Question:
Arun, Tarun and Varun work for 24, 21 and 15 days respectively and get paid 2160, 2400 and 2160 rupees respectively. They get paid the same even if they work for a partial day. If the work has to be completed within 10 days or less, what is the minimum amount that has to be paid to complete the entire task?
Arun, Tarun and Varun work for 24, 21 and 15 days respectively and get paid 2160, 2400 and 2160 rupees respectively. They get paid the same even if they work for a partial day. If the work has to be completed within 10 days or less, what is the minimum amount that has to be paid to complete the entire task?
Show Hint
For each worker, compare pay divided by the number of days they would take alone; this ratio equals their total quoted payment and shows who is most cost-efficient. Then check whether skipping the costliest worker still lets the remaining workers finish within the deadline.
Updated On: Aug 17, 2026
Show Solution
Verified By Collegedunia
Correct Answer: 2160
Approach Solution - 1
Approach: When wages are flat per job, the smart lens is cost per unit of work. Find each worker's cost-per-unit, pick the cheapest, but make sure the chosen team can still finish within 10 days.
Step 1: Take total work \(= \text{LCM}(24, 21, 15) = 840\) units. Daily output:
Arun \(= 840/24 = 35\), Tarun \(= 840/21 = 40\), Varun \(= 840/15 = 56\) units/day.
Step 2: Each man's total wage is fixed for the whole job, so his cost per unit is (total pay)/840:
Arun \(= 2160/840 = 18/7\), Tarun \(= 2400/840 = 20/7\), Varun \(= 2160/840 = 18/7\) rs/unit.
Step 3: Arun and Varun are the cheap pair at \(18/7\) per unit; Tarun is dearer. So build the team from Arun and Varun only and pay Tarun nothing.
Step 4: Can Arun + Varun finish in 10 days? Combined rate \(= 35 + 56 = 91\) units/day, so time \(= 840/91 \approx 9.23\) days \(\le 10\). The deadline is met.
Step 5: Since every one of the 840 units is done at \(18/7\) rs, cost \[ = 840 \times \frac{18}{7} = 120 \times 18 = 2160. \]
Final Answer: Minimum amount \(= \boxed{2160}\) rupees.
Step 1: Take total work \(= \text{LCM}(24, 21, 15) = 840\) units. Daily output:
Arun \(= 840/24 = 35\), Tarun \(= 840/21 = 40\), Varun \(= 840/15 = 56\) units/day.
Step 2: Each man's total wage is fixed for the whole job, so his cost per unit is (total pay)/840:
Arun \(= 2160/840 = 18/7\), Tarun \(= 2400/840 = 20/7\), Varun \(= 2160/840 = 18/7\) rs/unit.
Step 3: Arun and Varun are the cheap pair at \(18/7\) per unit; Tarun is dearer. So build the team from Arun and Varun only and pay Tarun nothing.
Step 4: Can Arun + Varun finish in 10 days? Combined rate \(= 35 + 56 = 91\) units/day, so time \(= 840/91 \approx 9.23\) days \(\le 10\). The deadline is met.
Step 5: Since every one of the 840 units is done at \(18/7\) rs, cost \[ = 840 \times \frac{18}{7} = 120 \times 18 = 2160. \]
Final Answer: Minimum amount \(= \boxed{2160}\) rupees.
Was this answer helpful?
2
2
Show Solution
Verified By Collegedunia
Approach Solution -2
Approach: Since each person is paid a flat daily wage but produces a fixed daily share of work, work out the cost per unit of total work for each, not the daily wage, and see who is cheapest.
Daily wage: Arun \(2160/24=90\), Tarun \(2400/21\approx114.3\), Varun \(2160/15=144\). Daily work rate (as a fraction of the whole job): Arun \(\tfrac1{24}\), Tarun \(\tfrac1{21}\), Varun \(\tfrac1{15}\).
