Problem
Fraction sequences and their tenth terms
Two lists of fractions, A and B, are given. Each list follows a rule of its own. Writing everything in sixths makes the rules visible. Find the gap between the 10th fraction of each list.
Reasoning · Grade 4-2 Adding and Subtracting Fractions
Your answer
Watch out A's rule tempts, but B adds the two terms before — a different rule entirely.
How to solve
Strategy Look for a Pattern — The two lists mix whole numbers, proper fractions and mixed numbers, which makes the rules invisible. So the first move is to rewrite every term the same way — as a number of sixths — and then look at what the numerators do. Once each rule is clear, I just list the numerators out to the 10th one for each list and subtract.
1STEP 1
Rewrite list A as a count of sixths
Rewrite A as counts of sixths.
2/6, 7/6, 12/6, 17/6, 22/6, 27/6
A mixed number is just a whole number plus a fraction, and one whole is 6 sixths, so counting everything in sixths is the same skill as counting everything in cents instead of dollars-and-cents.
4.NF.B.3Organize Information In More WaysRewriting every term as a count of sixths turns a list of mixed numbers into a list of plain whole numbers.
Why?
One whole is six sixths, so a mixed number can be counted entirely in pieces of the same size.
🧱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?
Renaming a number in smaller equal pieces does not change how big it is, so the rewritten list is the same list.
🧱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.
2STEP 2
Find the rule for A and reach its 10th term
A grows by 5 sixths, so its 10th is 47/6.
(2 + 5 × 9)/6 = 47/6 = 7 5/6
Counting up by 5s is a Grade 4 pattern rule; the only new part is remembering that from the 1st term to the 10th there are 9 steps, not 10.
4.OA.C.5Look For A Pattern3STEP 3
Rewrite list B as a count of sixths
Rewrite B in sixths as well.
2/6, 3/6, 5/6, 8/6, 13/6, 21/6, 34/6
Using the same denominator for every term of both lists means the numerators alone carry the whole story.
4.NF.B.3Organize Information In More Ways4STEP 4
Find the rule for B and list out to the 10th term
B adds the two before it, so its 10th is 144/6.
55/6 + 89/6 = 144/6 = 24
Once the rule is 'add the two before', writing the terms in a row and adding neighbours is bookkeeping a Grade 4 student can do without a single new idea.
4.OA.C.5Make A Systematic List5STEP 5
Subtract the two 10th fractions
The gap is 97/6 = 16 1/6.
144/6 - 47/6 = 97/6 = 16 1/6
Subtracting sixths from sixths never needs a common denominator to be found — both lists were built in sixths from the start.
4.NF.B.3Organize Information In More WaysAnswer
16 1/6
144/6 − 47/6 = 97/6
List A creeps up by 5/6 each time, so after 10 terms it should be somewhere near 8 — and 7 5/6 is. List B doubles up faster and faster, so its 10th term should be much bigger — 24 is. A difference of about 24 - 8 = 16 is exactly the size we got, and the leftover 1/6 is right because 24 has no sixths while 7 5/6 has five of them.
Takeaway
Write every fraction over the same denominator first — the hidden rule usually pops right out of the numerators!
- Rewrite list A as a count of sixths
- Find the rule for A and reach its 10th term
- Rewrite list B as a count of sixths
- Find the rule for B and list out to the 10th term
- Subtract the two 10th fractions