Operations with Positive and Negative Fractions

You can only add pieces that are the same size. Thirds and quarters are different sizes, so $\frac{2}{3}+\frac{3}{4}$ cannot be done until both are cut into a common size. The smallest size that works is the lowest common denominator (LCD), the smallest number both denominators divide into: for 3 and 4 it is 12. Rewrite each fraction as an equivalent fraction with that denominator, then add or subtract the numerators and keep the denominator. With negative fractions, put the sign in the numerator (write $-\frac{3}{4}$ as $\frac{-3}{4}$) and then add the numerators exactly as you add integers.

A mixed number such as $2\frac{1}{3}$ means $2+\frac{1}{3}$. Before multiplying or dividing, and usually before adding or subtracting, change it to an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the denominator. $2\frac{1}{3}=\frac{2\times 3+1}{3}=\frac{7}{3}$. A negative mixed number is negative as a whole: $-2\frac{1}{3}$ means $-\left(2+\frac{1}{3}\right)$, so it equals $-\frac{7}{3}$. To change an improper answer back, divide: $\frac{17}{12}$ is $17÷12=1$ remainder $5$, so $1\frac{5}{12}$.

To multiply fractions, multiply the numerators and multiply the denominators. No common denominator is needed. The area model shows why: $\frac{2}{3}\times \frac{3}{4}$ means "$\frac{2}{3}$ of $\frac{3}{4}$". Cut a square into 4 columns and 3 rows (that is $3\times 4=12$ equal parts, the new denominator), shade 3 columns, then take 2 of the 3 rows of that shading ($2\times 3=6$ parts, the new numerator). You get $\frac{6}{12}=\frac{1}{2}$. To keep numbers small, cancel a factor shared by any numerator and any denominator before multiplying. Then decide the sign with the integer rule: same signs give a positive product, different signs give a negative one.

The reciprocal of a fraction flips it: the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$, and the reciprocal of $-\frac{2}{5}$ is $-\frac{5}{2}$ (the sign stays). To divide by a fraction, multiply by its reciprocal. Why does that work? Division asks "how many of these fit?" How many $\frac{3}{4}$ cup scoops fit in 3 cups? Each cup holds $\frac{4}{3}$ scoops, so 3 cups hold $3\times \frac{4}{3}=4$ scoops, and indeed $3÷\frac{3}{4}=4$. Only flip the fraction you are dividing by (the second one). Zero has no reciprocal, so you can never divide by $0$.

Fraction Review | How to Add, Subtract, Multiply, and Divide Fractions

Formulas do not care whether you substitute whole numbers or fractions. A picture is $3\frac{1}{2}$ in by $2\frac{3}{4}$ in. Its perimeter is $P=2(l+w)=2\left(\frac{7}{2}+\frac{11}{4}\right)=2\left(\frac{14}{4}+\frac{11}{4}\right)=2\times \frac{25}{4}=\frac{25}{2}=12\frac{1}{2}$ in, and its area is $A=lw=\frac{7}{2}\times \frac{11}{4}=\frac{77}{8}=9\frac{5}{8}$ square inches. For temperature, $C=\frac{5}{9}(F-32)$. With $F=-4$, the bracket is $-4-32=-36$, and $\frac{5}{9}\times (-36)=-\frac{180}{9}=-20$, so $-4^{\circ }\text{F}$ is $-20^{\circ }\text{C}$. Follow the order of operations (brackets first), keep fractions exact while you work, and convert to a decimal only at the end if you need one.