Convert a number from a 7-base number system to an 11-base number system while maintaining trade accuracy | Step-by-Step Solution

Let's Break Down This Number System Conversion! 🔢

What We're Solving:

We need to convert 6532 tons of beef from a base-7 number system to a base-11 number system, then calculate how much silk we can get (1 kg beef = 2 meters silk). This combines number system conversion with unit conversion!

The Approach:

We'll:

• 1. Convert the base-7 number to our familiar base-10 (decimal)
• 2. Convert that base-10 number to base-11
• 3. Apply the trade conversion for the silk

Step-by-Step Solution:

Step 1: Convert 6532₇ to base-10 In base-7, each position represents powers of 7 (from right to left: 7⁰, 7¹, 7², 7³...)

6532₇ = 6×7³ + 5×7² + 3×7¹ + 2×7⁰ = 6×343 + 5×49 + 3×7 + 2×1 = 2058 + 245 + 21 + 2 = 2326₁₀ tons

Step 2: Convert 2326₁₀ to base-11 We divide by 11 repeatedly and collect remainders:

• 2326 ÷ 11 = 211 remainder 5
• 211 ÷ 11 = 19 remainder 2
• 19 ÷ 11 = 1 remainder 8
• 1 ÷ 11 = 0 remainder 1

Reading remainders from bottom to top: 1825₁₁ tons

Step 3: Calculate silk conversion

• 2326 tons = 2,326,000 kg of beef
• 2,326,000 kg × 2 meters/kg = 4,652,000 meters of silk

The Answer:

• The beef amount in base-11 is 1825₁₁ tons
• This converts to 4,652,000 meters of silk (or 4,652 kilometers!)

Memory Tip: 💡

Remember "Divide and Conquer" - when converting TO a base, Divide by that base and collect remainders. When converting FROM a base, Calculate using powers of that base. The remainders or coefficients always read in the direction you're working!

Great job working through this complex problem that combines multiple math concepts! You're becoming a real "mathematical translator"! 🌟