Answer: Option C
Let x be the greatest unit of time that can divide both 5 hours 15 minutes and 8 hours 24 minutes into integers.
We can write 5 hours 15 minutes as 5x + 15y, where x is the unit of time and y is the fraction of x. Similarly, 8 hours 24 minutes can be written as 8x + 24y.
We want both expressions to be integers, which means y must also be an integer.
To find the greatest unit of time that can divide both expressions into integers, we need to find the greatest common factor (GCF) of the two expressions.
Let's first convert the minutes to hours:5 hours 15 minutes = 5.25 hours8 hours 24 minutes = 8.4 hours
Now, we can write:5x + 15y = 5.25k ...(1)8x + 24y = 8.4k ...(2)
where k is a positive integer (since both expressions must be integers).
Let's simplify these expressions by multiplying them by 100 to get rid of the decimals:
500x + 1500y = 525k ...(3)800x + 2400y = 840k ...(4)
To find the GCF of these two expressions, we can subtract (3) from (4):
300x + 900y = 315k
We can see that 300 and 900 have a common factor of 300, so we can simplify the expression further:
x + 3y = 3.15k
Since x and y are integers, 3y must be a multiple of 3.
We can also see that x and 3y have a common factor of 3, so let's write:
x = 3m3y = 3n
where m and n are integers.
Substituting these values in the equation above, we get:
3m + 3n = 3.15km + n = 1.05k
We want to find the greatest unit of time, which means we want to find the greatest value of k for which m and n are integers.
The greatest common factor of 105 and 100 is 5.
So, let's try k = 5:
m + n = 5.25
The only integer values of m and n that satisfy this equation are:
m = 2n = 3
Substituting these values in x and y, we get:
x = 6y = 9
So, the greatest unit of time with which both 5 hours 15 minutes and 8 hours 24 minutes can be represented as integers is 63 minutes (Option C).If you think the solution is wrong then please provide your own solution below in the comments section .