Then convert to percent by multiplying by 100: 1.25 imes 100 = 125 ext{%} . So, 35 is 125% of 28.

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 values. 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 quantity—part, percent, or whole—is unknown and solve for it accordingly.

For example, if four is 5% of what number, translate this as:

\[4 = 0.05 \times x\]

Solving for \)x\( involves dividing both sides by 0.05:

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

This means four is 5% of 80.

In another case, if 35 is what percent of 28, the equation is:

\[35 = x \times 28\]

Solving for \)x\(:

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

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 more than the whole amount. Percents can exceed 100% when the part is greater than the whole, such as scoring extra credit on a test.

Visualizing percents helps clarify their meaning: 0% represents none of the whole, 100% represents the entire whole, and values between 0% and 100% represent partial amounts. Values above 100% indicate more than the whole.

In summary, solving percent problems involves translating word statements into equations using the relationships between part, percent, and whole, converting percents to decimals for calculation, and interpreting the results in context. This approach applies whether the percent is less than, equal to, or greater than 100%.

example

Translating Percent Problems Example 1

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Translating Percent Problems Example 1 Video Summary

When solving problems involving percentages, it's helpful to translate the words into mathematical expressions. For example, if a number is 50% of 18, we represent the unknown number as x. The phrase "is" translates to "equals," so the equation becomes x = 50% of 18. Since percentages are 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 think of it as 80% of 18, which can be easier to compute.

example

Translating Percent Problems Example 2

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44s

Translating Percent Problems Example 2 Video Summary

Caroline earns \$3,452 each month and pays \$1,142 in rent. To determine what percentage of her monthly paycheck goes toward rent, we compare the rent amount to her total income. A percent represents a ratio between a part and the whole, so dividing the rent by the total income gives the fraction of her paycheck spent on rent.

The calculation is:

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

Converting this decimal to a percentage by multiplying by 100 yields approximately 33%. This means that about one-third of Caroline's monthly income is allocated to rent expenses, illustrating how percentages can effectively represent proportions of a whole 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?

$dollar sign 412.50$

$dollar sign 384.62$

$dollar sign 162.50$

$dollar sign 337.50$

example

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 calculated as \$100\% - 59\% = 41\%\(. This means 41% of the global population resides outside Asia.

Given the total world population of 8,142,000,000, the number of people living in Africa can be found by multiplying this total by Africa's population percentage, 19%. Converting 19% to decimal form (0.19) and performing the multiplication yields:

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

Thus, 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%, 9%, x%, and 59% (Asia), where x represents the unknown percentage for Oceania. 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:

\[x = 1\]

This shows Oceania accounts for 1% of the world population.

Finally, South America's population percentage is expressed as \)5x$. Since $x = 1\%$, South America represents:

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

Therefore, 5% of the global 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.

$dollar sign 14562.5$

$dollar sign 12506.5$

$dollar sign 13312.5$

$dollar sign 14000$

example

Translating Percent Problems Example 4

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Translating Percent Problems Example 4 Video Summary

When purchasing three packs of laundry detergent priced at \$6.49 each, applying a coupon that offers \$1 off per pack significantly reduces the total cost. First, calculate the discounted price per pack by subtracting the \$1 coupon value from the original price, resulting in \$5.49 per pack. Multiplying this discounted price by three gives the subtotal before tax: \$16.47.

Next, to determine the sales tax amount, convert the sales tax rate of 7.5% to its decimal form, 0.075, and multiply it by the subtotal. This calculation yields a sales tax of \$1.24. Adding the sales tax to the subtotal provides the final total cost for the purchase, which is \$17.71.

Mathematically, the process can be expressed as:

\[\text{Discounted price per pack} = 6.49 - 1 = 5.49\]

\[\text{Subtotal for three packs} = 5.49 \times 3 = 16.47\]

\[\text{Sales tax} = 16.47 \times 0.075 = 1.23525 \approx 1.24\]

\[\text{Total cost} = 16.47 + 1.24 = 17.71\]

This approach highlights the importance of understanding how discounts and sales tax interact to affect the final purchase price, enabling better budgeting and financial planning when shopping.

concept

Percent Change

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Percent Change Video Summary

Percent change is a valuable way to describe how values vary over time by expressing the difference between two amounts as a percentage of the original value. This concept helps quantify increases or decreases in a way that is easy to understand and compare across different contexts. When the change is positive, it is called a percent increase; when negative, it is a percent decrease.

