Problem
Sum of every digit written
Write down every whole number from 0 to 99. Add up every single digit that appears. For instance 47 contributes 4 + 7 = 11. Find the total of all the digits.
Reasoning · Grade 4-1 Numbers and Digit Counts
Your answer
Watch out It reads like adding the numbers, but what gets added is the digits.
How to solve
Strategy Look for a Pattern — Adding 100 separate digit sums one at a time is slow and easy to slip up on, so I look for a pattern that collapses the work. First I rewrite every number with two digits, which makes all 100 numbers look alike. Then I try the same question on the much shorter list 0 to 9 to discover the pairing trick, and finally I carry that trick over to the full list. A second, completely different count — how many times each digit gets written — gives an independent check.
1STEP 1
Give every number two digits
Write every number with two digits, 00 to 99.
0 → 00, 7 → 07, 47 → 47
Writing 7 as 07 is the same trick you already use when you line numbers up in columns — the extra 0 holds a place but is worth nothing, so it cannot change the sum.
4.NBT.A.2Organize Information In More Ways2STEP 2
Try the easier list 0 to 9 first
The short list 0 to 9 adds to 45.
(0+9)+(1+8)+(2+7)+(3+6)+(4+5) = 5 × 9 = 45
A ten-number list is short enough to check by hand, so you can see the pairing trick really works before trusting it on the long list.
3.OA.D.9Solve An Easier Related Problem3STEP 3
Pair each number with its partner up to 99
Pairing 47 with 52 and so on, each pair gives 18.
(4+7)+(5+2) = 11 + 7 = 18
Partner numbers like 47 and 52 have digits that fill each other in — 4 and 5 make 9, 7 and 2 make 9 — so no matter which pair you pick the digits always total the same 18.
3.OA.D.9Look For A PatternPairing each number with its partner up to 99 makes every pair's digits total the same 18.
Why?
As one number climbs the other falls by the same amount, so the pair always lands on the same total.
🧱Pairing from both ends gives a constant sumIn an evenly spaced list the smallest with the largest, the next with the next, always make the same total, so a long sum becomes a multiplication.
Why?
The tens digits fill each other up to 9 and so do the ones digits, because each place is counted separately.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
4STEP 4
Count how many pairs there are
With 100 numbers there are 50 pairs.
100 ÷ 2 = 50
Naming the first pair and the last pair shows the pairing starts at 00 and stops at 49, which is a safe way to count without missing one or counting one twice.
4.OA.A.3Make A Systematic List5STEP 5
Multiply the number of pairs by the value of each pair
Multiplying gives 50 × 18 = 900.
50 × 18 = 900
Once every group is the same size, adding all the groups turns into one multiplication — exactly what multiplication was invented for.
4.NBT.B.5Look For A Pattern6STEP 6
Check it a completely different way
Counting by place, 20 × 45 = 900 too.
20 × 45 = 900
Counting digit by digit instead of number by number reorganizes the very same pile of digits, so it has to give the same total — which makes it a real check, not a repeat of the first method.
4.NBT.B.5Organize Information In More WaysAnswer
900
50 × 18 = 900
There are 100 numbers and each pair of numbers uses 18 digits' worth, so the average number contributes 9; that gives 100 times 9 = 900, matching the answer. The total also has to lie between 0 and 100 times 18 = 1800, because 99 has the largest possible digit sum of 18, and 900 sits comfortably inside that range. The answer is a plain total of digit values, so it carries no units.
Takeaway
Pair the first number with the last, the second with the second-last, and so on — when every pair has the same total, a huge addition turns into one small multiplication.
- Give every number two digits
- Try the easier list 0 to 9 first
- Pair each number with its partner up to 99
- Count how many pairs there are
- Multiply the number of pairs by the value of each pair
- Check it a completely different way