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Gmat750aspirant
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If x and y are positive integers such that 3x + 4y = 24, what is the value of x + y?
A) 4
B) 5
C) 6
D) 7
E) 8
A) 4
B) 5
C) 6
D) 7
E) 8
Gmat750aspirant
Both 4y and 24 are multiples of 4, meaning 3x must also be a multiple of 4. This implies that x must be a multiple of 4. Testing the smallest positive multiple, x = 4, gives 3(4) + 4y = 24, which simplifies to y = 3. Thus, x = 4 and y = 3, giving x + y = 7.
Answer: D.
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sujoykrdatta
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GMAT 1: 750 Q51 V39
Posts: 598
Gmat750aspirant
Method 1:
The easiest values of x and y that work are x = 8 and y = 0
Since y=0 isn't positive, we reduce x by the coefficient of y and increase y by the coefficient of x to get: x=4, y=3 - these values satisfy the equation.
(Think why this method works)
Here both are positive integers.
So answer = 4+3 = 7
Method 2
3x + 4y = 24
A) Divide by 3: x + 4y/3 = 8
=> x + y < 8
B) Divide by 4: 3x/4 + y = 6
=> x + y > 6
Thus the only value is 7
Method 3:
We can use the method shown by Bunuel
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Gmat750aspirant
3x & 4y both must be even or odd since their sum is even.
Since 4y can't be odd, it means both must be even.
3x+4y = 24; 4y = 24-3x
x=2; 4y = 24-6 = 18; can't be
x=4; 4y = 24-12 = 12; bingo; y=3
x=6; 4y = 24-18 = 6; can't be
x=8; 4y = 24-24 = 0; can't be since y is also +ve integer.
Sum 3+4=7;
Posts: 1,382
Alternate solution:
3x + 4y = 24
Since multiples of a number added/subtracted result in another multiple, y must be a multiple of 3
x = (24 - 4y)/3
x = 8 - (4y/3)
Substituting:
y= 3, makes x = 4
y = 6, makes x = 0 (not possible since x is positive integer)
All greater values of y will result in a -ve value for x
So, only y= 3, x = 4 works which makes x+y = 7
3x + 4y = 24
Since multiples of a number added/subtracted result in another multiple, y must be a multiple of 3
x = (24 - 4y)/3
x = 8 - (4y/3)
Substituting:
y= 3, makes x = 4
y = 6, makes x = 0 (not possible since x is positive integer)
All greater values of y will result in a -ve value for x
So, only y= 3, x = 4 works which makes x+y = 7
Gmat750aspirant
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