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 reflects how much the quantity has increased or decreased. 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. This means the price increased by 20% relative to its original cost.

Conversely, if a 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. Here, the negative sign indicates a reduction, so we say the population decreased by 33%.

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 by showing the relative size of the change.

Understanding percent change is essential for interpreting data in economics, biology, finance, and everyday life, as it provides a clear measure of how quantities evolve over time in a way that is easy to communicate and analyze.

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?

$0.15\%$

$15\%$

$13\%$

$0.13\%$

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 $500 kg$ 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?

$400 kg$

$500 kg$

$600 kg$

$520 kg$

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 solve a percent problem where you need to find the whole amount, start by identifying the parts given: the partial amount and the percent. Translate the problem into an equation using the fact that "is" means equals and "of" means multiplication. For example, if 4 is 5% of a number, write it as $4 = 0.05 \times x$ , where 0.05 is the decimal form of 5%. Then, solve for $x$ by dividing both sides by 0.05: $x = \frac{4}{0.05} = 80$ . So, 4 is 5% of 80. This method works for any similar problem where the whole amount is unknown.

To calculate percent change, first find the difference between the new value and the original value by subtracting the original from the new: $Change = New - Original$ . Next, divide this change by the original value to find the ratio: $Ratio = \frac{Change}{Original}$ . Finally, convert this ratio to a percent by multiplying by 100 or moving the decimal point two places to the right. For example, if the price changes from \$50 to \$60, the change is \$10, the ratio is $\frac{10}{50} = 0.2$ , and the percent change is 20%. A positive percent change indicates an increase, while a negative value indicates a decrease.

Converting between percents and decimals is essential in percent problem solving. To convert a percent to a decimal, divide the percent value by 100, which moves the decimal point two places to the left. For example, 10% becomes 0.10. This decimal form is used in calculations like multiplication. Conversely, to convert a decimal to a percent, multiply by 100 or move the decimal point two places to the right. For instance, 1.25 as a decimal becomes 125%. This conversion is important when interpreting results, especially when the percent is greater than 100%, indicating a value larger than the whole.

Percent values can be greater than 100% when the part being measured exceeds the whole. For example, if 35 is what percent of 28, the calculation yields 125%, meaning 35 is 125% of 28. In context, this means the quantity is larger than the original or whole amount. This often happens in real-world scenarios like scoring more than the total points possible due to extra credit or price increases beyond the original price. Understanding that percents over 100% represent more than the whole helps interpret such results accurately.

When the percent is unknown, set up an equation where the partial amount equals the percent (as a decimal) times the whole amount. For example, if 35 is what percent of 28, write $35 = x \times 28$ , where $x$ is the decimal form of the percent. Solve for $x$ by dividing both sides by 28: $x = \frac{35}{28} = 1.25$ . Then convert to percent by multiplying by 100: $1.25 imes 100 = 125ext{%}$ . So, 35 is 125% of 28.

Percent change provides context to the size of a change relative to the original value, which absolute change alone cannot. For example, a decrease of 4,000 rabbits might seem large, but if the original population was 120,000, it is a small change (about 3.33%). However, if the original population was 12,000, the same decrease is significant (33%). Percent change normalizes the difference, allowing meaningful comparisons across different scales. This is why percent change is widely used in economics, population studies, and other fields to measure increases or decreases effectively.

Your Beginning & 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 $dollar sign 17.65$ . 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 $500 kg$ 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 solve a word problem involving percents where you need to find the whole amount?

What is the process to calculate percent change between two values?

How do you convert a percent to a decimal and vice versa in percent problem solving?

Why can percent values be greater than 100%, and what does that mean in context?

How do you solve a percent problem when the percent is unknown?

What is the significance of using percent change instead of absolute change in real-world problems?

Your Beginning & 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 solve a word problem involving percents where you need to find the whole amount?

How do you solve a word problem involving percents where you need to find the whole amount?

What is the process to calculate percent change between two values?

What is the process to calculate percent change between two values?

How do you convert a percent to a decimal and vice versa in percent problem solving?

How do you convert a percent to a decimal and vice versa in percent problem solving?

Why can percent values be greater than 100%, and what does that mean in context?

Why can percent values be greater than 100%, and what does that mean in context?

How do you solve a percent problem when the percent is unknown?

How do you solve a percent problem when the percent is unknown?

What is the significance of using percent change instead of absolute change in real-world problems?

What is the significance of using percent change instead of absolute change in real-world problems?

Your Beginning & 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 $500 kg$ 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?