Numbers & Place Value

Problem

Factor out a common factor with parentheses

We are asked to rewrite the sum 423.4 × 9 + 42.34 × 8 + 4.234 × 20 as a simpler expression and then find its value. The three decimal factors 423.4, 42.34, and 4.234 are the same digits with the decimal point shifted, so we look for a way to write them all using one common number.
OperationsBase-ten numbers
Your answer
How to solve
Strategy Organize Information in More Ways — Adding three big decimal products head-on is messy. Instead we re-organize each term around the common number 4.234: since 423.4 = 4.234 × 100 and 42.34 = 4.234 × 10, every term becomes 4.234 times something. That lets us factor 4.234 out with parentheses (the distributive property) and the numbers inside add to a friendly 1000.
1STEP 1

Rewrite each decimal as a multiple of 4.234

The three decimals are the same digits with the point moved, so each is 4.234 times a power of ten.

423.4 = 4.234 × 100, 42.34 = 4.234 × 10
2STEP 2

Turn each product into 4.234 × (a number)

Combining the multipliers gives 4.234 x 900, 4.234 x 80 and 4.234 x 20.

4.234×900 + 4.234×80 + 4.234×20
3STEP 3

Factor out 4.234 with parentheses

Every term is 4.234 times something, so pull it out front: 4.234 x (900 + 80 + 20).

4.234 × (900 + 80 + 20)
4STEP 4

Add inside the parentheses, then multiply

900 + 80 + 20 = 1000, and multiplying 4.234 by 1000 just shifts the point three places.

4.234 × (900+80+20) = 4.234 × 1000 = 4234
Answer
4.234×(900+80+20)=42344.234 \times (900 + 80 + 20) = 4234
Each term is positive and the biggest is 423.4 × 9 ≈ 3811, plus 42.34 × 8 ≈ 339, plus 4.234 × 20 ≈ 85, which totals about 4235 — right next to our exact 4234, so the answer is the correct size.
Takeaway

When numbers share the same digits, rewrite them around one common factor and pull it out with parentheses — the messy sum collapses into a single easy multiplication.

  • Rewrite each decimal as a multiple of 4.234
  • Turn each product into 4.234 × (a number)
  • Factor out 4.234 with parentheses
  • Add inside the parentheses, then multiply
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