Find the rule in a fraction sequence

3.OA.D.93.NF.A.1 · take · grade 3

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

The fractions below are listed following a fixed rule. Find the fraction that will be placed in the 41st position.

$\frac{1}{2},\ \frac{1}{3},\ \frac{2}{3},\ \frac{1}{4},\ \frac{2}{4},\ \frac{3}{4},\ \frac{1}{5},\ \cdots$

Show solution

Understand

Fractions are listed by a rule: first all fractions with denominator 2, then denominator 3, then 4, and so on, with numerators counting up from 1 to one less than the denominator within each group. I need the fraction in the 41st position.

Givens

The list is 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, ...
Group with denominator (k+1) has k fractions, with numerators 1, 2, ..., k.
I want the 41st fraction.

Unknowns

The fraction at position 41.

Constraints

Within a group the denominator is fixed and numerators run 1 up to denominator minus 1.
Groups appear in order of increasing denominator.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list groups by denominator, and the group with denominator (k+1) contributes k terms. Adding up these group sizes (1+2+3+...) tells me which group position 41 falls in, then I count within that group.

Execute

#5 Look for a Pattern 3.OA.D.9

Denominator 2 has 1 fraction, denominator 3 has 2, denominator 4 has 3, and in general denominator d has d-1 fractions. The running totals are 1, 3, 6, 10, 15, 21, 28, 36, 45.

1,\ 1{+}2{=}3,\ {+}3{=}6,\ {+}4{=}10,\ {+}5{=}15,\ {+}6{=}21,\ {+}7{=}28,\ {+}8{=}36,\ {+}9{=}45

Each new denominator adds one more fraction than the previous group, a clear growing pattern.

#9 Solve an Easier Related Problem 3.OA.D.9

After denominator 9 the total is 36 fractions (positions 1 through 36). The next group is denominator 10 (9 fractions, positions 37 through 45). So position 41 is in the denominator-10 group.

36 < 41 \le 45 \Rightarrow \text{denominator } 10

Using the running totals reduces a far-off 41st term to 'which short group is it in'.

#5 Look for a Pattern 3.NF.A.1

Position 41 is the (41 - 36) = 5th fraction in the denominator-10 group, whose numerators count 1, 2, 3, ... So the numerator is 5.

41 - 36 = 5 \Rightarrow \frac{5}{10}

Within a group the numerator equals the place number, so the 5th term has numerator 5.

Answer: 5/10

Review

Positions 37-45 hold 1/10 through 9/10; the 41st is the 5th of these, which is 5/10. This fits the pattern (numerator counts up, denominator fixed at 10). The numerator 5 is between 1 and 9, as required for that group.

Make a systematic list (tool 2): keep listing groups 1/10, 2/10, 3/10, 4/10, 5/10 from position 37; the 5th lands on position 41, confirming 5/10.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding that group sizes grow by 1 and using running totals to locate position 41.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Reading off the numerator and denominator of the located fraction.

💡 This only needs Grade 3 pattern sense: add up the group sizes until you reach 41, then count inside that group!