Find a number from digit clues

4.NBT.A.2 · take · grade 4

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

Find the four-digit number that satisfies all of the conditions below.

It is greater than $4000$ and less than $5000$ .
The hundreds digit is $6$ .
The sum of all of its digits is $20$ .
The tens digit and the ones digit are equal.

Show solution

Understand

Find the four-digit number that is between 4000 and 5000, has hundreds digit 6, has digits adding to 20, and has equal tens and ones digits.

Givens

The number is greater than 4000 and less than 5000.
The hundreds digit is 6.
The sum of all four digits is 20.
The tens digit equals the ones digit.

Unknowns

The four-digit number meeting all conditions.

Constraints

Each digit is 0 through 9.
All four conditions must hold together.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems

Each clue pins down one place value, so I work through the conditions in order to nail the thousands and hundreds digits, then solve a small equation for the matching tens and ones digits.

Execute

#7 Identify Subproblems 4.NBT.A.2

Being between 4000 and 5000 forces the thousands digit to be 4. The third clue says the hundreds digit is 6. So far the number looks like 4 6 _ _.

4000 < n < 5000 \Rightarrow \text{thousands} = 4,\quad \text{hundreds} = 6

A number in the 4000s must start with 4, and a stated hundreds digit is just read straight off.

#6 Guess and Check 4.NBT.A.2

The four digits add to 20. We already have 4 + 6 = 10, so the tens and ones digits must add to 10. Since they are equal, each is 5.

4 + 6 + d + d = 20 \Rightarrow 2d = 10 \Rightarrow d = 5

Two equal digits that total 10 can only both be 5.

#7 Identify Subproblems 4.NBT.A.2

Thousands 4, hundreds 6, tens 5, ones 5 gives 4655.

4655

Placing each found digit in its place builds the answer.

Answer: 4655

Review

Check 4655: it is between 4000 and 5000, the hundreds digit is 6, the digits 4+6+5+5 = 20, and the tens and ones digits are both 5. All four conditions hold.

List every number 46dd between 4600 and 4699 with equal last two digits (4600, 4611, ..., 4699) and pick the one whose digits sum to 20; only 4655 works.

Standards · min grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading place values and assembling a four-digit number from digit conditions.

💡 Take the clues one at a time: each one fixes a digit, and two equal digits summing to 10 must both be 5!