Big digits in high places

2.NBT.A.43.NBT.A.2 · take · grade 3

Archetype: Build the Largest or Smallest Value from Digit Cards · step in a 7-type progression

Using the number cards $5$ , $3$ , and $7$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 5, 3, and 7 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 5, 3, and 7.
Each card is used exactly once across the two numbers... but with only three cards, the standard reading is to form the largest and smallest three-digit numbers from the same three digits and subtract.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 5, 3, 7 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 5, 3, and 7 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

To make a difference biggest, make the first number as large as possible and the second as small as possible; the place-value principle (big digits in high places) tells us how to arrange the cards, and a quick systematic check confirms it.

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 7, 5, 3 from hundreds down gives the largest three-digit number 753.

753

The hundreds place is worth the most, so the largest digit belongs there.

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

A number is smallest when the smallest digit sits in the highest place. Ordering 3, 5, 7 from hundreds down gives the smallest three-digit number 357.

357

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 753 minus 357. Subtracting with borrowing gives 396.

753 - 357 = 396

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 753 - 357 = 396

Review

753 and 357 use each card 5, 3, 7 exactly once, and 753 - 357 = 396 is positive and under 753, which is sensible. Any other pairing of these digits gives a smaller spread, so 396 is the maximum.

List all distinct three-digit numbers from 5, 3, 7 (357, 375, 537, 573, 735, 753) and check max minus min: 753 - 357 = 396 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!