Problem
Big digits in high places
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.
Base-ten numbers
Your answer
Watch out It's easy to think you need six different digits for two separate numbers — really the same three cards 5, 3, 7 are just rearranged twice: once for the largest number, once for the smallest.
How to solve
Strategy Guess and Check — 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.
1STEP 1
Largest number
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.
3.NBT.A.2Make A Systematic ListPutting the biggest card in the hundreds place, the next in the tens place, and the smallest in the units place builds the largest three-digit number you can make from 7, 5, and 3.
Why?
A three-digit number counts its hundreds digit a hundred times, its tens digit ten times, and its units digit once, so the hundreds place is worth far more than the tens and the tens far more than the units.
Why?
Each place to the left is worth ten of the place on its right, because ten ones bundle into one ten and ten tens bundle into one hundred.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The size of the number is only those three place-amounts added together, so the total reaches its biggest when the place worth the most holds the biggest digit, the next-worth place the next digit, and the least-worth place the smallest digit.
Why?
The hundreds part, the tens part, and the units part cover the whole number with nothing left out and nothing counted twice, so making each part as large as it can be makes their sum as large as it can be.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Smallest number
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.
2.NBT.A.4Make A Systematic List3STEP 3
Subtract for the largest difference
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.
3.NBT.A.2Guess And CheckAnswer
396
753 - 357 = 396
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.
Takeaway
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!
- Largest number
- Smallest number
- Subtract for the largest difference
Where next?
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