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If the least common multiple of 2x and y is 400 and the greatest commo
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02 Nov 2021, 07:47
02 Nov 2021, 07:47
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If the least common multiple of 2x and y is 400 and the greatest common divisor of 6x and 3y is 30, what is the value of \(\frac{x*y}{2} \) ?
A. 100
B. 500
C. 1000
D. 1500
E. 2000
A. 100
B. 500
C. 1000
D. 1500
E. 2000
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Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If the least common multiple of 2x and y is 400 and the greatest commo
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02 Nov 2021, 08:41
02 Nov 2021, 08:41
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Carcass wrote:
If the least common multiple of 2x and y is 400 and the greatest common divisor of 6x and 3y is 30, what is the value of \(\frac{x*y}{2} \) ?
A. 100
B. 500
C. 1000
D. 1500
E. 2000
A. 100
B. 500
C. 1000
D. 1500
E. 2000
If the least common multiple of 2x and y is 400, then if we multiply 2x and y each by 3, the resulting values will have a least common multiple of 1200.
In other words, the least common multiple of 6x and 3y is 1200
We're also told that the greatest common divisor of 6x and 3y is 30
--------ASIDE----------------------
There's a nice property that says:
(greatest common divisor of x and y)(least common multiple of x and y) = xy
Example: x = 10 and y = 15
Greatest common divisor of 10 and 15 = 5
Least common multiple of 10 and 15 = 30
Notice that these values satisfy the above rule, since (5)(30) = (10)(15)
--------BACK TO THE QUESTION! ----------------------
When we apply the above property, we get: (GCD of 6x and 3y)(LCM of 6x and 3y) = (6x)(3y)
Substitute values to get: (30)(1200) = (6x)(3y)
Simplify to get: 36,000 = 18xy
Divide both sides by 18 to get: 2000 = xy
So, \(\frac{xy}{2}=\frac{2000}{2} = 1000\)
Answer: C
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If the least common multiple of 2x and y is 400 and the greatest commo
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27 Oct 2022, 11:02
27 Oct 2022, 11:02
1
Given that LCM of 2x and y is 400 and GCD of 6x and 3y is 30 and we need to find the value of \(\frac{x*y}{2}\)
Lets solve the Problem using Two Methods:
Method 1:
LCM of 2x and y = 400
If we multiply 2x and y by 3 to get 6x and 3y then their LCM = (LCM of 2x and y) * 3 = 400*3 = 1200
LCM of two numbers * GCD of two numbers = Product of the two numbers
=> LCM(6x,3y) * GCD(6x,3y) = 6x * 3y
=> 1200 * 30 = 18 *xy
=> xy = \(\frac{1200*30}{18}\) = 2000
=> \(\frac{x*y}{2}\) = \(\frac{2000}{2}\) = 1000
So, Answer will be C
Method 2:
GCD of 6x and 3y = 30
If we divide 6x and 3y by 3 to get 2x and y then their GCD = (GCD of 6x and 3y) / 3 = \(\frac{30}{3}\) = 10
LCM of two numbers * GCD of two numbers = Product of the two numbers
=> LCM(2x,y) * GCD(2x,y) = 2x * y
=> 400 * 10 = 2xy
=> xy = 2000
=> \(\frac{x*y}{2}\) = \(\frac{2000}{2}\) = 1000
So, Answer will be C
Hope it helps!
To learn more about LCM and GCD watch the following videos
Lets solve the Problem using Two Methods:
Method 1:
LCM of 2x and y = 400
If we multiply 2x and y by 3 to get 6x and 3y then their LCM = (LCM of 2x and y) * 3 = 400*3 = 1200
LCM of two numbers * GCD of two numbers = Product of the two numbers
=> LCM(6x,3y) * GCD(6x,3y) = 6x * 3y
=> 1200 * 30 = 18 *xy
=> xy = \(\frac{1200*30}{18}\) = 2000
=> \(\frac{x*y}{2}\) = \(\frac{2000}{2}\) = 1000
So, Answer will be C
Method 2:
GCD of 6x and 3y = 30
If we divide 6x and 3y by 3 to get 2x and y then their GCD = (GCD of 6x and 3y) / 3 = \(\frac{30}{3}\) = 10
LCM of two numbers * GCD of two numbers = Product of the two numbers
=> LCM(2x,y) * GCD(2x,y) = 2x * y
=> 400 * 10 = 2xy
=> xy = 2000
=> \(\frac{x*y}{2}\) = \(\frac{2000}{2}\) = 1000
So, Answer will be C
Hope it helps!
To learn more about LCM and GCD watch the following videos
