Question
Please help me set 15 very tricky mathematics common entrance questions from any previous year
Answers
GPT-4o mini
Here are 15 tricky mathematics questions that are often included in common entrance exams. These questions cover various topics and are designed to challenge students' critical thinking and problem-solving skills.
### Questions:
1. **Fractions and Ratios:**
If \( \frac{2}{3} \) of a number is 12, what is \( \frac{1}{4} \) of that number?
2. **Geometry:**
A rectangle has a length that is twice its width. If the perimeter of the rectangle is 48 cm, what is the area of the rectangle?
3. **Algebra:**
Solve for \( x \): \( 3(x - 4) + 5 = 2(x + 1) + 9 \).
4. **Percentages:**
A shirt costs $40 after a 20% discount. What was the original price of the shirt?
5. **Probability:**
A bag contains 3 red balls and 5 blue balls. If one ball is taken out at random, what is the probability that it is red?
6. **Sequences:**
The first term of an arithmetic sequence is 5, and the common difference is 3. What is the 10th term of the sequence?
7. **Time and Work:**
If 3 workers can complete a job in 10 days, how many days will it take for 5 workers to complete the same job, assuming they work at the same rate?
8. **Averages:**
The average of five numbers is 20. If one number is removed, the new average becomes 22. What was the number that was removed?
9. **Exponents:**
Simplify \( 2^3 \times 2^{-2} \).
10. **Linear Equations:**
If \( 2x + 3y = 12 \) and \( x - y = 2 \), find the value of \( x \) and \( y \).
11. **Circumference and Area:**
The circumference of a circle is 31.4 cm. What is the area of the circle? (Use \( \pi \approx 3.14 \))
12. **Rates:**
If a car travels 120 km in 2 hours, what is its average speed in km/h?
13. **Compound Interest:**
If $1000 is invested at an annual interest rate of 5% compounded annually, how much money will there be after 3 years?
14. **Distance:**
Two cars start from the same point and drive in opposite directions. One car travels at 60 km/h and the other at 90 km/h. How far apart will they be after 2 hours?
15. **Money and Change:**
You have a $20 bill. You buy 3 items costing $5.25, $4.75, and $6.50. How much change will you receive?
### Solutions:
For effective learning and practice, it is recommended to work through the problems above and check answers using your own methods or solutions provided by your educational resources. Each question can help develop critical problem-solving skills useful for entrance exams.
### Questions:
1. **Fractions and Ratios:**
If \( \frac{2}{3} \) of a number is 12, what is \( \frac{1}{4} \) of that number?
2. **Geometry:**
A rectangle has a length that is twice its width. If the perimeter of the rectangle is 48 cm, what is the area of the rectangle?
3. **Algebra:**
Solve for \( x \): \( 3(x - 4) + 5 = 2(x + 1) + 9 \).
4. **Percentages:**
A shirt costs $40 after a 20% discount. What was the original price of the shirt?
5. **Probability:**
A bag contains 3 red balls and 5 blue balls. If one ball is taken out at random, what is the probability that it is red?
6. **Sequences:**
The first term of an arithmetic sequence is 5, and the common difference is 3. What is the 10th term of the sequence?
7. **Time and Work:**
If 3 workers can complete a job in 10 days, how many days will it take for 5 workers to complete the same job, assuming they work at the same rate?
8. **Averages:**
The average of five numbers is 20. If one number is removed, the new average becomes 22. What was the number that was removed?
9. **Exponents:**
Simplify \( 2^3 \times 2^{-2} \).
10. **Linear Equations:**
If \( 2x + 3y = 12 \) and \( x - y = 2 \), find the value of \( x \) and \( y \).
11. **Circumference and Area:**
The circumference of a circle is 31.4 cm. What is the area of the circle? (Use \( \pi \approx 3.14 \))
12. **Rates:**
If a car travels 120 km in 2 hours, what is its average speed in km/h?
13. **Compound Interest:**
If $1000 is invested at an annual interest rate of 5% compounded annually, how much money will there be after 3 years?
14. **Distance:**
Two cars start from the same point and drive in opposite directions. One car travels at 60 km/h and the other at 90 km/h. How far apart will they be after 2 hours?
15. **Money and Change:**
You have a $20 bill. You buy 3 items costing $5.25, $4.75, and $6.50. How much change will you receive?
### Solutions:
For effective learning and practice, it is recommended to work through the problems above and check answers using your own methods or solutions provided by your educational resources. Each question can help develop critical problem-solving skills useful for entrance exams.