This example keeps both operands exact from start to finish. The denominators 2 and 4 describe different part sizes, so the numerators cannot yet be combined. Four is the least common multiple and becomes the shared denominator. Rewriting 1/2 as 2/4 preserves its value, while 3/4 already uses fourths. The addition then produces 5/4. That reduced improper fraction is greater than one and can be displayed as 1 1/4 or the terminating decimal 1.25 without changing its value.
Rewrite 1/2 as an equivalent number of fourths
The least common multiple of 2 and 4 is 4. To change halves into fourths, multiply both parts of 1/2 by 2: (1 × 2)/(2 × 2) = 2/4. Multiplying by 2/2 is multiplying by one, so the rational value does not move. The operation is now 2/4 + 3/4. Both numerators count the same quarter-sized unit, which is the exact condition needed before addition. No decimal approximation has been introduced.
Add the numerators and test lowest terms
Combine the quarter counts while keeping their unit: (2 + 3)/4 = 5/4. The greatest common divisor of 5 and 4 is 1, so no nontrivial integer divides both parts and 5/4 is already in lowest terms. Adding the denominators would produce 5/8, which is smaller than either expected sum and fails the common-unit rule. The exact reduced fraction 5/4 remains the canonical calculation result even when a friendlier mixed form is also shown.
Verify 5/4, 1 1/4, and 1.25
Integer division gives 5 = 1 × 4 + 1, so 5/4 becomes 1 1/4. Because the denominator 4 equals 2², its only prime factor is 2 and the decimal terminates. Scaling by 25 gives 125/100, hence 1.25 exactly. A magnitude check agrees: one half plus three quarters must exceed one, and 0.5 + 0.75 = 1.25. These equivalent forms provide independent checks while exact fraction arithmetic remains the source of truth.