Problem
Division applications challenge
Four harder problems. An age gap stays fixed as both people age, and every flowchart step must stay a whole number. Both the stones and the weekdays repeat in cycles. Answer all four.
Reasoning · Grade 3-2 Applications of Division
Your answer
Watch out Time seems to widen the age gap, but the gap stays 32 — only the ratio moves.
How to solve
Strategy Identify Subproblems — This is four separate puzzles, so I solve each as its own subproblem. The age puzzle is a difference held constant (the gap 32 equals 2 parts when father is 3 parts). The flowchart is a chain of operations followed step by step from 720. The stones and the calendar both rest on a repeating cycle, so I look for the pattern and use division with remainder.
1STEP 1
P1: read the age difference as parts
At three times, the gap is two parts, so one part is 16 — 16 years old.
3 parts - 1 part = 2 parts = 32 → 1 part = 32 ÷ 2 = 16
The age gap never changes, and '3 times' makes that gap equal to 2 of Minsu's ages, so Minsu = half of 32.
3.OA.A.3Guess And CheckThe gap between the two ages never changes, so when one age is three times the other that gap equals two of the smaller age.
Why?
Both people grow older by the same amount every year, so sliding both ages up together leaves the distance between them untouched.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
Why?
Three of the smaller age is the smaller age together with two more of it, so the gap is exactly those two extra copies.
🧱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
P2: follow the flowchart from 720
The flowchart applies each arrow's operation in order from 720.
720 → 240 → 120 → 60 → 30 → 15 → 45 → 135 → 405 → 810 → 1620 → 540 → 180 → 90
A flowchart is just a recipe: do each arrow's operation in turn, and every step lands on a whole number, ending at 90.
4.OA.A.3Work Backwards3STEP 3
P3: split 145 stones into repeating blocks
The stones cycle in sevens: 145 = 7 × 20 + 5.
145 ÷ 7 = 20 remainder 5; leftover 5 = (° ● ● ° ●) = 2 white, 3 black
Breaking the long row into equal blocks of 7 turns 'count to 145' into 'count 20 blocks plus a few extras'.
4.NBT.B.6Look For A Pattern4STEP 4
P3: total each color and compare
Twenty cycles plus the five give white 42 and black 103, so black leads by 61.
white = 40 + 2 = 42, black = 100 + 3 = 103, 103 - 42 = 61
Each block already has 3 more black than white, so the big difference comes mostly from 20 blocks (20 x 3 = 60) plus 1 from the leftovers.
3.OA.A.3Identify Subproblems5STEP 5
P4: count the days from 2019 to 2025
From 2019 to 2025 the days total 2192, two leap years included.
366 + 365 + 365 + 365 + 366 + 365 = 2192 days
Only the two leap-day crossings add an extra day, so the total is six years of 365 plus 2.
4.NBT.B.4Identify Subproblems6STEP 6
P4: use the 7-day cycle
Counting the days with two leap years leaves 1, giving Saturday.
2192 ÷ 7 = 313 remainder 1; Fri → Sat
Whole weeks keep the weekday; the single leftover day pushes Friday to Saturday.
4.NBT.B.6Look For A PatternAnswer
16 years old, the flowchart value, black leads by 61, Saturday
P1: if Ethan is 16, father is 16 + 32 = 48 = 3 x 16, correct. P2: every flowchart step stayed a whole number and the operations balanced (five divisions and... checking 720 - ＞ 90 is one-eighth, consistent with the mix of x and / ), and reversing from 90 back through the chain returns 720. P3: 42 + 103 = 145 stones total, matching, and 5 black per block over 2 white per block makes black the majority. P4: remainder 1 is in range 0-6; six ordinary years would be remainder 6 (Fri+6=Thu), and the two extra leap days add 2 more (Thu+2 = Sat), agreeing.
Takeaway
Break a scary multi-part page into small puzzles: equal-parts for ages, follow-the-arrows for the flowchart, and divide-with-remainder for the repeating stones and calendar -- all Grade 3-4 moves!
- P1: read the age difference as parts
- P2: follow the flowchart from 720
- P3: split 145 stones into repeating blocks
- P3: total each color and compare
- P4: count the days from 2019 to 2025
- P4: use the 7-day cycle