Percent Problems in Context - ISEE Middle Level Quantitative Reasoning

1. After a 20% price increase, the new price of a bicycle is $300. The store owner then offers a 10% discount on the new price. What was the original price of the bicycle before the increase?
   Answer: $250 The question asks for the original price, so the 10% discount is extra information not needed to solve the problem. Let the original price be P. The new price, $300, is 120% of the original price. The equation is $1.20 \times P = \$300$. To find P, divide $300 by 1.20: $P = 300 \div 1.2 = \$250$.
2. In a bag of 80 marbles, 25% are red. One-half of the remaining marbles are blue. How many marbles are blue?
   Answer: 30 First, calculate the number of red marbles: 25% of 80 is $0.25 \times 80 = 20$. Next, find the number of remaining marbles: $80 - 20 = 60$. Finally, calculate the number of blue marbles, which is one-half of the remaining: $60 \div 2 = 30$.
3. The number of tickets sold for a play on Friday was 400. The number of tickets sold on Saturday was 25% more than on Friday. How many tickets were sold in total over the two days?
   Answer: 900 First, calculate the number of tickets sold on Saturday. The increase was 25% of 400, which is $0.25 \times 400 = 100$ tickets. So, on Saturday, $400 + 100 = 500$ tickets were sold. The total for the two days is the sum of Friday's and Saturday's sales: $400 + 500 = 900$ tickets.
4. A jacket is on sale for $60 after a 25% discount. What was the original price of the jacket?
   Answer: $80.00 The sale price of $60 represents 100% - 25% = 75% of the original price. Let P be the original price. The equation is $0.75 \times P = \$60$. To find the original price, divide the sale price by 0.75: $P = \$60 \div 0.75 = \$80$.
5. The price of a stock increased by 20% on Monday and then decreased by 20% on Tuesday. If the price of the stock was $100 at the start of Monday, what was its price at the end of Tuesday?
   Answer: $96 First, calculate the price after the 20% increase on Monday: $\$100 \times 1.20 = \$120$. Next, calculate the price after the 20% decrease on Tuesday. The decrease is based on the new price of $120: $\$120 \times 0.20 = \$24$. The final price is $\$120 - \$24 = \$96$.
6. In a survey, 110 students, representing 40% of the seventh-grade class, said they prefer comedy movies. How many students are in the entire seventh-grade class?
   Answer: 275 Let T be the total number of students in the class. We are given that 40% of T is 110. The equation is $0.40 \times T = 110$. To find T, divide 110 by 0.40: $T = 110 \div 0.40 = 275$. There are 275 students in the seventh-grade class.
7. The population of a town was 8,000 in 2020. By 2021, the population had increased by 5%. What was the population of the town in 2021?
   Answer: 8,400 First, calculate the increase in population: 5% of 8,000 is $0.05 \times 8000 = 400$. Then, add this increase to the original population: $8000 + 400 = 8,400$. Alternatively, a 5% increase means the new population is 105% of the original: $8000 \times 1.05 = 8,400$.
8. A test has two sections. A student answered 80% of the 30 questions in the first section correctly. If the student scored 70% on the entire 50-question test, what percentage of the 20 questions in the second section did the student answer correctly?
   Answer: 55% First, find the number of questions answered correctly in section 1: $30 \times 0.80 = 24$. Next, find the total number of questions answered correctly on the whole test: $50 \times 0.70 = 35$. The number of questions answered correctly in section 2 is the difference: $35 - 24 = 11$. Finally, find the percentage for section 2, which has 20 questions: $\frac{11}{20} \times 100\% = 55\%$.
9. A salesperson earns a monthly salary of $2,000 plus a 4% commission on all sales over $10,000. If their total sales for the month were $40,000, what were their total earnings for the month?
   Answer: $3,200 First, determine the amount of sales eligible for commission: $\$40,000 - \$10,000 = \$30,000$. Next, calculate the commission earned on this amount: 4% of $30,000 is $0.04 \times 30000 = \$1,200$. Finally, add the salary to the commission for total earnings: $\$2,000 + \$1,200 = \$3,200$.
10. A tablet computer is priced at $400. A store offers a 15% discount. Sales tax of 8% is calculated on the discounted price. What is the total cost of the tablet?
    Answer: $367.20 First, calculate the discounted price. The discount is $0.15 \times 400 = \$60$. The discounted price is $\$400 - \$60 = \$340$. Next, calculate the sales tax on this discounted price: $0.08 \times 340 = \$27.20$. Finally, add the tax to the discounted price to get the total cost: $\$340 + \$27.20 = \$367.20$.
