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The ratio of the duration required for a senior technician, an associate technician, and an apprentice technician to finish a maintenance service individually is 1 : 2 : 3 respectively. If 10 senior, 15 associate, and 18 apprentice technicians work together to complete a service and the total service charge they received is Rs. 1410, then what is the total service charge for 10 senior, 12 associate, and 15 apprentice technicians?

Correct Answer: 1260 Rs.

Step-by-step Explanation:

Step 1: Determine the ratio of efficiencies of the technicians
The ratio of the duration required for a senior technician, an associate technician, and an apprentice technician to finish a maintenance service individually is given as 1 : 2 : 3.

Efficiency (work done per unit time) is inversely proportional to the duration of time taken. Therefore, the ratio of their individual work efficiencies is:

$\text{Efficiency Ratio}=\frac{1}{1}:\frac{1}{2}:\frac{1}{3}$

To convert this into whole numbers, multiply each term by the least common multiple (LCM) of 1, 2, and 3, which is 6:

$\text{Efficiency Ratio}=6:3:2$

Let:
Efficiency of 1 Senior technician = $6x$
Efficiency of 1 Associate technician = $3x$
Efficiency of 1 Apprentice technician = $2x$

Step 2: Calculate the total efficiency and value of $x$ for the first group
The first group consists of 10 senior, 15 associate, and 18 apprentice technicians.

The total work rate (total efficiency) of this group is:

$\text{Total Efficiency}=(10\times 6x)+(15\times 3x)+(18\times 2x)$

$\text{Total Efficiency}=60x+45x+36x=141x$

Since the total service charge received by this group is Rs. 1410, the earnings correspond directly to their combined efficiency:

$141x=1410$

$x=\frac{1410}{141}=10$

Step 3: Calculate the service charge for the second group
The second group consists of 10 senior, 12 associate, and 15 apprentice technicians.

The total work rate (total efficiency) of the second group is:

$\text{Total Efficiency}=(10\times 6x)+(12\times 3x)+(15\times 2x)$

$\text{Total Efficiency}=60x+36x+30x=126x$

Now, substitute the value of $x=10$ to find the total service charge:

$\text{Total Service Charge}=126\times 10=1260\text{Rs.}$

Therefore, the total service charge for 10 senior, 12 associate, and 15 apprentice technicians is 1260 Rs.