Compare place by place to bound an unknown digit

4.NBT.A.2 · adapt · grade 4

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

▶ Practice — 10 problems

Find every digit from $0$ to $9$ that can go in the $\square$ to make the statement true.

$4{,}541{,}592 > 45\square6{,}719$

Show solution

Understand

Find every single digit 0-9 that can replace the box so that 4,541,592 is greater than 45□6,719.

Givens

The fixed number is 4,541,592 (seven digits).
The compared number is 45□6,719 (seven digits) with one unknown digit in the hundred-thousands place.
We need 4,541,592 > 45□6,719.

Unknowns

All digits 0-9 that make the inequality true

Constraints

The box holds a single digit from 0 to 9.

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

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.

Execute

#1 Draw a Diagram 4.NBT.A.2

Both numbers are seven digits long: 4 541 592 versus 4 5□ 6 719. Compare from the highest place. The millions digit (4 = 4) and the hundred-thousands... first compare the leading digits: 4 = 4, then 5 = 5.

4\,5\,4\,1\,5\,9\,2 \;\text{vs}\; 4\,5\,\square\,6\,7\,1\,9

When two numbers have the same length, you compare digit by digit from the left, exactly like reading them aloud.

#2 Make a Systematic List 4.NBT.A.2

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 \;?\; \square

The leftmost place where two numbers differ is the one that decides which is larger.

#2 Make a Systematic List 4.NBT.A.2

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 < 4{,}546{,}719

If the deciding digits tie, you peek at the next place; here it shows 4 is too big.

#2 Make a Systematic List 4.NBT.A.2

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.

\square \in \{0,1,2,3\}

Any ten-thousands digit below 4 keeps the second number clearly smaller.

Answer: 0, 1, 2, 3

Review

With box = 3, 45□6,719 = 4,536,719 < 4,541,592 (true). With box = 4, 4,546,719 > 4,541,592 (false). The cutoff at 4 is correct, so 0,1,2,3 are exactly the digits that work.

Guess and check (tool 6): plug each digit 0-9 into the box and compare; you will find the inequality holds only for 0, 1, 2, and 3.

Standards · min grade 4

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

💡 This only needs Grade 4 comparing: line the numbers up and the first place they differ tells you the answer!