Times ten adds one place

4.NBT.A.1 · adapt · grade 4

Archetype: Place-Value Regrouping · step in a 7-type progression

▶ Practice — 10 problems

Multiplying $4{,}500$ by $100$ gives a number that equals $45$ multiplied by how many?

Show solution

Understand

The number 4,500 × 100 equals 45 multiplied by some factor; find that factor.

Givens

4,500 × 100 is one value.
That value also equals 45 × (unknown factor).

Unknowns

The factor that 45 is multiplied by

Constraints

Multiplying by 10 or 100 shifts digits to higher places (adds zeros).

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Use the place-value pattern that multiplying by 100 appends two zeros to compute 4,500 × 100, then work backwards to see how many times 45 fits into it.

Execute

#5 Look for a Pattern 4.NBT.A.1

Multiplying by 100 adds two zeros, so 4,500 × 100 = 450,000.

4{,}500 \times 100 = 450{,}000

Times one hundred shifts every digit two places to the left, tacking on two zeros.

#11 Work Backwards 4.NBT.A.1

Start from 45. To reach 450,000 from 45, the digits 4 and 5 must move four places left, which is multiplying by 10,000 (four zeros). So 45 × 10,000 = 450,000.

45 \times 10{,}000 = 450{,}000

Going from 45 to 450,000 means appending four zeros, i.e. multiplying by 10,000.

#9 Solve an Easier Related Problem 4.NBT.A.1

Since 4,500 × 100 = 450,000 = 45 × 10,000, the missing factor is 10,000.

4{,}500 \times 100 = 45 \times 10{,}000

Both sides equal 450,000, so 45 is multiplied by 10,000.

Answer: 10,000

Review

Check: 45 × 10,000 = 450,000 and 4,500 × 100 = 450,000, so both products match. The factor 10,000 is correct.

Guess and check with place value (tool 6): 4,500 is 45 × 100, so 4,500 × 100 = 45 × 100 × 100 = 45 × 10,000.

Standards · min grade 4

4.NBT.A.1 Recognize that a digit represents ten times what it represents in place to its right — Using the times-ten place-value shift to relate 4,500 × 100 and 45 × 10,000.

💡 This only needs Grade 4 place-value sense: every times-ten just slides the digits one place and adds a zero!