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Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectively as follows:
a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tn
If [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,
Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?
a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tn
If [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,
Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?
(2016)
- a)6
- b)10
- c)8
- d)9
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Two positive real numbers, a and b, are expressed as the sum of m posi...
If a positive number a is expressed as the sum of two positive numbers s1 and s2 then [a] could be at the most 1 more than [s1] + [s2], i.e., the fractional parts of s1 and s2 together, can provide at most 1.
Similarly, the fractional parts of s1, s2, s3, s4, s5 can together, provide at most 4.
Conversely, if [a] is 4 more than [s1] + [s2] + [s3] + [s4] + [sm], then m has to be at least 5.
Similarly, the least value of n is 4.
∴ (m + n)min = 5 + 4 = 9
Similarly, the fractional parts of s1, s2, s3, s4, s5 can together, provide at most 4.
Conversely, if [a] is 4 more than [s1] + [s2] + [s3] + [s4] + [sm], then m has to be at least 5.
Similarly, the least value of n is 4.
∴ (m + n)min = 5 + 4 = 9
Most Upvoted Answer
Two positive real numbers, a and b, are expressed as the sum of m posi...
... sm
b = t1 t2 ... tn
where s1, s2, ..., sm and t1, t2, ..., tn are positive real numbers.
The sum of a and b can be expressed as:
a + b = (s1 + s2 + ... + sm) + (t1 + t2 + ... + tn)
Since addition is associative, we can group the terms together:
a + b = (s1 + t1) + (s2 + t2) + ... + (sm + tn)
Therefore, the sum of a and b can be expressed as the sum of (m + n) positive real numbers:
a + b = u1 + u2 + ... + u(m+n)
where u1 = s1 + t1, u2 = s2 + t2, ..., u(m+n) = sm + tn.
b = t1 t2 ... tn
where s1, s2, ..., sm and t1, t2, ..., tn are positive real numbers.
The sum of a and b can be expressed as:
a + b = (s1 + s2 + ... + sm) + (t1 + t2 + ... + tn)
Since addition is associative, we can group the terms together:
a + b = (s1 + t1) + (s2 + t2) + ... + (sm + tn)
Therefore, the sum of a and b can be expressed as the sum of (m + n) positive real numbers:
a + b = u1 + u2 + ... + u(m+n)
where u1 = s1 + t1, u2 = s2 + t2, ..., u(m+n) = sm + tn.
Community Answer
Two positive real numbers, a and b, are expressed as the sum of m posi...
If a positive number a is expressed as the sum of two positive numbers s1 and s2 then [a] could be at the most 1 more than [s1] + [s2], i.e., the fractional parts of s1 and s2 together, can provide at most 1.
Similarly, the fractional parts of s1, s2, s3, s4, s5 can together, provide at most 4.
Conversely, if [a] is 4 more than [s1] + [s2] + [s3] + [s4] + [sm], then m has to be at least 5.
Similarly, the least value of n is 4.
∴ (m + n)min = 5 + 4 = 9
Similarly, the fractional parts of s1, s2, s3, s4, s5 can together, provide at most 4.
Conversely, if [a] is 4 more than [s1] + [s2] + [s3] + [s4] + [sm], then m has to be at least 5.
Similarly, the least value of n is 4.
∴ (m + n)min = 5 + 4 = 9
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Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectivelyas follows:a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tnIf [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?(2016)a)6b)10c)8d)9Correct answer is option 'D'. Can you explain this answer?
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Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectivelyas follows:a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tnIf [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?(2016)a)6b)10c)8d)9Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectivelyas follows:a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tnIf [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?(2016)a)6b)10c)8d)9Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectivelyas follows:a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tnIf [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?(2016)a)6b)10c)8d)9Correct answer is option 'D'. Can you explain this answer?.
Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectivelyas follows:a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tnIf [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?(2016)a)6b)10c)8d)9Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectivelyas follows:a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tnIf [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?(2016)a)6b)10c)8d)9Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two positive real numbers, a and b, are expressed as the sum of m positive real numbers and n positive real numbers respectivelyas follows:a = s1 + s2 +…+ sm and b = t1 + t2 +…+ tnIf [a] = [s1] + [s2] +…+ [sm] + 4 and [b] = [t1 ] + [t2 ] +…+ [tn] + 3,Where [x] denotes the greatest integer less than or equal to x, what is the minimum possible value of m + n?(2016)a)6b)10c)8d)9Correct answer is option 'D'. Can you explain this answer?.
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