Numbers & Place Value

Problem

Decimal point position predicts the product

We have two products: ⓐ is 21 × 14 (whole numbers) and ⓑ is 0.21 × 0.14 (the same digits, but each factor is a decimal). We need to find how many times as large ⓐ is compared to ⓑ.
Base-ten numbers
Your answer
How to solve
Strategy Look for a Pattern — Going from 21 to 0.21 divides by 100, and going from 14 to 0.14 also divides by 100. Looking at this pattern of how each factor shrinks tells us how the product shrinks, so we can find the ratio without relying only on the full multiplication. A simpler related case (like 0.2 × 0.1 versus 2 × 1) makes the rule easy to see.
1STEP 1

Compare each factor

Moving the point two places left divides by 100, so 21 is 100 times 0.21 and 14 is 100 times 0.14.

21 = 0.21 × 100, 14 = 0.14 × 100
2STEP 2

See how the product changes

Both factors grow by 100, so the product grows by 100 x 100 = 10000.

(21 × 14)/(0.21 × 0.14) = 100 × 100 = 10000
3STEP 3

Check by direct calculation

21 x 14 = 294 and 0.21 x 0.14 = 0.0294: the same digits, four places apart.

294 ÷ 0.0294 = 10000
Answer
10000 times
294 and 0.0294 share the same digit string 294 but the decimal point sits four places apart, so the ratio must be a power of ten with four zeros: 10000. The magnitude makes sense.
Takeaway

When both numbers you multiply shrink by 100, the answer shrinks by 100 × 100 = 10000.

  • Compare each factor
  • See how the product changes
  • Check by direct calculation
Where next?
Another one like thissuggested
12 practice problems
Same look, different logicsuggested
Place big digits in high places to maximize product
Step back and solidifysuggested
Carry ten units to the next placeTen hundreds make one thousand
3 reasoning problems grow out of this type — the bridge toward competition →

▶ Practice — 12 problems