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Fish181
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Last Year a Company Gave Bonuses to a number of employees 27 Apr 2024, 12:57
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Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­

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Last Year a Company Gave Bonuses to a number of employees Updated on: 23 Oct 2025, 12:25
Originally posted by Bunuel on 27 Apr 2024, 13:18.
Last edited by Bunuel on 23 Oct 2025, 12:25, edited 1 time in total.
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Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­
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Assume x $750 bonuses, y $1500 bonuses, and z $7,350 bonuses were given. Then, we'd have:

750x + 1,500y + 7,350z = 64,800

Reduce by 150:

5x + 10y + 49z = 432

Our goal is to minimize x + y + z. To achieve this, we need to maximize the number of the largest bonus, z. The value of z cannot be 9 or higher, so we try z = 8. In this case, we have:

5x + 10y + 392 = 432
5x + 10y = 40

Now, let's maximize the number of the next largest bonus, y. Since x > 0, the maximum value of y is 3, making x = 2.

Therefore, x + y + z = 2 + 3 + 8 = 13.

Answer: D.­
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Last Year a Company Gave Bonuses to a number of employees 29 Apr 2024, 21:42
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Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­
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Last Year a Company Gave Bonuses to a number of employees 28 Apr 2024, 05:40
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Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­
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­Three amounts of Bonus are $750, $1500, and $7,350

Total Bonus = $64800

i.e $750a + $1500b + $7,350c = $64800

Now we need to find \((a+b+c)_{Min} = ?\)

i.e $750a + $1500b + $7,350c = $64800
i.e. $75a + $150b + $7,35c = $6480

i.e. 5a + 10b + 49c = 432

IMPORTANT: Now to minimize a+b+c, the value of c MUST be Maximized so that values of a and b are the least

Also, 5a + 10b = 432 - 49c

IMPORTANT: also, 5a+10b must be multiple of 5 so we need to fund a multiple of 49 (i.e. 49c) which when subtracted from 432 leaves a result that is divisible by 5

IMPORTANT: so the unit digit of 49c must be either 2 or 7

for max value of x=8, 49x = 49*8 = 392

i.e. 5a + 10b = 432 - 49c = 432 - 392 = 40

i.e. a+2b = 8

now, for Max value of b, b = 3 and then a = 2

so \((a+b+c)_{Min} = (2+3+8) = 13\)

Answer: Option D
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Last Year a Company Gave Bonuses to a number of employees 29 Apr 2024, 13:37
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Bunuel KarishmaB , Can you please explain to me why you say that the  value of z cannot be 9 or higher,  Thank you in advance
Bunuel

Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­
Assume x $750 bonuses, y $1500 bonuses, and z $7,350 bonuses were given. Then, we'd have:



750x + 1,500y + 7,350z = 64,800
Reduce by 150:



5x + 10y + 49z = 432
Our goal is to minimize x + y + z. To achieve this, we need to maximize the number of the largest bonus, z. The value of z cannot be 9 or higher, so we try z = 8. In this case, we have:



5x + 10y + 392 = 432
5x + 10y = 40
Now, let's maximize the number of the next largest bonus, y. Since x > 0, the maximum value of y is 3, making x = 2.

Therefore, x + y + z = 2 + 3 + 8 = 13.

Answer: D.­
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Last Year a Company Gave Bonuses to a number of employees 29 Apr 2024, 13:41
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sayan640
Bunuel KarishmaB , Can you please explain to me why you say that the  value of z cannot be 9 or higher,  Thank you in advance
Bunuel

Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­
Assume x $750 bonuses, y $1500 bonuses, and z $7,350 bonuses were given. Then, we'd have:




750x + 1,500y + 7,350z = 64,800
Reduce by 150:




5x + 10y + 49z = 432
Our goal is to minimize x + y + z. To achieve this, we need to maximize the number of the largest bonus, z. The value of z cannot be 9 or higher, so we try z = 8. In this case, we have:




5x + 10y + 392 = 432
5x + 10y = 40
Now, let's maximize the number of the next largest bonus, y. Since x > 0, the maximum value of y is 3, making x = 2.

Therefore, x + y + z = 2 + 3 + 8 = 13.

Answer: D.­
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­Because 49 * 9 = 441 > 432. 
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Last Year a Company Gave Bonuses to a number of employees 05 Jun 2024, 10:13
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Bunuel

Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­

 
Our goal is to minimize x + y + z. To achieve this, we need to maximize the number of the largest bonus, z. The value of z cannot be 9 or higher, so we try z = 8. In this case, we have:


 
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­Hi Bunuel
Can you explain this line "Our goal is to minimize x + y + z. To achieve this, we need to maximize the number of the largest bonus, z"?

I did not understand this fact that in order to minimise x+y+z, we need to maximise z!

 
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Last Year a Company Gave Bonuses to a number of employees 05 Jun 2024, 10:44
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Bunuel

Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­


 
Our goal is to minimize x + y + z. To achieve this, we need to maximize the number of the largest bonus, z. The value of z cannot be 9 or higher, so we try z = 8. In this case, we have:



 

 
­Hi Bunuel
Can you explain this line "Our goal is to minimize x + y + z. To achieve this, we need to maximize the number of the largest bonus, z"?

I did not understand this fact that in order to minimise x+y+z, we need to maximise z!


 
Show more

We have 750x + 1,500y + 7,350z = 64,800

­The strategy to minimize the total number of bonuses (x + y + z) by maximizing z is based on efficiency. Since z represents the highest bonus amount, giving more of these reduces the total number of bonuses needed to reach the sum of $64,800. Each higher bonus contributes significantly more to the total, meaning fewer total bonuses are needed.
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Last Year a Company Gave Bonuses to a number of employees 22 Jun 2024, 03:49
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Fish181
Last Year a company gave bonuses to a number of employees, but only in the three amounts of $750, $1500, and $7,350. If the total amount of the bonuses was $64800 and each of the three amounts was given to at least one employee, what is the fewest number of bonuses that the company could have given to employees last year?­

A 10
B 11
C 12
D 13
E 14­

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We need the minimum number of bonuses to let's try to maximize the number of the big bonus of $7350.
We know that total amount was 64,800. 
7350 * 10 will be too much. Even 7350 * 9 will be much greater than 63000. 
Hence maximum we can have 8 bonuses of the amount 7350.

7350*8 = 58,800

Leftover amount = 64,800 - 58,800 = 6000

Now we go on to the next big amount of 1500. We can give 4 bonuses of 1500 but we must give at least 1 bonus of each amount. Hence we give 3 bonuses of $1500 each and 2 bonuses of $750 each. 

Total number of bonuses = 8 + 3 + 2 = 13

Answer (D)
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Last Year a Company Gave Bonuses to a number of employees 23 Oct 2025, 12:16
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This takes more than 2 mins
VivekSri

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Last Year a Company Gave Bonuses to a number of employees 06 Nov 2025, 12:15
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Computer Science students will know that this is the greedy method of solving the coin changing problem! (Ofc you don't need to know any of that to solve this)
But note that the greedy soln need not always be optimal XD
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