Common fractions: simplifying, comparing, and the four operations

In a common fraction the denominator cannot be zero: for the notation $\frac{a}{b}$ it is required that $b\ne0$. When simplifying, the numerator and the denominator are divided by the same number that is not zero; when adding and subtracting, a common denominator is found; when dividing, only the divisor fraction is inverted.

Below, simplifying fractions, comparing them, and the four arithmetic operations are shown with worked examples. In each rule the restrictions on the denominator and the divisor are also noted separately.

Simplifying fractions: we work with the greatest common divisor EKUB

By the basic property of a fraction, if the numerator and the denominator are divided by one and the same number not equal to zero, the value of the fraction does not change. Simplifying a fraction means dividing the numerator and the denominator by their common divisor. The fastest way is to divide at once by the greatest common divisor (EKUB): the fraction then reaches irreducible form in one step.

Example 1. Simplify the fraction $\frac{36}{48}$.

Solution: we notice that $36 = 12 \cdot 3$ and $48 = 12 \cdot 4$, so $\mathrm{EKUB}(36;\,48) = 12$. We divide the numerator and the denominator by 12: $\frac{36}{48} = \frac{36:12}{48:12} = \frac{3}{4}$.

If the greatest common divisor EKUB is not immediately apparent, simplify step by step: first by 2 ($\frac{18}{24}$), again by 2 ($\frac{9}{12}$), and then by 3 ($\frac{3}{4}$). The result is the same, only there are more steps. For large numbers it is convenient to factor the numerator and the denominator into prime factors: the common factors are crossed out, and the remaining ones give the answer.

Methods of comparing fractions

For comparing, it is enough to know three cases. If the denominators are equal and positive, the fraction with the greater numerator is greater. If the numerators are the same positive number and the denominators are also positive, the fraction with the smaller denominator is greater, because the parts are larger. In the general case, first bring the denominators to positive form. After that, for $\frac{a}{b}$ and $\frac{c}{d}$ one may compare $a\cdot d$ with $c\cdot b$; here $b>0$ and $d>0$.

Example 2. Compare the fractions $\frac{5}{8}$ and $\frac{7}{12}$.

Solution: we take $\mathrm{EKUK}(8;\,12) = 24$ as the common denominator. Then $\frac{5}{8} = \frac{15}{24}$ and $\frac{7}{12} = \frac{14}{24}$. Because $15 > 14$, we have $\frac{5}{8} > \frac{7}{12}$.

We check (cross-multiplication method): $5 \cdot 12 = 60$ and $7 \cdot 8 = 56$; because $60 > 56$, the conclusion is correct.

Another useful method is to compare the fractions with a «benchmark» number. For example, one can see at once that $\frac{4}{9} < \frac{1}{2}$ and $\frac{6}{11} > \frac{1}{2}$, so it is clear that $\frac{6}{11}$ is greater even without looking for a common denominator.

Addition and subtraction: a common denominator is required

When fractions have equal denominators, the numerators are added or subtracted, and the denominator remains unchanged. If the denominators are different, we first bring them to the least common multiple (EKUK) and then perform the operation. At the end we always write the answer in simplified form.

Example 3. Calculate: $\frac{3}{4} + \frac{5}{6} - \frac{7}{12}$.

Solution: $\mathrm{EKUK}(4;\,6;\,12) = 12$. We bring each fraction to the denominator 12: $\frac{3}{4} = \frac{9}{12}$, $\frac{5}{6} = \frac{10}{12}$, and the third fraction stays as it is. Now $\frac{9}{12} + \frac{10}{12} - \frac{7}{12} = \frac{9 + 10 - 7}{12} = \frac{12}{12} = 1$.

Answer: $1$. Note: taking the least common multiple, and not the product of the denominators, as the common denominator makes the calculation much easier.

Multiplication and division: the simplest operations

In multiplication, the numerator is multiplied by the numerator and the denominator by the denominator: $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$, where $b\ne0$ and $d\ne0$. In division, the first fraction is multiplied by the reciprocal of the second: $\frac{a}{b}:\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$. For this operation the divisor fraction must not be zero, that is, $c\ne0$ is also required. Simplifying before multiplying makes the numbers smaller and the calculation easier.

Example 4. Calculate: $\frac{4}{9} : \frac{8}{15}$.

Solution: we replace division by multiplication: $\frac{4}{9} : \frac{8}{15} = \frac{4}{9} \cdot \frac{15}{8}$. Before multiplying we simplify: $4$ and $8$ by $4$ ($1$ and $2$ remain), $15$ and $9$ by $3$ ($5$ and $3$ remain). The result is $\frac{1}{3} \cdot \frac{5}{2} = \frac{5}{6}$.

We check: calculating directly gives $\frac{4 \cdot 15}{9 \cdot 8} = \frac{60}{72}$, and simplifying this by 12 again gives $\frac{5}{6}$.

If mixed numbers are involved, first convert them to improper fractions: for example, $2\frac{1}{3} = \frac{2 \cdot 3 + 1}{3} = \frac{7}{3}$. After that, all the rules work as usual: $2\frac{1}{3} \cdot \frac{9}{14} = \frac{7}{3} \cdot \frac{9}{14} = \frac{63}{42} = \frac{3}{2} = 1\frac{1}{2}$. If a natural number is involved, treat it as a fraction whose denominator is 1: $5 = \frac{5}{1}$, and then examples such as $5 : \frac{2}{3} = \frac{5}{1} \cdot \frac{3}{2} = \frac{15}{2} = 7\frac{1}{2}$ also follow one and the same rule.

Independent exercise

Calculate: $\frac{5}{6}-\frac{1}{4}:\frac{3}{8}$. Short answer: first the division is performed, $\frac{1}{4}:\frac{3}{8}=\frac{2}{3}$; then $\frac{5}{6}-\frac{2}{3}=\frac{1}{6}$.

Typical mistakes and how to avoid them

The following mistakes arise when a rule is applied mechanically; beside each of them there is a correct check.

• Adding by «numerator to numerator, denominator to denominator». $\frac{1}{2} + \frac{1}{3}$ is never equal to $\frac{2}{5}$. The correct way is a common denominator: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.
• «Simplifying» an addend. The threes in $\frac{3+5}{3}$ cannot be cancelled: simplifying applies only to factors, not to a single term of a sum.
• Inverting the wrong fraction in division. Only the second fraction, that is, the divisor, is inverted; if, on passing to multiplication, you invert the first fraction and do not change the second, then, when the dividend is not zero, the reciprocal of the correct result is obtained. If the dividend is zero, it cannot be inverted.
• Looking at only one part in a comparison. The conclusion «the denominator is larger — so the fraction is larger» is wrong: when the numerators are the same positive number and the denominators are also positive, the fraction with the larger denominator is smaller.
• Leaving the answer unsimplified. $\frac{60}{72}$ is arithmetically correct, but $\frac{5}{6}$ is expected as the final answer; in tests the choice given is often exactly the simplified one.

Conclusion and the next step

• In every fraction, check that the denominator is not zero.
• In addition and subtraction, bring the fractions to a common denominator; in multiplication, simplify beforehand.
• Before a cross-multiplication comparison, bring the denominators to positive form.
• In division, invert only the second fraction and make sure that the divisor is not zero.
• Repeat the rules of the greatest common divisor EKUB and the least common multiple EKUK or solve the tests on fractions.