11. A student estimated the length of a rope to be 80 cm. The actual length was 64 cm. What was the percent error of the student's estimate relative to the actual length?
    Answer: 25% Percent error is calculated as $\frac{|\text{Estimated Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100\%$. The error in measurement is $|80 - 64| = 16$ cm. The percent error is $\frac{16}{64} \times 100\% = \frac{1}{4} \times 100\% = 25\%$.
12. Sarah deposits $600 into a savings account that earns 2.5% simple annual interest. If she does not make any other deposits or withdrawals, how much interest will she have earned after 3 years?
    Answer: $45.00 Simple interest is calculated using the formula I = Prt, where I is interest, P is principal, r is the rate, and t is time. First, find the interest for one year: $\$600 \times 0.025 = \$15$. Then, multiply by the number of years to find the total interest: $\$15 \times 3 = \$45$.
13. A student's score on a test improved from 60 points to 75 points. What was the percent increase in the student's score?
    Answer: 25% The formula for percent increase is $\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$. The increase in score is $75 - 60 = 15$ points. The percent increase is $\frac{15}{60} \times 100\% = 0.25 \times 100\% = 25\%$.
14. A fundraising event raised a total of $5,000. 30% of the money will be used for administrative costs. The remaining money will be split equally among 4 charities. How much money will each charity receive?
    Answer: $875 First, calculate the amount of money remaining after administrative costs. If 30% is used for costs, then 70% remains for charities. The amount for charities is $0.70 \times \$5,000 = \$3,500$. This amount is split equally among 4 charities, so each charity receives $\$3,500 \div 4 = \$875$.
15. A recipe for a smoothie requires 20 ounces of liquid. 75% of the liquid is yogurt. If the rest of the liquid is juice, how many ounces of juice are required?
    Answer: 5 If 75% of the liquid is yogurt, then the remaining percentage must be juice. The percentage of juice is $100\% - 75\% = 25\%$. To find the amount of juice, calculate 25% of the total liquid volume: $0.25 \times 20 = 5$ ounces.
16. A family's dinner bill is $120. They want to leave a 20% tip on the bill before tax is added. The sales tax is 10% of the bill. What is the total amount they will pay?
    Answer: $156.00 First, calculate the tip: 20% of $120 is $0.20 \times 120 = \$24$. Next, calculate the sales tax: 10% of $120 is $0.10 \times 120 = \$12$. The total amount is the sum of the bill, the tip, and the tax: $\$120 + \$24 + \$12 = \$156$.
17. In a company, 60% of the employees are men. If 25% of the men and 50% of the women have worked at the company for more than 10 years, what percentage of all employees have worked there for more than 10 years?
    Answer: 35% Assume there are 100 employees. Then there are 60 men and 40 women. The number of men who have worked for more than 10 years is $0.25 \times 60 = 15$. The number of women who have worked for more than 10 years is $0.50 \times 40 = 20$. The total number of employees who have worked for more than 10 years is $15 + 20 = 35$. Since the total number of employees is 100, this represents 35% of all employees.
18. A video game originally priced at $50 is on sale for 20% off. A week later, the store offers an additional 10% off the sale price. What is the final price of the game?
    Answer: $36.00 First, calculate the initial sale price. A 20% discount on $50 is $0.20 \times 50 = \$10$. The sale price is $\$50 - \$10 = \$40$. Next, calculate the additional 10% discount on the sale price. A 10% discount on $40 is $0.10 \times 40 = \$4$. The final price is $\$40 - \$4 = \$36$.
19. A school library has 1,200 books. 45% of the books are fiction. Of the fiction books, 20% are science fiction. How many science fiction books are in the library?
    Answer: 108 First, find the number of fiction books: $1200 \times 0.45 = 540$. Then, find the number of science fiction books, which is 20% of the fiction books: $540 \times 0.20 = 108$.
20. A club has 15 members who are in 7th grade and 10 members who are in 8th grade. What percentage of the club members are in 7th grade?
    Answer: 60% First, find the total number of members in the club: $15 + 10 = 25$. Next, calculate the percentage of members who are in 7th grade by dividing the number of 7th graders by the total number of members and multiplying by 100: $\frac{15}{25} \times 100\% = 0.60 \times 100\% = 60\%$.

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