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The greatest common divisor (GCD) and least common multiple (LCM) of two numbers are 12 and 480, respectively, and the numbers are in the ratio of 5 : 8. Find the difference between the two numbers.
36
12
24
42
36
Since the ratio is 5:8, the numbers are 5x and 8x. The GCD (x) is 12. So, the numbers are 5(12)=60 and 8(12)=96. Their difference is 96 - 60 = 36.
GCD of two numbers = 12, LCM of two numbers = 480, and the ratio of the numbers is 5 : 8.
a├Чb=GCD├ЧLCM
Since the ratio is 5:8, the numbers are 5x and 8x. The GCD (x) is 12. So, the numbers are 5(12)=60 and 8(12)=96. Their difference is 96 - 60 = 36.
Students often try to solve using the LCM formula (5x├Ч8x)=12├Ч480 which is correct but redundant, as the ratio already contains the multiplier (GCD).
Define the numbers
Since the ratio of the numbers is 5:8, we can represent the numbers as 5x and 8x.
n1тАЛ=5x,n2тАЛ=8x
Use the GCD
The Greatest Common Divisor (GCD) of 5x and 8x is x. Given that the GCD is 12, we find x=12.
x=12
Calculate the two numbers
Substitute x=12 into our expressions for the numbers.
n1тАЛ=5├Ч12=60,n2тАЛ=8├Ч12=96
Calculate the difference
Subtract the smaller number from the larger number to find the difference.
96тИТ60=36
A is correct because the numbers are 60 and 96, and their difference is 36.
This concept is a foundational part of Number Systems; remember that for any two numbers a and b, the product of their ratio parts multiplied by their GCD will always yield the LCM.
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