Cost per unit of the whole job \(=\dfrac{\text{daily wage}}{\text{daily work rate}}\), which simplifies back to each person's total quoted payment: Arun \(2160\), Tarun \(2400\), Varun \(2160\) \(-\) meaning finishing the ENTIRE job using only Arun, only Varun, or any split between the two always costs exactly Rs. \(2160\), while any contribution from Tarun costs more per unit of work.
So the cheapest plan uses only Arun and Varun together. Their combined daily rate is \(\tfrac{1}{24}+\tfrac{1}{15}=\tfrac{13}{120}\), so together they finish in \(\tfrac{120}{13}\approx9.23\) days, within the \(10\)-day limit, at the minimum possible cost of \[ \boxed{\text{Rs. }2160} \]
Daily wage: Arun \(2160/24=90\), Tarun \(2400/21\approx114.3\), Varun \(2160/15=144\). Daily work rate (as a fraction of the whole job): Arun \(\tfrac1{24}\), Tarun \(\tfrac1{21}\), Varun \(\tfrac1{15}\).
Cost per unit of the whole job \(=\dfrac{\text{daily wage}}{\text{daily work rate}}\), which simplifies back to each person's total quoted payment: Arun \(2160\), Tarun \(2400\), Varun \(2160\) \(-\) meaning finishing the ENTIRE job using only Arun, only Varun, or any split between the two always costs exactly Rs. \(2160\), while any contribution from Tarun costs more per unit of work.
So the cheapest plan uses only Arun and Varun together. Their combined daily rate is \(\tfrac{1}{24}+\tfrac{1}{15}=\tfrac{13}{120}\), so together they finish in \(\tfrac{120}{13}\approx9.23\) days, within the \(10\)-day limit, at the minimum possible cost of \[ \boxed{\text{Rs. }2160} \]
Was this answer helpful?
0
0
Show Solution
Verified By Collegedunia
Approach Solution -3
Concept:
Step 1: Find daily wage and daily rate for each person.
Arun: wage $=2160/24=90$ rupees/day, rate $=1/24$ of the job per day.
Tarun: wage $=2400/21=800/7$ rupees/day, rate $=1/21$ of the job per day.
Varun: wage $=2160/15=144$ rupees/day, rate $=1/15$ of the job per day.
Step 2: Notice a shortcut: wage divided by rate always gives back the total pay.
For one person, (daily wage)/(daily rate) equals (pay/days)/(1/days), which is just pay. So Arun and Varun both give $2160$, while Tarun gives $2400$. This ratio measures how expensive it is to get one full job done by that person alone, and Tarun is the costliest.
Step 3: Exclude the costliest worker and check the deadline.
Since Tarun costs more per unit of work, use only Arun and Varun. Their combined rate is $\dfrac{1}{24}+\dfrac{1}{15}=\dfrac{5+8}{120}=\dfrac{13}{120}$ of the job per day, so together they finish in $\dfrac{120}{13}\approx9.23$ days, which is within the $10$-day limit.
Step 4: Compute the total cost for this pair.
Total cost $=\dfrac{\text{sum of daily wages}}{\text{sum of daily rates}}=\dfrac{90+144}{13/120}=\dfrac{234\times120}{13}=18\times120=2160$.
Final Answer: The minimum amount to be paid is $2160$ rupees.
- Think in terms of two separate quantities for each person: a daily wage (pay divided by the number of days they need to finish the job alone) and a daily work rate (the fraction of the job completed in one day).
- For any set of workers working together for $T$ days, total cost equals $T$ times the sum of their daily wages, and the job finishes when $T$ times the sum of their daily rates equals $1$.
- Combining these two facts, the cost of using a particular set of workers to finish the whole job equals (sum of daily wages) divided by (sum of daily rates); comparing this ratio for different sets shows which combination is cheapest.
Step 1: Find daily wage and daily rate for each person.
Arun: wage $=2160/24=90$ rupees/day, rate $=1/24$ of the job per day.
Tarun: wage $=2400/21=800/7$ rupees/day, rate $=1/21$ of the job per day.
Varun: wage $=2160/15=144$ rupees/day, rate $=1/15$ of the job per day.
Step 2: Notice a shortcut: wage divided by rate always gives back the total pay.
