A can complete a piece of work in 26 days and B can complete it in 16 days. If…

For two people who work on alternate full days, first write each person's one-day share as a fraction of the whole job — 1 divided by the number of days that person needs working alone — and then add the work up day by day in the order the turns actually fall, until the running total reaches the whole job. The day on which the work still outstanding is cleared in less than a full day fixes the answer: the whole days already completed, plus that fraction of a day.

1. A's one-day work is \(\frac{1}{26}\) of the job and B's one-day work is \(\frac{1}{16}\) of the job. Taking LCM(26, 16) = 208 as the total work in units, A does 208 ÷ 26 = 8 units per day and B does 208 ÷ 16 = 13 units per day.

2. The turns start with B and alternate, so each two-day cycle (B's day, then A's day) completes 13 + 8 = 21 units.

3. After 9 complete cycles (18 days) the work done is 9 × 21 = 189 units, leaving 208 − 189 = 19 units.

4. The 19th day is B's turn, since every odd-numbered day belongs to B; B's full day adds 13 units, taking the total to 189 + 13 = 202 units and leaving 208 − 202 = 6 units.

5. The 20th day is A's turn; A's full-day rate is 8 units, but only 6 units remain, so A needs \(\frac{6}{8}=\frac{3}{4}\) of that day to clear them.

6. Total time = 19 complete days + \(\frac{3}{4}\) of the 20th day = \(19\frac{3}{4}\) days.

Cross-check with plain fractions, without the 208-unit scaling: 9 cycles contribute \(9\times\left(\frac{1}{16}+\frac{1}{26}\right)=9\times\frac{21}{208}=\frac{189}{208}\) of the job; adding B's 19th day \(\left(\frac{1}{16}=\frac{13}{208}\right)\) gives \(\frac{202}{208}\) and leaves \(\frac{6}{208}\). At A's rate of \(\frac{1}{26}\) per day, clearing \(\frac{6}{208}\) of the job needs \(\frac{6}{208}\div\frac{1}{26}=\frac{6}{208}\times 26=\frac{3}{4}\) of a day — the same value the unit method gave, so the arithmetic is confirmed independently.

So the pair finishes the work in \(19\frac{3}{4}\) days: 19 complete days followed by three-quarters of the 20th day.