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Percentage – Word Problems in GRE
Introduction: The GRE (Graduate Record Examination) is a standardized test required for admission into many graduate schools in the United States and other countries. One of the critical topics in the GRE quantitative reasoning section is percentage word problems. In this extended concept, we will discuss various percentage word problems frequently appearing on the GRE and strategies to solve them effectively and efficiently.
Percentage Basics: Before diving into word problems, review the basics. A percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is “%,” which means “per hundred.” For example, 25% can be written as 25/100, which reduces to 1/4 or 0.25.
To convert a fraction to a percentage, divide the numerator by the denominator and multiply the result by 100. To convert a percentage to a fraction, divide the percentage by 100 and then simplify the resulting fraction if possible.
Common Types of Percentage Word Problems in GRE:
- Finding the percentage of a given value
- Calculating the original value given the percentage and the final value
- Determining the percentage change (increase or decrease) between two values
- Solving problems involving discounts, markups, or sales tax
- Comparing percentages in context
Strategies for Solving Percentage Word Problems:
- Read the problem carefully: Ensure you understand the problem and identify what is being asked. Look for key information and determine the unknown value(s) you need to find.
- Set up the problem: Translate the word problem into a mathematical expression or equation using the given information.
- Solve the problem: Use appropriate methods, such as proportions, cross-multiplication, or algebraic equations, to solve the problem.
- Check your answer: Make sure your answer is reasonable and in the correct units, if applicable.
Now, let’s discuss each type of percentage word problem in more detail and provide examples and solutions.
1. Finding the percentage of a given value: In these types of problems, you will be given a value and a percentage, and you will need to find a certain percentage of that value.
Example: What is 20% of 150?
Solution: To find 20% of 150, multiply the percentage (as a decimal) by the given value: 20% of 150 = (20/100) × 150 = 0.20 × 150 = 30. Answer: 30
2. Calculating the original value given the percentage and the final value: These problems involve finding the original value when given a percentage and the final value after the percentage has been applied.
Example: If a shirt is on sale for $45 after a 25% discount, what was the original price of the shirt?
Solution: Let x be the original price of the shirt. Since the shirt is on sale for 25% off, it is now being sold for 75% (100% – 25%) of its original price: 0.75x = $45. To find the original price, divide both sides of the equation by 0.75: x = $45 / 0.75 = $60. Answer: $60
3. Determining the percentage change (increase or decrease) between two values: In these problems, you need to calculate the percentage increase or decrease between an initial value and a final value.
Example: If the population of a town increased from 12,000 to 15,000, what is the percentage increase in population?
Solution: First, find the difference between the initial and final populations: 15,000 – 12,000 = 3,000. Next, divide the difference by the initial population to see the proportion of the increase: 3,000 / 12,000 = 0.25. Finally, convert the proportion to a percentage by multiplying by 100: 0.25 × 100 = 25% Answer: 25%
4. Solving problems involving discounts, markups, or sales tax: These problems typically involve calculating the final price of an item after applying a discount, markup, or sales tax.
Example: A pair of shoes originally cost $80. If there is a 15% discount and a 10% sales tax applied, what is the final price?
Solution: First, find the price after the 15% discount: Discount amount = 0.15 × $80 = $12 Discounted price = $80 – $12 = $68 Next, find the amount of sales tax: Sales tax amount = 0.10 × $68 = $6.80 Finally, add the sales tax amount to the discounted price: Final price = $68 + $6.80 = $74.80 Answer: $74.80
4. Comparing percentages in context: These problems involve comparing percentages in different situations, such as comparing the success rates of two treatments or the pass rates of two exams.
Example: In a school, 80% of the students passed the math exam, while 70% of the students passed the science exam. What is the percentage difference between the pass rates of the two exams?
Solution: First, find the difference between the two pass rates: 80% – 70% = 10%. The percentage difference between the pass rates of the two exams is 10%. Answer: 10%
In conclusion, percentage word problems are standard on the GRE and require a strong understanding of percentages and the ability to translate word problems into mathematical expressions or equations. By practicing various types of percentage word problems and using the above strategies, you can improve your problem-solving skills and increase your chances of success in the GRE quantitative reasoning section.
Additional Tips for Solving Percentage Word Problems:
- Pay attention to units: Ensure you are working with the correct units throughout the problem and that your answer is expressed in the appropriate units.
