Exact Calculations with Fractions

What an exact calculation means

A calculation is exact when its answer has not been rounded. For example, $\frac{2}{3}$ is exact, whereas $0.67$ is only an approximation to it. Fractions are therefore useful for preserving exact values throughout a calculation.

A fraction $\frac{a}{b}$ represents $a\div b$, where $b\ne0$. Multiplying or dividing both the and by the same non-zero number does not change its value:

$$\frac{3}{5}=\frac{3\times4}{5\times4}=\frac{12}{20}.$$

This idea of equivalent fractions is what makes both common denominators and cancellation work.

Simplifying fractions

To give a fraction in its simplest form, divide its numerator and denominator by their highest common factor. For instance,

$$\frac{42}{56}=\frac{42\div14}{56\div14}=\frac{3}{4}.$$

A negative sign may be placed before the fraction, in the numerator or in the denominator: $-\frac35=\frac{-3}{5}=\frac3{-5}$. It is clearest to put it before the fraction. Two negative signs cancel, so .

Adding and subtracting

Fractions can be added or subtracted directly only when they describe parts of the same size—that is, when they have a common denominator. If the denominators already match, keep that denominator and combine the numerators:

$$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}, \qquad \frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}.$$

For different denominators, use a common multiple of the denominators. The lowest common multiple usually keeps the arithmetic smallest. To calculate $\frac34+\frac56$, the lowest common denominator is :

$$\frac34+\frac56 =\frac9{12}+\frac{10}{12} =\frac{19}{12}.$$

Each conversion multiplies the top and bottom by the same number, so the fractions keep their original values. Only after the denominators match can the numerators be combined.

Subtraction can produce a negative exact value. For example,

$$\frac7{13}-\frac57 =\frac{49}{91}-\frac{65}{91} =-\frac{16}{91}.$$

The denominator $91$ stays fixed because the calculation is counting signed numbers of ninety-firsts.

Multiplying fractions

To multiply fractions, multiply the numerators and multiply the denominators:

$$\frac ab\times\frac cd=\frac{ac}{bd}.$$

A common denominator is not needed. It is often easier to cancel common factors across the multiplication first. For example,

$$\frac{14}{15}\times\frac9{28} =\frac{1}{15}\times\frac9{2} =\frac9{30} =\frac3{10}.$$

Here $14$ and $28$ were divided by $14$ before multiplication. This is valid because it removes a factor from both the numerator and denominator of the overall product.

Dividing fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal:

$$\frac ab\div\frac cd =\frac ab\times\frac dc,$$

provided the divisor $\frac cd$ is not zero. For example,

$$\frac58\div\frac{15}{16} =\frac58\times\frac{16}{15} =\frac{2}{3}.$$

This rule works because multiplying by $\frac{16}{15}$ reverses multiplication by $\frac{15}{16}$. The first fraction remains unchanged; only the fraction being divided by is inverted.

Mixed numbers and improper fractions

A mixed number combines a whole number and a proper fraction. Convert it to an improper fraction before applying the fraction rules. For example,

$$2\frac13=\frac{2\times3+1}{3}=\frac73.$$

Thus

$$2\frac13\times1\frac12 =\frac73\times\frac32 =\frac72.$$

The exact answer may be left as the improper fraction $\frac72$, or written as $3\frac12$ if a mixed number is requested.

For a negative mixed number, $-2\frac13$ means the negative of the entire mixed number, so it equals $-\frac73$.

Calculations containing several operations

Use the normal order of operations: brackets first, then multiplication and division, followed by addition and subtraction. Work exactly at every stage. For example,

$$\frac23-\frac14\times\frac85 =\frac23-\frac25 =\frac{10}{15}-\frac6{15} =\frac4{15}.$$

The multiplication is completed before the subtraction. Keeping fractional values throughout avoids introducing rounding error, and simplifying the final answer completes the exact calculation.

Exact form

Keep values as fractions: $\frac23$ is exact, while $0.67$ is rounded. Simplify the final fraction using common factors.

Add or subtract

Use a common denominator, then combine only the :