Problem
Simplify a long fraction sum
Add 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64. A square of area 1 cut in half over and over gives one region per term. Find the sum of the six fractions.
Reasoning · Grade 5-1 Fraction Calculations
Your answer
Watch out The square looks full, but 1/64 is still empty, so the sum is not 1.
How to solve
Strategy Draw a Diagram — The picture converts the sum into an area. Once every term is a region of a square whose total area is 1, adding the six terms is the same as measuring how much of the square is coloured — and the fast way to measure that is to look at what is NOT coloured. Only one small cell is left blank, so instead of six additions I do one subtraction. Afterwards I use the pattern in the denominators to redo the same calculation without the picture, by rewriting each term as a difference so that neighbouring pieces cancel; that confirms the answer and shows why the trick is not special to this particular square.
1STEP 1
Count the terms before anything else
The bottoms double, so there are 6 terms.
2=2¹, 4=2², 8=2³, 16=2⁴, 32=2⁵, 64=2⁶ → 6 terms
Doubling denominators means halving the piece each time, so counting the terms is the same as counting how many times the square has been cut in half.
4.NF.A.1Look For A Pattern2STEP 2
Match each term to its region in the square
Each term matches one region of the square.
1 = 1/2+1/4+1/8+1/16+1/32+1/64+(blank cell)
Partitioning a shape into equal parts and calling one part a unit fraction of the whole is Grade 3 work; the only new thing here is doing it six times in a row.
3.G.A.2Draw A Diagram3STEP 3
Identify the blank cell
The unfilled region is 1/64.
blank cell = 1/64
Same size means same fraction of the whole — no measuring needed, just the fact that the cut was a halving.
3.G.A.2Draw A Diagram4STEP 4
Change focus: total minus the one piece left over
So the sum is 1 − 1/64 = 63/64.
1/2+1/4+1/8+1/16+1/32+1/64 = 1-1/64 = 64/64-1/64 = 63/64
Counting what is missing rather than what is there is the whole point of the picture: six pieces to add become one piece to remove.
5.NF.A.1Change Focus Count The Complement5STEP 5
Redo it without the picture: rewrite each term as a difference
Rewriting each term as a difference also gives 63/64.
(1-1/2)+(1/2-1/4)+(1/4-1/8)+(1/8-1/16)+(1/16-1/32)+(1/32-1/64)=1-1/64=63/64
Writing a term as a difference of two neighbours makes the middle of the sum collapse; this is the same 'simplify before you compute' move the unit is built around, and it works for sums far too long to draw.
5.NF.A.1Look For A PatternRewriting each term as a difference of two fractions makes the middle pieces cancel and leaves only the ends.
Why?
Each term's second piece is exactly the next term's first piece, so the pair adds to nothing and drops out of the sum.
🧱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?
Two fractions with the same denominator subtract by counting pieces, which is what makes the cancelling exact and not approximate.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
6STEP 6
Check the long way, with a common denominator
The long way with a common bottom gives 63/64.
32/64+16/64+8/64+4/64+2/64+1/64=63/64
The slow method agrees with the fast one, and the numerators 32, 16, 8, 4, 2, 1 are themselves a halving pattern — one short of 64, which is exactly why the answer is one sixty-fourth short of the whole square.
5.NF.A.1Look For A PatternAnswer
63/64
1 − 1/64 = 63/64
The answer is a fraction just under 1, which is exactly what the picture shows: almost all of the unit square is coloured, with only a single tiny cell left blank. It also passes a lower bound check, since the first term alone is 1/2 and the sum must therefore exceed 1/2 — and 63/64 does. It can never reach 1, because at every stage exactly half of what remains is left uncoloured. Three independent routes agree on 63/64: the area argument (1 minus the blank 1/64 cell), the difference rewrite in which everything between the first and last term cancels, and the plain common-denominator addition 32 + 16 + 8 + 4 + 2 + 1 = 63 over 64. The term count is six, matching the six labelled regions in the square.
Takeaway
When each piece is half of what is left, the pieces almost fill the whole — so add up what is missing instead: just one 1/64 cell, leaving 63/64.
- Count the terms before anything else
- Match each term to its region in the square
- Identify the blank cell
- Change focus: total minus the one piece left over
- Redo it without the picture: rewrite each term as a difference
- Check the long way, with a common denominator
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