- Use estimation: In some cases, estimating an answer can help determine if your final result is reasonable. This can be especially helpful when working under time pressure, as it can help you identify if you’ve made a calculation error.
- Simplify complex problems: Break down complex percentage word problems into smaller, more manageable steps. This can help you avoid confusion and ensure you address all aspects of the problem.
- Practice, practice, practice: The more percentage word problems you work through, the more confident and skilled you will become in solving them. Make sure to practice problems that cover a wide range of topics and scenarios to build a strong foundation in percentage word problems.
Now, let’s look at some more advanced percentage word problems that may appear on the GRE.
- Solving problems involving compound percentages involves applying multiple percentages to a given value in succession.
Example: A store has a 20% off sale on all items. After the sale, they increase the prices by 10%. What is the final price of an item that originally cost $100?
Solution: First, find the price after the 20% discount: Discount amount = 0.20 × $100 = $20 Discounted price = $100 – $20 = $80 Next, find the price after the 10% increase: Increase amount = 0.10 × $80 = $8 Final price = $80 + $8 = $88 Answer: $88
2. Solving problems involving relative percentages: These problems involve comparing two or more percentages in relation to one another.
Example: In a class of 60 students, 70% are boys, and 80% of the boys passed the exam. If 60% of the class passed the exam, what percentage of the girls passed the exam?
Solution: First, find the number of boys and girls in the class: Number of boys = 0.70 × 60 = 42 Number of girls = 60 – 42 = 18 Next, find the number of students who passed the exam: Number of students who passed = 0.60 × 60 = 36 Now, find the number of boys who passed the exam: Number of boys who passed = 0.80 × 42 = 33.6 (approximately 34) Next, find the number of girls who passed the exam: Number of girls who passed = 36 – 34 = 2 Finally, find the percentage of girls who passed the exam: Percentage of girls who passed = (2 / 18) × 100 = 11.11% Answer: Approximately 11.11%
These additional tips and advanced problem types will help you further develop your skills in solving percentage word problems. Remember that practice is essential for mastering this topic, and remember to review your work and learn from any mistakes to improve your problem-solving abilities continuously.
Common Misconceptions and Errors in Percentage Word Problems:
- Misinterpreting “of” in percentage word problems: In percentage word problems, “of” typically means multiplication. For example, “20% of 150” should be interpreted as (20/100) × 150. Misinterpreting “of” as addition or subtraction can lead to incorrect solutions.
- Confusing percentage increase and percentage decrease: When solving percentage increase or decrease problems, it’s essential to understand the context. A percentage increase involves adding the percentage to the original value, while a percentage decrease involves subtracting the percentage from the original value.
- Incorrectly converting percentages to decimals or fractions: To convert a percentage to a decimal, divide by 100; to convert a percentage to a fraction, place the percentage over 100 and simplify, if possible. Errors in converting percentages can lead to incorrect calculations and solutions.
- Neglecting to check for reasonableness: Always check that your answer is reasonable and makes sense in the context of the problem. If your answer seems unreasonable, review your calculations and the problem statement to identify any errors.
Now let’s look at some strategies for avoiding these common errors and misconceptions in percentage word problems:
- Read the problem carefully: Make sure to read the problem thoroughly and understand the context. Identify what is being asked and determine the unknown value(s) you need to find.
- Write down the known information: Jot down the given information and relationships in the problem. This will help you avoid confusion and make identifying errors or misconceptions easier.
- Double-check your calculations: After solving the problem, review your calculations to ensure you have accurately converted percentages to decimals or fractions and have performed the correct operations.
- Practice, practice, practice: As with any topic, practice is essential to develop a strong understanding and avoid common misconceptions and errors. Work through a variety of percentage word problems to solidify your understanding and improve your problem-solving skills.
In conclusion, the GRE quantitative reasoning section’s percentage word problems are critical. By understanding the common misconceptions and errors, practicing various problem types, and using the strategies outlined in this extended concept, you will be well-prepared to tackle percentage word problems on the GRE and improve your overall performance.
Practice questions with an answer key.
Question types: Multiple Choice
Question 1: A car dealer offers a 15% discount on a car priced at $25,000. After the discount, the dealer adds a 5% sales tax to the discounted price. What is the final price of the car? A) $21,500 B) $22,575 C) $23,375 D) $24,850
Question 2: The price of a computer decreases from $1,200 to $900. What is the percentage decrease in the price of the computer?
