Problem
Find patterns in mixed-number sequences and sum
A list of mixed numbers follows a rule: 1 2/13, 3 4/13, 5 6/13, ..., up to 11 12/13. I must figure out the rule, list every term, and add them all up.
FractionsOperations
Your answer
Watch out Don't stop after adding the whole-number parts — the fraction parts still need to be added in and combined with the whole-number total.
How to solve
Strategy Look for a Pattern — Find the rule (whole parts are odd numbers, numerators are even numbers), list all six terms, then split the sum into a whole-number part and a fraction part — two easy subproblems.
1STEP 1
Find the rule and list the terms
The whole parts go 1, 3, 5, 7, 9, 11 (odd numbers) and the numerators go 2, 4, 6, 8, 10, 12 (the next even number), all over 13. So the full list is 1 2/13, 3 4/13, 5 6/13, 7 8/13, 9 10/13, 11 12/13 — six terms.
1 2/13, 3 4/13, 5 6/13, 7 8/13, 9 10/13, 11 12/13
Spotting that each whole part jumps by 2 and each numerator jumps by 2 makes the pattern obvious.
4.OA.C.5Look For A Pattern2STEP 2
Add the whole-number parts
Add the whole parts: 1 + 3 + 5 + 7 + 9 + 11 = 36.
1+3+5+7+9+11=36
Pairing 1+11, 3+9, 5+7 gives three 12s = 36 — a quick whole-number sum.
4.OA.A.3Identify Subproblems3STEP 3
Add the fraction parts
Add the fraction parts (same denominator 13): (2+4+6+8+10+12)/13 = 42/13. Since 42/13 = 3 with remainder 3, this is 3 3/13.
(2+4+6+8+10+12)/13=42/13=3 3/13
Like denominators let you add numerators, then regroup the improper fraction into a mixed number.
4.NF.B.3Identify Subproblems4STEP 4
Combine the two parts
Add the whole-part total to the fraction-part total: 36 + 3 3/13 = 39 3/13.
36+3 3/13=39 3/13
Putting the whole and fractional totals back together gives the full mixed-number sum.
4.NF.B.3Identify SubproblemsAdding the whole-part subtotal to the fraction-part subtotal gives the list's true total, so the whole parts and the fraction parts may be summed in separate groups and then joined.
Why?
The whole list is really one long addition in which every term hands over a whole part and a fraction part.
Why?
A mixed number such as 3 4/13 is exactly its whole part 3 joined with its fraction part 4/13, with nothing left out and nothing counted twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
In that long addition the pieces can be reordered to bring every whole part to the front and every fraction part behind them, and the total stays the same.
🧱Commutativity of additionPutting a with b or b with a piles up the same total — order does not change a sum.
Why?
The reordered pieces can then be bundled and added group by group — the whole parts into one subtotal and the fraction parts into another — before those two subtotals are joined.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Answer
39 3/13
36 + 3 3/13 = 39 3/13
Six terms averaging about 6.5 each give roughly 6 × 6.5 ≈ 39, which matches 39 3/13. The fraction parts stay consistent in thirteenths, and the leftover 3/13 is a proper fraction, so the answer is sensible.
Takeaway
This only needs Grade 4 pattern-spotting plus same-denominator fraction adding — split the wholes from the thirteenths!
- Find the rule and list the terms
- Add the whole-number parts
- Add the fraction parts
- Combine the two parts
Where next?
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