Problem
Rename large numbers by place value
Korean names a new unit every four digits, the West every three. A is 3000 ten-millions. B is 3 billion. Find how many times B the number A is.
Reasoning · Grade 4-1 Large Numbers
Your answer
Watch out The two groupings confuse the count, but a billion carries 9 zeros, not 12.
How to solve
Strategy Analyze the Units — A and B are described in two different naming systems, so they cannot be compared until both are written the same way. The safest common ground is plain digits: turn each unit name into its number of zeros, write A and B out, then line the two numerals up and count how many places apart they are.
1STEP 1
Write A in digits
3000 ten-millions makes A a 3 with 10 zeros.
A = 3000 × 10 000 000 = 30 000 000 000
Multiplying by a power of ten never changes the digit 3; it only slides it left, so all I have to track is how many zeros pile up behind it.
5.NBT.A.2Look For A Pattern2STEP 2
Write B in digits
A billion has 9 zeros, so B is a 3 with 9 zeros.
B = 3 × 1 000 000 000 = 3 000 000 000
Reading the chart's bottom row from the right — thousand, million, billion — is the same as counting off three zeros at a time.
5.NBT.A.2Analyze The Units3STEP 3
Compare the two numerals
Same leading digit, just one extra zero.
30 000 000 000 ÷ 3 000 000 000 = 10
Each step left along the chart multiplies the value by ten, so a gap of exactly one column means a factor of exactly ten.
5.NBT.A.1Analyze The UnitsBecause each step left along the chart multiplies by ten, a gap of exactly one column means a factor of exactly ten.
Why?
Every place bundles ten of the place to its right, so moving a digit one column left multiplies what it is worth by ten.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The same multiplier applies wherever you stand on the chart, so the ratio between the two numbers is fixed by the column gap alone.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
4STEP 4
Sanity-check on a smaller version of the same question
One zero of difference means 10 times.
3000 × 100 = 300 000 = 10 × (3 × 10 000)
Testing the zero-counting rule on numbers small enough to write out by hand shows it was applied correctly before trusting it on eleven-digit numbers.
5.NBT.A.2Solve An Easier Related ProblemAnswer
10 times
30000000000 ÷ 3000000000 = 10
Both numbers begin with the digit 3, so the answer had to be a power of ten rather than an awkward fraction. A has 11 digits and B has 10, a difference of exactly one digit, which confirms a factor of 10. The answer is a whole number greater than 1, as it must be since A is the larger of the two.
Takeaway
When two big numbers start with the same digit, just count their zeros — every extra zero means ten times bigger!
- Write A in digits
- Write B in digits
- Compare the two numerals
- Sanity-check on a smaller version of the same question