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Last Year a Company Gave Bonuses to a number of employees

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Fish181
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Last Year a Company Gave Bonuses to a number of employees [#permalink] 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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D
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Bunuel
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Re: Last Year a Company Gave Bonuses to a number of employees [#permalink] Last Year a Company Gave Bonuses to a number of employees   27 Apr 2024, 13:18
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Fish181 wrote:
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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Re: Last Year a Company Gave Bonuses to a number of employees [#permalink] Last Year a Company Gave Bonuses to a number of employees   28 Apr 2024, 05:40
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Fish181 wrote:
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

­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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Re: Last Year a Company Gave Bonuses to a number of employees [#permalink] Last Year a Company Gave Bonuses to a number of employees   29 Apr 2024, 13:37
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 wrote:
Fish181 wrote:
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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Re: Last Year a Company Gave Bonuses to a number of employees [#permalink] Last Year a Company Gave Bonuses to a number of employees   29 Apr 2024, 13:41
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sayan640 wrote:
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 wrote:
Fish181 wrote:
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.­

­

­Because 49 * 9 = 441 > 432. 
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Re: Last Year a Company Gave Bonuses to a number of employees [#permalink] Last Year a Company Gave Bonuses to a number of employees   29 Apr 2024, 21:42
Fish181 wrote:
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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Re: Last Year a Company Gave Bonuses to a number of employees [#permalink]
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