The rules for adding and subtracting integers work for every rational number: negative fractions, negative decimals and negative mixed numbers. You also need what you know about unlike denominators.
Idea clave
Change subtraction into adding the opposite. Then use the sign rules. For fractions, use a common denominator; for decimals, line up the decimal points.
Por qué funciona
The sign rules come from moves on the number line. A move does not have to be a whole number of steps. Moving $\frac{3}{4}$ to the left and then $\frac{1}{6}$ to the right partly cancels, the same as with integers.
Worked examples
Ejemplo resuelto: Fractions with different signs
Find $-\frac{3}{4} + \frac{1}{6}$.
- Common denominator 12: $-\frac{9}{12} + \frac{2}{12}$.
- Different signs: $9 - 2 = 7$, and the negative part is bigger.
- The sum is $-\frac{7}{12}$.
Respuesta: $-\frac{7}{12}$
Ejemplo resuelto: Decimals
Find $-2.5 - 1.75$.
- Add the opposite: $-2.5 + (-1.75)$.
- Both negative: $2.5 + 1.75 = 4.25$, so the answer is $-4.25$.
Respuesta: $-4.25$
Ejemplo resuelto: Mixed numbers
Find $-1\frac{1}{2} - \left(-3\frac{1}{4}\right)$.
- Add the opposite: $-1\frac{1}{2} + 3\frac{1}{4}$.
- As fractions: $-\frac{6}{4} + \frac{13}{4} = \frac{7}{4}$.
- $\frac{7}{4} = 1\frac{3}{4}$. Check the sign: $3\frac{1}{4}$ is farther from $0$, so the answer is positive ✓.
Respuesta: $1\frac{3}{4}$
Error común
Two slips: forgetting to change "minus a negative" into "plus a positive," and adding denominators ($\frac{1}{2} + \frac{1}{4} \ne \frac{2}{6}$). Check with an estimate: $-\frac{3}{4} + \frac{1}{6}$ should be a little more than $-\frac{3}{4}$, and $-\frac{7}{12}$ is ✓.