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

Percent decrease = 0.15 % 0.15\%

Percent decrease = 0.18 % 0.18\%

Percent decrease = 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\%

Percent decrease = 0.125 % 0.125\%

Percent decrease = 62.5 % 62.5\%

Percent decrease = 6.25 % 6.25\%

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

Percent change measures how much a value has increased or decreased relative to its original amount. To calculate it, first find the change by subtracting the original value from the new value: change = new − original . Then, divide this change by the original value to get a decimal: decimalChange = change original . Finally, convert the decimal 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, and the percent change is 10 50 = 0.2 , which is 20%. A positive percent change indicates an increase, while a negative value indicates a decrease.

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 for calculations, such as 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 skills enhances problem-solving with ratios, decimals, and percent applications in real-world scenarios.

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 multiplication with 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:

\[x = 125\%\]

This indicates that 35 is 125% of 28, which is possible because the part can be greater than the whole, resulting in a percent greater than 100%. For example, scoring 125% on a test could mean achieving full marks plus extra credit.

Understanding percents in context helps clarify their meaning: 0% means none of the whole, values between 0% and 100% represent a part of the whole, 100% means the entire whole, and values above 100% indicate more than the whole. Regardless of the percent value, the problem-solving process 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 and write x = 18% of 50. 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 percentages represent a fraction out of 100, converting 50% to decimal is equivalent to dividing 50 by 100, and similarly for 18%. 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 a percentage by a number is commutative in this context, leading to the same product regardless of the order. Understanding this property can simplify mental math, especially when calculating percentages without a calculator. For instance, calculating 18% of 50 might seem complex, but recognizing it as the same as 50% of 18 makes the problem easier to solve quickly.

example

Translating Percent Problems Example 2

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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

To determine if the percentages in each pie graph are correct, we need to verify that the sum of all the percentages equals 100%, since 100% represents the whole pie.

For part A, the percentages are 30%, 20%, and 40%. Adding these gives:

\$30 + 20 = 50\(

\)50 + 40 = 90\(

Since 90% does not equal 100%, part A does not correctly represent a pie graph.

For part B, the percentages are 30%, 30%, and 30%. Adding these gives:

\)30 + 30 = 60\(

\)60 + 30 = 90\(

Again, 90% is less than 100%, so part B also does not correctly represent a pie graph.

For part C, the percentages are 25%, 25%, 10%, and 40%. Adding these gives:

\)25 + 25 = 50\(

\)10 + 40 = 50\(

\)50 + 50 = 100$

Since the total is exactly 100%, part C correctly represents a pie graph.

example

Translating Percent Problems Example 4

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Translating Percent Problems Example 4 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 (59%) from the total 100%, resulting in 41%. This means 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:

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

Thus, approximately 1.55 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% (Europe), 5x% (South America), 9% (North America), 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:

\[x = 1\]

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

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

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

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 5

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Translating Percent Problems Example 5 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. To calculate the discounted price for three packs, subtract the \$1 discount from the original price per pack, resulting in \$5.49 per pack. Multiplying this discounted price by three gives the subtotal before tax: \$16.47.

Next, to determine the total cost including sales tax, apply the sales tax rate of 7.5%, expressed as the decimal 0.075, to the subtotal. Multiplying \$16.47 by 0.075 calculates the sales tax amount, which is \$1.24. Adding this sales tax to the subtotal yields the final total cost of \$17.71.

This process highlights the importance of understanding how discounts and sales tax interact in real-world purchasing scenarios. The key formulas used are:

Subtotal with discount: \[(6.49 - 1) \times 3 = 5.49 \times 3 = 16.47\]

Sales tax amount: \[16.47 \times 0.075 = 1.24\]

Total cost including tax: \[16.47 + 1.24 = 17.71\]

By breaking down the problem into calculating the discounted subtotal and then applying the sales tax, one can accurately determine the final amount charged for multiple items with discounts and taxes involved.

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.

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?

$0.15 percent sign$

$15 percent sign$

$13 percent sign$

$0.13 percent sign$

Problem

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

Percent decrease = $15 percent sign$

Percent decrease = $0.15 percent sign$

Percent decrease = $0.18 percent sign$

Percent decrease = $17.6 percent sign$

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 = $0.125 percent sign$

Percent decrease = $62.5 percent sign$

Percent decrease = $6.25 percent sign$

Percent decrease = $12.5 percent sign$

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 partial amount and the percent given. 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 $4 = 0.05 \times x$ , where 0.05 is the decimal form of 5%. Then, solve for $x$ by dividing both sides by 0.05: $x = \frac{4}{0.05}$ . This gives $x = 80$ , meaning 4 is 5% of 80. Always convert percents to decimals before multiplying, and isolate the variable to find the whole amount.

Percent change measures how much a value has increased or decreased relative to its original amount. To calculate it, first find the change by subtracting the original value from the new value: $change = new - original$ . Then, divide this change by the original value to get a decimal: $decimalChange = \frac{change}{original}$ . Finally, convert the decimal 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, and the percent change is $\frac{10}{50} = 0.2$ , which is 20%. A positive percent change indicates an increase, while a negative value indicates a decrease.

Converting between percents and decimals is essential in solving percent problems. 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 $0.10$ . Conversely, to convert a decimal to a percent, multiply the decimal by 100 or move the decimal point two places to the right. For instance, $0.75$ becomes 75%. This conversion allows you to perform multiplication and division accurately when solving equations involving percents, such as finding parts of a whole or calculating percent changes.

A percent value greater than 100% means the part is larger than the whole. For example, if 35 is what percent of 28, the calculation gives $\frac{35}{28} = 1.25$ or 125%. This indicates that 35 is 125% of 28, meaning it exceeds the whole amount by 25%. In real-world terms, this could represent extra credit on a test or an increase beyond the original quantity. Understanding that percents can exceed 100% helps interpret situations where the quantity surpasses the reference 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. When stating the result, you can say the population decreased by 33%, dropping the negative sign because "decrease" implies a negative change. This helps quantify how significant the reduction is relative to the original amount, which is more informative than just stating the absolute change.

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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

Translating Percent Problems Example 4

Translating Percent Problems Example 4 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 5

Translating Percent Problems Example 5 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 method to calculate percent change between two values?

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

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

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

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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

Translating Percent Problems Example 4

Translating Percent Problems Example 4 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 5

Translating Percent Problems Example 5 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 method to calculate percent change between two values?

What is the method 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?

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

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

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

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

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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?