Problem
Compare place by place to bound an unknown digit
Find every single digit 0-9 that can replace the box so that 4,541,592 is greater than 45□6,719.
Base-ten numbers
Your answer
Watch out It's easy to assume the boundary digit works too without checking the next place — but plugging in 4 for the box actually makes the inequality false, so 4 must be excluded.
How to solve
Strategy Make a Systematic List — Both numbers have the same number of digits, so I line them up place by place from the left and compare; the first place where they differ decides which is larger, which bounds the box digit. I then list every digit that fits.
1STEP 1
Line up the digits
Both numbers are seven digits: 4 541 592 versus 4 5□ 6 719. Compare from the top and the leading digits agree: 4 = 4, then 5 = 5.
4 5 4 1 5 9 2 vs 4 5 □ 6 7 1 9
When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.
4.NBT.A.2Draw A Diagram2STEP 2
Reach the deciding place
The first two leading digits match (4=4, 5=5). The next place is the ten-thousands place: the fixed number has 4 there, the other has the box. To keep 4,541,592 the bigger number, 4 must be at least as large as the box. If the box were also 4, we would have to look further; if it is smaller than 4, the fixed number already wins here.
4 ? □
The leftmost place where two numbers differ is the one that decides which is larger.
4.NBT.A.2Make A Systematic ListWhen two seven-digit numbers already share their top two place digits, the ten-thousands place becomes the deciding place: the number with the larger digit there is the larger number.
Why?
The millions and hundred-thousands places hold the same digit in both numbers, so those parts are equal on each side and the ten-thousands place is the highest place where the two can still differ.
Why?
A whole number is just the sum of the values sitting in each of its places, so when a place holds the same digit in both numbers it contributes the same amount to each and drops out of the comparison.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
One unit in the ten-thousands place is worth ten thousand, while everything in the thousands, hundreds, tens, and ones together can reach at most nine thousand nine hundred ninety-nine, so a single larger digit up there cannot be caught up from below.
Why?
Each place is worth ten of the place to its right, so the ten-thousands place outranks the thousands, hundreds, tens, and ones stacked together.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
3STEP 3
Test the tie case box = 4
If the box is 4, the other number is 4,546,719. Comparing 4,541,592 with 4,546,719 the next differing place gives 1 versus 6, so 4,541,592 is smaller. So box = 4 fails, and box must be strictly less than 4.
4,541,592 less than 4,546,719
If the deciding digits tie, you peek at the next place; here it shows 4 is too big.
4.NBT.A.2Make A Systematic List4STEP 4
List the working digits
The box must be smaller than 4, so it can be 0, 1, 2, or 3. Each makes 45□6,719 at most 4,536,719, which is less than 4,541,592.
□ ∈ {0,1,2,3}
Any ten-thousands digit below 4 keeps the second number clearly smaller.
4.NBT.A.2Make A Systematic ListAnswer
0, 1, 2, 3
With box = 3, 45□6,719 = 4,536,719 less than 4,541,592 (true). With box = 4, 4,546,719 greater than 4,541,592 (false). The cutoff at 4 is correct, so 0,1,2,3 are exactly the digits that work.
Takeaway
This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!
- Line up the digits
- Reach the deciding place
- Test the tie case box = 4
- List the working digits
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