A fraction becomes 1/2 when 3 is subtracted from its numerator. If 5 and 7 are respectively added to its numerator and denominator, it becomes 3\4 then, the LCM of the numerator and denominator of the said fraction,must be
let the original fraction be a/b
equation #1:
(a - 3)/b = 1/2
b = 2a - 6
equation #2:
(a+5)/(b+7) = 3/4
4a + 20 = 3b + 21
4a - 3b = 1
sub in #1
4a - 3(2a-6) = 1
4a-6a+18 = 1
-2a = -17
a = 17/2
b = 2(17/2)-6 = 11
our fraction is (17/2)/11 = 17/22
When we check with 17/22, we fail
When we check with (17/2) / 11 it works
that is:
(17/2 - 3)/11 = 1/2
(17/2 + 5)/(11+7) = 3/4
but for 17/22, which is the same as (17/2) / 11
if we have (17-3)/22 ≠ 1/2
and (15+5)/(22+7) ≠ 3/4
They should have specified to take the fraction in its
original complex form
johnny
diggie
biggums
To find the least common multiple (LCM) of the numerator and denominator of the given fraction, we first need to determine the fraction itself.
Let's assume the fraction is represented as a/b, where 'a' is the numerator and 'b' is the denominator.
According to the given information:
- When 3 is subtracted from the numerator, the fraction becomes 1/2. This can be expressed as (a - 3)/b = 1/2.
- When 5 is added to the numerator and 7 is added to the denominator, the fraction becomes 3/4. This can be expressed as (a + 5)/(b + 7) = 3/4.
Now, we can solve these two equations simultaneously to find the values of 'a' and 'b'.
Equation 1: (a - 3)/b = 1/2
Cross-multiplying gives us: 2(a - 3) = b
Equation 2: (a + 5)/(b + 7) = 3/4
Cross-multiplying gives us: 4(a + 5) = 3(b + 7)
Expanding and simplifying both equations:
2a - 6 = b (equation 1)
4a + 20 = 3b + 21 (equation 2)
Rearranging equation 1, we get b = 2a - 6. Substituting b in equation 2:
4a + 20 = 3(2a - 6) + 21
4a + 20 = 6a - 18 + 21
4a + 20 = 6a + 3
Bringing like terms together:
4a - 6a = -20 + 3
-2a = -17
a = -17 / -2
a = 17/2
Using the value of 'a' in equation 1:
b = 2(17/2) - 6
b = 17 - 6
b = 11
Therefore, the fraction is 17/11.
To find the LCM of the numerator (17) and denominator (11) of the fraction, we can simply multiply them as there are no common factors:
17 * 11 = 187
So, the LCM of the numerator and denominator of the given fraction is 187.