stonecold wrote:
Which of the following fractions can be written as the difference of reciprocals of two consecutive integers?
A. 1/100
B. 1/121
C. 1/45
D. 1/56
E. 3/72
Great question!
First, let x and x+1 be two consecutive integers.
So, 1/x and 1/(x+1) are the reciprocals.
So, their difference = 1/x - 1/(x+1)
To subtract these fractions we must find common denominators.
So, 1/x - 1/(x+1) = (x+1)/(x)(x+1) - x/(x+1)(x)
= 1/(x)(x+1)
Notice that the denominator is the
product of two consecutive integers AND the numerator is 1
Check the denominators of the answer choices:
A. 1/100 We can't write 100 as the product of two consecutive integers. ELIMINATE.
B. 1/121 We can't write 121 as the product of two consecutive integers. ELIMINATE.
C. 1/45 We can't write 45 as the product of two consecutive integers. ELIMINATE.
D. 1/56 We CAN write 56 as the product of two consecutive integers (7x8). KEEP
E. 3/72 We CAN write 72 as the product of two consecutive integers (8x9). KEEP
We're left with D and E.
Since E is not in the correct format (with 1 in the numerator), we can ELIMINATE E
Answer: D
ASIDE: 1/56 = 1/(7)(8) = 1/(x)(x+1)
So, x = 7 and x+1 = 8
Notice that 1/7 - 1/8 = 8/56 - 7/56 = 1/56!
_________________