Understanding the Power of Multiplication: Solving ×16,774 with Smart Breakdowns (31 × 16,000 = 496,000; 31 × 700 = 21,700; 31 × 74)

Multiplication lies at the heart of mathematics, helping us simplify complex problems through smart calculations. Today, we explore the breakdown of the equation ×16,774 = 31 × 16,000 + 31 × 700 + 31 × 74 — a brilliant example of splitting large multiplications for easier understanding and computation.

Understanding the Context

Breaking Down 31 × 16,774 with Strategic Multiplication

Let’s start by analyzing the expression:
31 × 16,774 = 31 × (16,000 + 700 + 74)

Using the distributive property of multiplication:
31 × 16,774 = (31 × 16,000) + (31 × 700) + (31 × 74)

This method makes large multiplications more manageable — perfect for both mental math and educational purposes.

× 16774 = 31×16000=496,000; 31×774=31×700=21,700; 31×74

Key Insights

Step 1: Compute 31 × 16,000

Multiplying by 16,000 simply shifts the digits:
31 × 16,000 = 496,000

This quick conversion highlights how factoring out 10,000 (or in this case 16,000) simplifies big-digit multiplication.

Step 2: Compute 31 × 700

With 700, multiplication becomes straightforward:
31 × 700 = 31 × 7 × 100 = 217 × 100 = 21,700

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5Question: A science journalist tracks an experiment where a nanobot completes 40 tasks with 85% success. After 10 more tasks, its success rate rises to 88%. How many additional successful tasks were achieved?
Solution: Let $ x $ be the number of successful tasks in the additional 10 attempts. Initially, successful tasks = $ 0.85 imes 40 = 34 $. Total tasks after 10 more = $ 50 $. Total successful tasks = $ 34 + x $. New success rate: $ rac{34 + x}{50} = 0.88 $. Solving: $ 34 + x = 44 $, so $ x = 10 $.
oxed{10}

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Final Thoughts

This shows how grouping factors enhances accuracy — factoring out 100 not only speeds up calculation but also reduces cognitive load.

Step 3: Compute 31 × 74

Now for this intermediate step:
31 × 74
Break this down further:

  • 31 × 70 = 31 × 7 × 10 = 217 × 10 = 2,170
  • 31 × 4 = 124
  • Add: 2,170 + 124 = 2,294

So, 31 × 74 = 2,294

Final Calculation: Combine All Parts

Now sum up the components:
496,000 + 21,700 + 2,294 = 519,994

So,
31 × 16,774 = 519,994 (Note: While the original problem stated 496,000, this detailed breakdown clarifies the accurate result using smart multiplication steps.)

Why This Method Works Well