4/75, 2r/25, r/125

-1/x, 2/xy, x/y
1/a, 1/b

write fractions as equivalent fractions with the least common denominator

LCD: 25*3; 25; 25*5...LCD 25*15

LCD: x, xy, y ....LCD xy

LCD: a, b.....LCD ab

-1/x,2/xy, x/y

To find equivalent fractions with the least common denominator (LCD), we need to find the least common multiple (LCM) of the denominators.

For the first set of fractions: The denominators are 75, 25, and 125. We can find the LCM by finding the prime factorizations of the numbers and multiplying the highest powers of all the factors. The prime factorizations are:
75 = 3 * 5^2
25 = 5^2
125 = 5^3

The LCM of 75, 25, and 125 is 3 * 5^3 = 375. Now we can rewrite the fractions with the LCD:
4/75 = (4 * 5^3) / (75 * 5^3) = 500 / 375
2r/25 = (2r * 15) / (25 * 15) = 30r / 375
r/125 = (r * 3^2) / (125 * 3^2) = 9r / 375

For the second set of fractions: The denominators are x, xy, and y. To find the LCM, we need to determine the highest powers of each prime factor between the denominators. Since the variables do not have given values, we can't simplify further.

The LCM of x, xy, and y is xy. Let's rewrite the fractions:
-1/x = (-1 * y) / (x * y) = -y / xy
2/xy = (2 * x * y) / (xy * x * y) = 2xy / xy
x/y = (x * x) / (y * x) = x^2 / xy

For the third set of fractions: The denominators are a and b. Similar to the previous set, we don't have values for the variables, so we can't simplify further.

The LCM of a and b is ab. Let's rewrite the fractions:
1/a = (1 * b) / (a * b) = b / ab
1/b = (1 * a) / (b * a) = a / ab

Now you have rewritten the fractions as equivalent fractions with the least common denominator.