Problem
Compare numbers with an unknown digit
Find every digit from 0 to 9 that can be placed in the box so that both 4□63 ＜ 4759 and 76□8 ＞ 7665 are true at the same time.
Base-ten numbers
Your answer
Watch out When □ is 6 the tens digits tie, so compare the ones place — 7668 is greater than 7665, so 6 still counts.
How to solve
Strategy Eliminate Possibilities — The box has a genuinely finite set of candidates, 0 through 9. I compare place by place to find the range each inequality allows, then keep only the digits that satisfy both.
1STEP 1
Solve the first inequality
The thousands tie (4 = 4), so the hundreds place decides: □ must be under 7, leaving 0 to 6.
4 □ 63 ＜ 4759 → □ ≤ 6
When the top place is tied, the next place down decides which number is larger.
4.NBT.A.2Eliminate PossibilitiesIn 4□63 < 4759 the thousands digits are both 4, so the comparison comes down to the hundreds place, and □ must be less than 7.
Why?
The two numbers are equal in the thousands place, so the leading 4000 is the same in each and cannot make one larger than the other; whatever settles the comparison must come from the places below the thousands.
Why?
Each number is its thousands part plus its hundreds-tens-ones part, and when one shared piece is identical in both, only the remaining pieces can make them differ.
🧱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?
Among the places below the thousands the hundreds place is the strongest, so its digit decides the comparison: a hundreds digit below 7 keeps 4□63 under 4759.
Why?
One hundred is worth more than every ten and one beneath it put together, which can pile up to at most 99, so a gap of one in the hundreds place can never be overturned by the lower places.
Why?
Ten tens bundle into one hundred, so nine tens and nine ones reach only 99 — one short of a single hundred.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
2STEP 2
Solve the second inequality
The thousands and hundreds tie (7 = 7, 6 = 6), so the tens place decides: □ must be at least 6, leaving 6 to 9.
76 □ 8 ＞ 7665 → □ ≥ 6
With the higher places equal, the tens digit is what tips one number above the other.
4.NBT.A.2Eliminate Possibilities3STEP 3
Keep only digits that satisfy both
The first allows 0 to 6, the second allows 6 to 9 — the only digit in both is 6.
{0,1,2,3,4,5,6} ∩ {6,7,8,9} = {6}
A digit must obey both rules, so only the overlap of the two allowed ranges survives.
4.NBT.A.2Look For A PatternAnswer
6
{0, 1, 2, 3, 4, 5, 6} ∩ {6, 7, 8, 9} = {6}
Check 6 directly: 4663 ＜ 4759 is true and 7668 ＞ 7665 is true, so 6 works and it is the single overlap of the two ranges.
Takeaway
Compare numbers one place at a time from the left, find each rule's range, then keep the digit both rules share!
- Solve the first inequality
- Solve the second inequality
- Keep only digits that satisfy both
Where next?
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