A) 20% B) 25% C) 33.33% D) 40%
Question 3: In a group of 120 students, 60% are girls. If 75% of the girls and 50% of the boys pass an exam, what percentage of the total students pass the exam?
A) 60% B) 62.5% C) 65% D) 70%
Question 4: A store is having a 30% off sale on all items. After the sale, they increase the prices by 15%. What is the final price of an item that originally cost $80?
A) $62 B) $69 C) $74 D) $78
Question 5: A factory produces 1,500 items per day. If the production increases by 20% on Monday and then decreases by 10% on Tuesday, how many items are produced on Tuesday? A) 1,500 B) 1,620 C) 1,650 D) 1,800
- B) $22,575
- B) 25%
- C) 65%
- C) $74
- B) 1,620
Question types: Multiple Choice
Question 1: A company’s revenue increased by 20% in the first quarter and then decreased by 15% in the second quarter. If the company’s initial revenue was $50,000, what was the company’s revenue at the end of the second quarter?
A) $53,000 B) $55,000 C) $57,000 D) $59,000
Question 2: In a class of 80 students, 40% are boys. If the pass rate for boys is 70% and the overall pass rate for the class is 78%, what is the pass rate for girls in the class?
A) 80% B) 85% C) 90% D) 95%
Question 3: A store sells a shirt at a 30% markup over its cost price. During a sale, the store offers a 25% discount on the selling price. If the final sale price of the shirt is $45, what was the original cost price of the shirt?
A) $40 B) $50 C) $60 D) $80
Question 4: A project was completed in 8 days. On the first day, 5% of the project was completed, and on the second day, 10% was completed. If the remaining work was distributed evenly across the remaining 6 days, what percentage of the project was completed daily during those last 6 days?
A) 11.67% B) 12.5% C) 14.17% D) 15.83%
Question 5: A factory produces 2,000 units of a product. Due to an increase in demand, production needs to be increased by 25%. However, 5% of the produced units are defective and cannot be sold. How many non-defective units will be produced after the increase in production?
A) 2,375 B) 2,450 C) 2,500 D) 2,525
- C) $57,000
- D) 95%
- B) $50
- B) 12.5%
- A) 2,375
Question types: Quantity Comparison
Column A: The final price of a $1,000 item after a 10% discount and then a 5% sales tax
Column B: The final price of a $1,000 item after a 5% sales tax and then a 10% discount
Question 2: A company has 100 employees. If 60% of the employees are male and 70% of the male employees have a college degree, then:
Column A: The number of male employees with a college degree Column
B: The number of female employees
Column A: The value of a 25% increase followed by a 25% decrease on a number x
Column B: The original value of x
Question 4: The price of a product is first increased by 20% and then decreased by 30%. The price of another product is first increased by 30% and then decreased by 20%. If the original price of both products is the same, then:
Column A: The final price of the first product Column
B: The final price of the second product
Question 5: An investor invests $10,000 in a stock that increases in value by 50% in the first year and then decreases in value by 40% in the second year.
Column A: The value of the investment at the end of the second year
Column B: The initial investment amount
- Column B
- Column A
- Column B
- Column B
- Column A
Question types: Multiple Select
Question 1: A product’s price increases by 20% and then decreases by 20%. Which of the following statements are true?
A) The final price is equal to the original price.
B) The final price is less than the original price.
C) The final price is greater than the original price.
D) The percentage change between the final and original price is 0%.
Question 2: A company has 120 employees. If 40% of the employees are women, and 80% of the women and 60% of the men have a college degree, which of the following are true?
A) More than 50% of the total employees have a college degree.
B) At least 70 employees have a college degree.
C) The number of men with a college degree is greater than the number of women with a college degree.
D) The number of women with a college degree is equal to the number of men with a college degree.
Question 3: A store sells an item at a 30% markup over its cost price. During a sale, the store offers a 20% discount on the selling price. If the final sale price of the item is $60, which of the following could be the original cost price of the item?
A) $50 B) $55 C) $62.50 D) $70
Question 4: A project is completed in 6 days. On the first day, 10% of the project is completed, and on the second day, 20% is completed. If the remaining work is distributed evenly across the remaining 4 days, which of the following statements are true?
A) The project was completed faster in the last 4 days than in the first 2 days.
B) The percentage of the project completed daily during the last 4 days was 17.5%.
C) More than half of the project was completed in the first 2 days.
D) The percentage of the project completed daily during the last 4 days was greater than the percentage completed on the first day.
- B, D
- A, B, C
- A, D
- A, D
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