Working alone, the times taken by Anu, Tanu and Manu to complete any job are in the ratio 5:8:10. They accept a job which they can finish in 4 days if they all work together for 8 hours per day. However, Anu and Tanu work together for the first 6 days, working 6 hours 40 minutes per day. Then, the number of hours that Manu will take to complete the remaining job working alone is
Question:
Working alone, the times taken by Anu, Tanu and Manu to complete any job are in the ratio 5:8:10. They accept a job which they can finish in 4 days if they all work together for 8 hours per day. However, Anu and Tanu work together for the first 6 days, working 6 hours 40 minutes per day. Then, the number of hours that Manu will take to complete the remaining job working alone is
Working alone, the times taken by Anu, Tanu and Manu to complete any job are in the ratio 5:8:10. They accept a job which they can finish in 4 days if they all work together for 8 hours per day. However, Anu and Tanu work together for the first 6 days, working 6 hours 40 minutes per day. Then, the number of hours that Manu will take to complete the remaining job working alone is
Options
Answer: (1) 6
Explanation:
Step 1 — Let times (working alone) Let Anu : Tanu : Manu = 5 : 8 : 10. Put times = 5k, 8k, 10k hours respectively. Step 2 — Use the “all three together” information They finish the job in 4 days when working 8 hours per day, so total working hours together = 4 × 8 = 32 hours. Hence combined rate = 1 job / 32 hours. Combined rate in terms of k: 1/(5k) + 1/(8k) + 1/(10k) = (1/k)·(1/5 + 1/8 + 1/10) = (1/k)·(17/40). So (17/40)·(1/k) = 1/32 ⇒ 1/k = (40)/(17·32) ⇒ k = (17·32)/40 = 68/5 hours = 13.6 hours. Thus individual times: Anu = 5k = 68 hours, Tanu = 8k = 544/5 = 108.8 hours, Manu = 10k = 136 hours. Step 3 — Work done by Anu and Tanu in the first 6 days They work 6 days at 6 hours 40 minutes per day. 6 h 40 min = 6 + 40/60 = 6 + 2/3 = 20/3 hours per day. Total hours they worked = 6 × (20/3) = 40 hours. Their combined rate = 1/68 + 1/(544/5) = 1/68 + 5/544 = 13/544 job/hour. Work done in 40 hours = 40 × (13/544) = 65/68 of the job Step 4 — Remaining work and time for Manu alone Remaining = 1 − 65/68 = 3/68 of the job. Manu’s rate = 1/136 job/hour. Time Manu needs = (remaining) ÷ (Manu’s rate) = (3/68) ÷ (1/136) = (3/68) × 136 = 6 hours. Answer: Manu will take 6 hours to complete the remaining job
Explanation:
Step 1 — Let times (working alone) Let Anu : Tanu : Manu = 5 : 8 : 10. Put times = 5k, 8k, 10k hours respectively. Step 2 — Use the “all three together” information They finish the job in 4 days when working 8 hours per day, so total working hours together = 4 × 8 = 32 hours. Hence combined rate = 1 job / 32 hours. Combined rate in terms of k: 1/(5k) + 1/(8k) + 1/(10k) = (1/k)·(1/5 + 1/8 + 1/10) = (1/k)·(17/40). So (17/40)·(1/k) = 1/32 ⇒ 1/k = (40)/(17·32) ⇒ k = (17·32)/40 = 68/5 hours = 13.6 hours. Thus individual times: Anu = 5k = 68 hours, Tanu = 8k = 544/5 = 108.8 hours, Manu = 10k = 136 hours. Step 3 — Work done by Anu and Tanu in the first 6 days They work 6 days at 6 hours 40 minutes per day. 6 h 40 min = 6 + 40/60 = 6 + 2/3 = 20/3 hours per day. Total hours they worked = 6 × (20/3) = 40 hours. Their combined rate = 1/68 + 1/(544/5) = 1/68 + 5/544 = 13/544 job/hour. Work done in 40 hours = 40 × (13/544) = 65/68 of the job Step 4 — Remaining work and time for Manu alone Remaining = 1 − 65/68 = 3/68 of the job. Manu’s rate = 1/136 job/hour. Time Manu needs = (remaining) ÷ (Manu’s rate) = (3/68) ÷ (1/136) = (3/68) × 136 = 6 hours. Answer: Manu will take 6 hours to complete the remaining job
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