For one person, (daily wage)/(daily rate) equals (pay/days)/(1/days), which is just pay. So Arun and Varun both give $2160$, while Tarun gives $2400$. This ratio measures how expensive it is to get one full job done by that person alone, and Tarun is the costliest.
Step 3: Exclude the costliest worker and check the deadline.
Since Tarun costs more per unit of work, use only Arun and Varun. Their combined rate is $\dfrac{1}{24}+\dfrac{1}{15}=\dfrac{5+8}{120}=\dfrac{13}{120}$ of the job per day, so together they finish in $\dfrac{120}{13}\approx9.23$ days, which is within the $10$-day limit.
Step 4: Compute the total cost for this pair.
Total cost $=\dfrac{\text{sum of daily wages}}{\text{sum of daily rates}}=\dfrac{90+144}{13/120}=\dfrac{234\times120}{13}=18\times120=2160$.
Final Answer: The minimum amount to be paid is $2160$ rupees.
Was this answer helpful?
0
0
Top CAT Quantitative Aptitude Questions
- In a triangle ABC,AB=AC=8cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through B and C. Then the area,in sq.cm,of the overlapping region between the two circles is
- The simple interest accrued on an amount of Rs.25,000 at the end of three years is Rs.7,500. What would be the compound interest accrued on the same amount at the same rate in the same period?
- What is the sum of all roots of the equation \(|x+4|^2–10|x+4|=24\)?
- A solid metal sphere is melted and smaller spheres of equal radii are formed. 10% of the volume of the sphere is lost in the process. The smaller spheres have a radius which is \(\frac{1}{9}\text{th}\) the larger sphere. If 10 litres of paint were needed to paint the larger sphere,how many litres are needed to paint all the smaller spheres?
- The ratio of the ages of a husband and his wife when they got married was 6:5. 4 years and 6 years after their marriage they had their 1st and 2nd children. The sum of the present ages of the husband and wife is 6.4 times the sum of the present ages of their children. The average age of the family at present is 18.5 years. Find the ratio of the ages of the husband and wife when their second child was born.
View More Questions
Top CAT Time and Work Questions
- At their usual efficiency levels, A and B together finish a task in 12 days. If A had worked half as efficiently as she usually does, and B had worked thrice as efficiently as he usually does, the task would have been completed in 9 days. How many days would A take to finish the task if she works alone at her usual efficiency?
- Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?
- A cyclist leaves A at 10 am and reaches B at 11 am. Starting from 10:01 am, every minute a motor cycle leaves A and moves towards B. Forty-five such motor cycles reach B by 11 am. All motor cycles have the same speed. If the cyclist had doubled his speed, how many motor cycles would have reached B by the time the cyclist reached B?
- John jogs on track A at 6 kmph and Mary jogs on track B at 7.5 kmph. The total length of tracks A and B is 325 metres. While John makes 9 rounds of track A, Mary makes 5 rounds of track B. In how many seconds will Mary make one round of track A?
- Anil alone can do a job in 20 days while Sunil alone can do it in 40 days. Anil starts the job, and after 3 days, Sunil joins him. Again, after a few more days, Bimal joins them and they together finish the job. If Bimal has done 10% of the job, then in how many days was the job done?
View More Questions
Top CAT Questions
- Compare and contrast between Portia and Jessica characters in Merchant of Venice.
- In a triangle ABC,AB=AC=8cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through B and C. Then the area,in sq.cm,of the overlapping region between the two circles is
- The simple interest accrued on an amount of Rs.25,000 at the end of three years is Rs.7,500. What would be the compound interest accrued on the same amount at the same rate in the same period?
- What is the sum of all roots of the equation \(|x+4|^2–10|x+4|=24\)?
- A solid metal sphere is melted and smaller spheres of equal radii are formed. 10% of the volume of the sphere is lost in the process. The smaller spheres have a radius which is \(\frac{1}{9}\text{th}\) the larger sphere. If 10 litres of paint were needed to paint the larger sphere,how many litres are needed to paint all the smaller spheres?
View More Questions