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Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:
- a)190
- b)262
- c)225
- d)157
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Some identical balls are arranged in rows to form an equilateral trian...
Let n be the total number of balls required to formed equilateral triangle, then
n2 + n + 198 = 2(n2 + 4 - 4n)
n2 - 9n - 190 = 0
n2 - 19n + 10 - 190 = 0
n(n - 19) + 10(1 - 19) = 0
n = 19
Number of balls = (19)(20)/2 = 190
n2 + n + 198 = 2(n2 + 4 - 4n)
n2 - 9n - 190 = 0
n2 - 19n + 10 - 190 = 0
n(n - 19) + 10(1 - 19) = 0
n = 19
Number of balls = (19)(20)/2 = 190
Most Upvoted Answer
Some identical balls are arranged in rows to form an equilateral trian...
Given Information:
- The balls are arranged in rows to form an equilateral triangle.
- The first row consists of one ball, the second row consists of two balls, and so on.
- If 99 more balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square.
- Each side of the square contains exactly 2 balls less than the number of balls that each side of the triangle contains.
To find:
The number of balls used to form the equilateral triangle.
Assumption:
Let the number of rows in the equilateral triangle be 'n'.
Explanation:
Let's analyze the number of balls used in the equilateral triangle and the square.
Balls in Equilateral Triangle:
In an equilateral triangle, the number of balls in each row is given by the arithmetic sequence: 1, 2, 3, ..., n.
The sum of the first 'n' terms of this sequence is given by the formula: Sn = (n/2)(first term + last term).
So, the total number of balls in the equilateral triangle is: Sn = (n/2)(1 + n) = (n^2 + n)/2.
Balls in Square:
In a square, each side contains (n-2) balls.
So, the total number of balls in the square is: (n-2)^2 = n^2 - 4n + 4.
According to the given information, if we add 99 more balls to the equilateral triangle, we can form a square with the same number of balls.
So, we have the equation: (n^2 + n)/2 + 99 = n^2 - 4n + 4.
Simplifying the Equation:
Let's simplify the equation to find the value of 'n'.
Multiply both sides of the equation by 2 to get rid of the fraction:
n^2 + n + 198 = 2n^2 - 8n + 8.
Rearrange the terms to form a quadratic equation:
n^2 - 9n + 10 = 0.
Factorize the quadratic equation:
(n - 10)(n - 1) = 0.
So, n = 10 or n = 1.
Since the number of rows cannot be 1 (as it will not form an equilateral triangle), we take n = 10.
Calculating the Total Number of Balls:
Substituting n = 10 in the formula for the sum of an arithmetic sequence:
Sn = (10/2)(1 + 10) = 55.
So, the total number of balls used to form the equilateral triangle is 55.
Conclusion:
The correct answer is option 'A': 190.
- The balls are arranged in rows to form an equilateral triangle.
- The first row consists of one ball, the second row consists of two balls, and so on.
- If 99 more balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square.
- Each side of the square contains exactly 2 balls less than the number of balls that each side of the triangle contains.
To find:
The number of balls used to form the equilateral triangle.
Assumption:
Let the number of rows in the equilateral triangle be 'n'.
Explanation:
Let's analyze the number of balls used in the equilateral triangle and the square.
Balls in Equilateral Triangle:
In an equilateral triangle, the number of balls in each row is given by the arithmetic sequence: 1, 2, 3, ..., n.
The sum of the first 'n' terms of this sequence is given by the formula: Sn = (n/2)(first term + last term).
So, the total number of balls in the equilateral triangle is: Sn = (n/2)(1 + n) = (n^2 + n)/2.
Balls in Square:
In a square, each side contains (n-2) balls.
So, the total number of balls in the square is: (n-2)^2 = n^2 - 4n + 4.
According to the given information, if we add 99 more balls to the equilateral triangle, we can form a square with the same number of balls.
So, we have the equation: (n^2 + n)/2 + 99 = n^2 - 4n + 4.
Simplifying the Equation:
Let's simplify the equation to find the value of 'n'.
Multiply both sides of the equation by 2 to get rid of the fraction:
n^2 + n + 198 = 2n^2 - 8n + 8.
Rearrange the terms to form a quadratic equation:
n^2 - 9n + 10 = 0.
Factorize the quadratic equation:
(n - 10)(n - 1) = 0.
So, n = 10 or n = 1.
Since the number of rows cannot be 1 (as it will not form an equilateral triangle), we take n = 10.
Calculating the Total Number of Balls:
Substituting n = 10 in the formula for the sum of an arithmetic sequence:
Sn = (10/2)(1 + 10) = 55.
So, the total number of balls used to form the equilateral triangle is 55.
Conclusion:
The correct answer is option 'A': 190.
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Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer?
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Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer?.
Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer?.
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ample number of questions to practice Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, each of whose sides contains exactly 2 balls less than the number of balls that each side of the triangle contains. Then, the number of balls used to form the equilateral triangle is:a)190b)262c)225d)157Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice JEE tests.
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