Problem Solving

A man purchased $510 worth savings bonds in denominations of $15 and $30, including at least 1 of each denomination. He gave away 8 of the bonds as gifts but then lost all the rest of the bonds he had purchased. If the number of $30 bonds he gave away was a multiple of the number of $15 bonds he gave away, what was maximum possible value of the bonds that he lost?

(A) 180
(B) 225
(C) 285
(D) 300
(E) 330

OA: E

Source: Kaplan

Total value of bonds = value of bonds that he lost + value of bonds that he gave away

so to find max possible value of the bonds he lost we need to find the minimum possible value of the bonds he gave away.

Total # of bonds that he gave away = 8
# of $30 bonds + # of $15 bonds = 8
x+y= 8
given that x is a multiple of y
so the possible combinations and values of x & y will be
7+1=8, value => 7*30 + 1*15= 225
6+2=8, value => 6*30 + 2*15= 210
and
4+4=8, value => 4*30 + 4*15= 180
Select the min value and subtract it from the total value

530-180 = 330
Hence the answer is E

Let the total number of $15 bonds the man gave away be F and
Let the total number of $30 bonds the man gave away be T, Then

F + T = 8 and T = X*F, Where X is a positive integer (From the question)
F + X*F = 8
F*(1+X) = 8 = 4*2 = 2*4 = 1*8
So (F,X,T)=(4,1,4) Or (2,3,6) Or (1,7,7) are the only possible cases. For the value of the bonds that he lost to be maximum, the value of the bonds he gave away should be minimum. For the value of the bonds he gave away to be minimum, the number of $30 bonds should be less than or equal to the number of $15 bonds.

So(F,T)=(4,4) gives you the minimum value for the bonds he gave away.
The value of the bonds that he lost = Total value of the bonds - Total value of the bonds he lost = 510 - 4*15 - 4*30 = 510 - 180 = 330

ikaplan wrote:A man purchased $510 worth savings bonds in denominations of $15 and $30, including at least 1 of each denomination. He gave away 8 of the bonds as gifts but then lost all the rest of the bonds he had purchased. If the number of $30 bonds he gave away was a multiple of the number of $15 bonds he gave away, what was maximum possible value of the bonds that he lost?

(A) 180
(B) 225
(C) 285
(D) 300
(E) 330

OA: E

Source: Kaplan

Let x = the number of $15 bonds given away and y = the number of $30 bonds given away.
Since he gave away 8 bonds:
x+y = 8.

Since y must be a multiple of x, there are only three possibilities:
Case 1: x=1 and y=7
Case 2: x=2 and y=6
Case 3: x=4 and y=4.

To MAXIMIZE the total given away, we must MINIMIZE the total lost.
x=4 and y=4 will yield the minimum amount lost, since this combination offers the lowest ratio of $30 bonds to $15 bonds:
4*15 + 4*30 = 180.

Thus, the maximum possible amount given away = 510-180 = 330.

The correct answer is E.