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تمت الإجابة باستخدام OpenAI 01 Preview بواسطة OpenAI01

سؤال

Rata-rata hitung dari 15 bilangan dalam suatu daftar bilangan adalah 16. Jika rata-rata dari 12 bilangan dari daftar itu antara 8 dan 14. Maka rata-rata dari 3 bilangan lain yang mungkin adalah (1) 10 (2) 26 (3) 80 (4) 40 (1) dan (3) SAJA yang benar (1), (2), dan (3) SAJA yang benar SEMUA pilihan benar (2) dan (4) SAJA yang benar HANYA (4) yang benar

الفكرة

Translating the problem

I'm working through translating the problem from Bahasa Indonesia to English to ensure comprehension.

Identifying possible averages

I’m mapping out which of the given averages (10, 26, 80, 40) could realistically represent the average of the other three numbers, considering the constraints of the problem.

Calculating sums

I’m determining the total sum of the 15 numbers, finding it to be 240. Let’s break it down to sum of 12 numbers and the remaining 3 numbers.

Calculating S3

I’m mapping out S3 by subtracting S12 from 240, leading to S3 = 240 - S12, which ranges between 72 and 144.

Determining range

I’m figuring out the possible values for S3, knowing S12 is between 96 and 168, and S3 should be between 72 and 144.

Breaking down the average

OK, let's clarify the average of three numbers, S3S3, which ranges from just over 24 to just below 48.

Assessing the range

I’m thinking through the average of the three numbers, which must be between just above 24 and just below 48. Valid options are 26 and 40.

الإجابة

To solve this problem, we need to determine which of the given averages for the other three numbers are possible based on the provided conditions.

Given:

  • The average of 15 numbers is 16.
  • The average of 12 of those numbers is between 8 and 14.

Calculations:

  1. Total Sum of All 15 Numbers:

    Total Sum (S)=15×16=240\text{Total Sum (S)} = 15 \times 16 = 240

  2. Sum of the 12 Numbers: The average of these numbers is between 8 and 14, so:

    8<S1212<148 < \frac{S_{12}}{12} < 14

    Multiplying both sides by 12:

    96<S12<16896 < S_{12} < 168

  3. Sum of the Remaining 3 Numbers:

    S3=S−S12=240−S12S_3 = S - S_{12} = 240 - S_{12}

    Substituting the range for S12S_{12}:

    240−168<S3<240−96  ⟹  72<S3<144240 - 168 < S_3 < 240 - 96 \implies 72 < S_3 < 144

  4. Average of the 3 Numbers:

    Average=S33\text{Average} = \frac{S_3}{3}

    So the average ranges:

    24<Average<4824 < \text{Average} < 48

Evaluating the Given Options:

  • Option (1) 10: This is below the minimum average of 24. Not possible.
  • Option (2) 26: Falls within the range (24, 48). Possible.
  • Option (3) 80: Exceeds the maximum average of 48. Not possible.
  • Option (4) 40: Falls within the range (24, 48). Possible.

Conclusion: Only options (2) and (4) are possible averages for the other three numbers.

Answer: (2) dan (4) SAJA yang benar

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