At Buddys Car Wash, the prices for a hand wash and a machin

Question

At Buddy’s Car Wash, the prices for a hand wash and a machine wash are $25 and $16, respectively. Last year, Ramiro spent $840 getting his car washed at Buddy’s, where he had his car hand washed more often than he had it machine washed. How many times did he have his car washed at Buddy’s last year?

(A) 35
(B) 36
(C) 37
(D) 38
(E) 39

Show Answer

Official Answer

E

(A) 35
(B) 36
(C) 37
(D) 38
(E) 39

GridLock · Answer

Let no. of handwashes in the year = H
Let no. of machine washes in the year. = M

We know
H>M (1)

Also,
25H + 16M = 840
=> H = (840 - 16M)/25 (2)

From (1)
(840 - 16M)/25 > M
=> 840 - 16M > 25M
=> 840 > 41M
=> M < 20.something

We also know that both H and M need to be integers.

From (2) for H to be an integer, 840 - 16M must be a multiple of 25
Therefore, 840 - 16M must result in an integer ending with 0 or 5

Checking with M = 20, 840 - 320 = 520 => ends with 0 but not multiple of 25
19, 18, 17, 16 will not produce a difference from 840 that will end in 0 or 5

Checking with M = 15, 840 - 240 = 600 => multiple of 25; Therefore, let's take M = 15

H = 600/25 = 24

M + H = 24 + 15 = 39

Therefore, answer is (E) 39

GridLock · Answer

Thank you for posting the question and the kind comment

kevincan · Answer

One shortcut is to realize that 840 is a multiple of 8 but not of 16

kevincan · Answer

Suppose he had his car washed at Buddy’s a total of n times, x of which were by hand. We know that x > n -x, ie. x > n/2

25x + 16(n-x) = 840

9x = 840-16n=8(105-2n)

9x/8 = 105-2n, which is an odd number
Thus x must be a multiple of 8 but not of 16 I.e 8,24,40,...

If x=8, n=48

But 8< 48/2 , contrary to the fact that he had his car washed by hand more often

If x=24, n=39 and n-x = 15

24> 39/2 , so n ,the number of cash washes in total during the year, must be 39

Note that greater possible values of x (40 for example) would produce a negative value of n-x

At Buddys Car Wash, the prices for a hand wash and a machin

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At Buddys Car Wash, the prices for a hand wash and a machin

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