A is 30% more efficient than B. If A alone can complete a job in 23 days, how long will A and B together take to finish it?

Problem restatementA is 30% more efficient than B. A alone takes 23 days. Find the joint time of A and B working together.

Given data

• A's time alone = 23 days ⇒ A's rate = 1÷23 job/day.
• A is 30% more efficient than B ⇒ r_A = 1.3 × r_B.

Concept/ApproachConvert the efficiency statement into rates. From r_A = 1.3 r_B, obtain r_B. Then add rates to get the combined time.

Step-by-step calculationr_A = 1÷23r = r÷1.3 = (1÷23)÷1.3 = 1÷(23 × 1.3)Combined rate = r + r = 1÷23 + 1÷(23 × 1.3) = (1 + 1÷1.3) ÷ 231 + 1÷1.3 = 1 + 10÷13 = 23÷13So combined rate = (23÷13) ÷ 23 = 1÷13 job/dayTime together = 1 ÷ (1÷13) = 13 days

Verification/AlternativeLet B's rate be b. Then A's rate = 1.3b. If A = 1÷23, then 1.3b = 1÷23 ⇒ b = 1÷29.9. Sum = 1÷23 + 1÷29.9 ≈ 0.076923… ⇒ time ≈ 13 days.

Common pitfallsInterpreting “30% more efficient” as time reduction instead of rate increase. The percentage applies to rate, not days.

Final Answer13 days