Compare numbers with an unknown digit

4.NBT.A.2 · take · grade 4

Archetype: Pin Down a Number from Digit and Range Conditions · step in a 9-type progression

Among the digits from $0$ to $9$ , find every digit that can go in $\square$ to make both inequalities true.

$4\,\square\,63 < 4759 \qquad 76\,\square\,8 > 7665$

Show solution

Understand

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.

Givens

The box holds a single digit from 0 to 9.
First inequality: 4□63 < 4759 (box is the hundreds digit).
Second inequality: 76□8 > 7665 (box is the tens digit).

Unknowns

Which digit(s) make both inequalities true.

Constraints

The same digit goes in both boxes.
The digit is a whole number between 0 and 9.

Plan

#3 Eliminate Possibilities · also uses: #5 Look for a Pattern

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.

Execute

#3 Eliminate Possibilities 4.NBT.A.2

In 4□63 < 4759 the thousands digits match (4 = 4), so compare the hundreds digit: the box must be less than 7. If the box were 7, we would get 4763, which is greater than 4759, so 7 fails. Thus the box can be 0, 1, 2, 3, 4, 5, or 6.

4\,\square\,63 < 4759 \Rightarrow \square \leq 6

When the top place is tied, the next place down decides which number is larger.

#3 Eliminate Possibilities 4.NBT.A.2

In 76□8 > 7665 the thousands and hundreds digits match (7 = 7 and 6 = 6), so compare the tens digit: the box must be at least 6. If the box were 6 we get 7668 > 7665, which is true, so 6 works. Thus the box can be 6, 7, 8, or 9.

76\,\square\,8 > 7665 \Rightarrow \square \geq 6

With the higher places equal, the tens digit is what tips one number above the other.

#5 Look for a Pattern 4.NBT.A.2

The first inequality allows 0 to 6 and the second allows 6 to 9. The only digit in both lists is 6.

\{0,1,2,3,4,5,6\} \cap \{6,7,8,9\} = \{6\}

A digit must obey both rules, so only the overlap of the two allowed ranges survives.

Answer: 6

Review

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.

Test all ten digits 0 through 9 one at a time in both inequalities (guess and check) and keep the ones that pass both; only 6 survives.

Standards · min grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing four-digit numbers place by place to bound the unknown digit.

💡 Compare numbers one place at a time from the left, find each rule's range, then keep the digit both rules share!