Problem
Find the rule in a fraction sequence
Three rows of fractions each hide a rule. A missing fraction must continue that rule and agree with both neighbours. Track how the numerator and the denominator each move, then fill the blanks.
Reasoning · Grade 3-2 Applying Fractions
Your answer
Watch out 18/20 begs to become 9/10, but the rule lives in the unreduced form.
How to solve
Strategy Look for a Pattern — Each row is a sequence, so the key move is spotting the pattern. I treat the numerator and the denominator as two separate little sequences (and for row 2 I rewrite every term with the same denominator) so the rule becomes obvious.
1STEP 1
(1) Track numerator and denominator separately
Row 1 drops the top by 2 and lifts the bottom by 3: 18/20.
(20-2)/(17+3) = 18/20
Splitting a fraction sequence into a top sequence and a bottom sequence turns one hard pattern into two easy counting patterns.
4.OA.C.5Look For A Pattern2STEP 2
(2) Rewrite every term in eighths
Over a denominator of 8, row 2 climbs by 3: 21/8.
18/8 → (18+3)/8 = 21/8 = 2 5/8
Giving every fraction the same denominator lets me compare them as plain whole-number counts of eighths, so the steady +3 jump pops out.
4.NF.A.2Organize Information In More WaysRewriting every term in eighths turns the fractions into plain counts of eighths, and the steady jump of 3 becomes visible.
Why?
Once every fraction is cut into pieces of the same size, comparing them is just comparing how many pieces each one holds.
🧱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.
Why?
Multiplying a fraction's top and bottom by the same number renames it without changing its size, so the rewrite is safe.
🧱Scaling top and bottom keeps a fraction's valueMultiplying or dividing a fraction's top and bottom by the same number only renames it — the pieces change size but the amount you hold does not.
3STEP 3
(3) Spot the Fibonacci build-up
Row 3 adds the two before it: 34/55.
(13+21)/(21+34) = 34/55
When a list grows by adding the two previous terms, you just keep adding neighbours to make the next one.
4.OA.C.5Look For A PatternAnswer
18/20, 21/8, 34/55
Each answer sits neatly between its neighbours: in (1) 18/20 has numerator between 20 and 16 and denominator between 17 and 23; in (2) 21/8 lies between 18/8 and 24/8; in (3) 34/55 continues both Fibonacci lists, and its denominator 55 equals 21+34 as expected. All three fit the established rules.
Takeaway
Watch the top numbers and the bottom numbers as their own little patterns, and a row of tricky fractions turns into easy counting!
- (1) Track numerator and denominator separately
- (2) Rewrite every term in eighths
- (3) Spot the Fibonacci build-up