Mathematical Reasoning
Tom and Sam are playing a game with numbers.
Question 1 · Multiple choice
Question
Tom and Sam are playing a game with numbers.
Tom takes every whole number starting from 198 and ending at 219 (including both 198 and 219). He breaks each number into its separate digits and adds them all together.
For example: if Tom had the number 198, he would calculate 1 + 9 + 8.
Sam does the exact same thing, but he uses every whole number starting from 366 and ending at 387 (including both 366 and 387).
Options
After they both finish their calculations, what is the difference between Sam's total sum and Tom's total sum?
- A.148
- B.149
- C.151
- D.152
- E.150
Correct answer
Explanation
Solution — Digit Sum Over a Range
Step 1 — Calculate Tom's sum (numbers 198 to 219)
First, count how many numbers Tom is adding:
219 − 198 + 1 = 22 numbers
Now sum the digits in each place value.
Hundreds digits:
- 198 and 199 start with a 1 (two 1s) → 2 × 1 = 2
- 200 to 219 start with a 2 (twenty 2s) → 20 × 2 = 40
- Hundreds total = 42
Tens digits:
- 198 and 199 have a 9 in the tens place (two 9s) → 2 × 9 = 18
- 200 to 209 have a 0 in the tens place (ten 0s) → 10 × 0 = 0
- 210 to 219 have a 1 in the tens place (ten 1s) → 10 × 1 = 10
- Tens total = 28
Units digits:
- 198 and 199 end in 8 and 9 → 8 + 9 = 17
- 200 to 209 cover all digits from 0 to 9 → 0 + 1 + 2 + … + 9 = 45
- 210 to 219 also cover all digits from 0 to 9 → sum is 45
- Units total = 17 + 45 + 45 = 107
Tom's total sum = 42 + 28 + 107 = 177
Step 2 — Calculate Sam's sum (numbers 366 to 387)
Sam is also adding 387 − 366 + 1 = 22 numbers.
Hundreds digits:
- Every number from 366 to 387 starts with a 3 (twenty-two 3s) → 22 × 3 = 66
- Hundreds total = 66
Tens digits:
- 366 to 369 have a 6 in the tens place (four 6s) → 4 × 6 = 24
- 370 to 379 have a 7 in the tens place (ten 7s) → 10 × 7 = 70
- 380 to 387 have an 8 in the tens place (eight 8s) → 8 × 8 = 64
- Tens total = 24 + 70 + 64 = 158
Units digits:
- 366 to 369 end in 6, 7, 8, 9 → 6 + 7 + 8 + 9 = 30
- 370 to 379 cover all digits from 0 to 9 → sum is 45
- 380 to 387 end in 0 to 7 → 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
- Units total = 30 + 45 + 28 = 103
Sam's total sum = 66 + 158 + 103 = 327
Step 3 — Find the difference
Subtract Tom's sum from Sam's sum:
327 − 177 = 150
The difference in their sums is 150.
Exam tip: Whenever a question asks for the sum of digits over a large range of numbers, grouping them into blocks of ten (e.g. 200–209) is the intended shortcut. It reveals the recurring sum of 45 for the units column and lets you tally up the tens and hundreds much faster than doing it manually.