Problem
Compare numbers with an unknown digit
Among 0 to 9, find every digit for □ that makes 4□63 less than 4759 and, at the same time, 76□8 greater than 7665.
Base-ten numbers
Your answer
Watch out Don't merge the two answer lists — only digits that satisfy both count.
How to solve
1STEP 1
Find the first inequality's range
4□63 and 4759 share the thousands digit, so the hundreds place decides — □ is 0 through 6.
4□63 vs 4759 ⇒ □ ∈ {0,1,2,3,4,5,6}
With the thousands equal, the leading 4000 is the same in both and can't decide; below it the hundreds place is strongest and settles the comparison. A hundreds digit under 7 can't be overturned since the lower places reach only 99. If □ were 7, we'd get 4763, which exceeds 4759, so 7 fails.
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
Find the second inequality's range
76□8 and 7665 match in the thousands and hundreds, so the tens place decides — □ is 6 through 9.
76□8 vs 7665 ⇒ □ ∈ {6,7,8,9}
When the higher places all match, that part is identical in both numbers, so the remaining tens digit is what tips one number above the other. If □ is 6, we get 7668, which exceeds 7665, so 6 is included.
4.NBT.A.2Eliminate Possibilities3STEP 3
Keep only the overlapping digit
Only one digit sits in both 0-6 and 6-9 — 6.
{0,1,2,3,4,5,6} ∩ {6,7,8,9} = {6}
A digit must satisfy both conditions at once, so only where the two allowed ranges overlap can it be the answer — here just 6.
4.NBT.A.2Eliminate PossibilitiesAnswer
6
{0,1,2,3,4,5,6} ∩ {6,7,8,9} = {6}
Check — put 6 in □: 4663 is less than 4759 and 7668 is greater than 7665, so both hold.
Takeaway
Compare from the highest place down, dropping to the next place when tied — and only the overlap of both rules is the answer.
- 4□63 less than 4759 → □ is 0-6
- 76□8 greater than 7665 → □ is 6-9
- The overlap → 6
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested