Add the reciprocal of 1 1/5 to the sum of the numbers (+1.25) and (−1 3/4 ).

To find the reciprocal of a mixed number, we first convert it to an improper fraction.  1 1/5 = 5/5 + 1/5 (remember that 1 whole would have the same numerator and denominator) = 6/5.
The reciprocal is this number flipped, so 5/6.
To find the sum of 1.25 and -1 3/4, we will convert the decimal to a fraction. 0.25 is read as "twenty-five hundredths," so the fraction would be 25/100.  This will simplify to 1/4, so 1.25 = 1 1/4.
1 1/4 + -1 3/4 = -2/4 = -1/2.
Now we add 5/6 + -1/2:
Find the least common denominator (LCD).  6 is the first number that both 6 and 2 will divide into evenly.  This gives us 5/6 + -3/6 (we multiply 2 by 3 to get 6, so we multiply the numerator, -1, by 3 as well).  This equals 2/6 which simplifies to 1/3.