Rational numbers are the foundation of school-level arithmetic and appear often on the YÖS exam. The ability to work with fractions, transform and simplify them is a key skill for success.
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Definition: A rational number is a number that can be represented as the ratio of two integers.
$$\mathbb{Q} = \left\{\frac{p}{q} : p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0\right\}$$
Examples:- $\frac{3}{4}$ — proper fraction (numerator less than denominator)- $\frac{7}{3}$ — improper fraction (numerator greater than denominator)- $2 = \frac{2}{1}$ — an integer is also rational- $-\frac{5}{2}$ — a negative rational number- $0 = \frac{0}{1}$ — zero is a rational number
Property: If the numerator and denominator of a fraction are multiplied or divided by the same nonzero number, the value of the fraction does not change.
$$\frac{a}{b} = \frac{a \cdot k}{b \cdot k} = \frac{a : d}{b : d}$$
where $k \neq 0$ and $d \neq 0$
Examples:$$\frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}$$$$\frac{12}{18} = \frac{12 : 6}{18 : 6} = \frac{2}{3}$$
Definition: A fraction $\frac{a}{b}$ is called irreducible if $\gcd(a, b) = 1$ (numerator and denominator are coprime).
How to reduce a fraction to its irreducible form:1. Find $\gcd(a, b)$2. Divide numerator and denominator by $\gcd(a, b)$
Example:$$\frac{36}{48} = \frac{36 : 12}{48 : 12} = \frac{3}{4}$$
(since $\gcd(36, 48) = 12$)
Definition: A mixed number consists of an integer part and a proper fraction.
Converting an improper fraction to a mixed number:$$\frac{17}{5} = 3\frac{2}{5}$$
because $17 = 5 \cdot 3 + 2$
Converting a mixed number to an improper fraction:$$2\frac{3}{4} = \frac{2 \cdot 4 + 3}{4} = \frac{11}{4}$$
$$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$$$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$
Example:$$\frac{3}{8} + \frac{5}{8} = \frac{3 + 5}{8} = \frac{8}{8} = 1$$$$\frac{7}{10} - \frac{3}{10} = \frac{7 - 3}{10} = \frac{4}{10} = \frac{2}{5}$$
Algorithm:1. Find the LCM of the denominators2. Bring both fractions to a common denominator3. Add or subtract the numerators
Example:$$\frac{1}{3} + \frac{1}{4}$$
LCM(3, 4) = 12
$$\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}$$
$$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$$
Example with mixed numbers:$$2\frac{1}{3} + 1\frac{1}{4} = \frac{7}{3} + \frac{5}{4} = \frac{28}{12} + \frac{15}{12} = \frac{43}{12} = 3\frac{7}{12}$$
Rule: The product of fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators.
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$
Examples:$$\frac{2}{3} \cdot \frac{5}{7} = \frac{2 \cdot 5}{3 \cdot 7} = \frac{10}{21}$$
$$\frac{3}{4} \cdot \frac{4}{5} = \frac{3 \cdot 4}{4 \cdot 5} = \frac{12}{20} = \frac{3}{5}$$
(you can cancel 4 before multiplying)
Multiplication by an integer:$$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$$
Example:$$3 \cdot \frac{2}{5} = \frac{3 \cdot 2}{5} = \frac{6}{5} = 1\frac{1}{5}$$
Rule: To divide one fraction by another, multiply the first fraction by the reciprocal of the second.
$$\frac{a}{b} : \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c}$$
where $c \neq 0$
Reciprocal: The fraction $\frac{d}{c}$ is the reciprocal of $\frac{c}{d}$, since $\frac{c}{d} \cdot \frac{d}{c} = 1$
Examples:$$\frac{3}{4} : \frac{2}{5} = \frac{3}{4} \cdot \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$
$$\frac{5}{6} : 2 = \frac{5}{6} \cdot \frac{1}{2} = \frac{5}{12}$$
$$4 : \frac{2}{3} = 4 \cdot \frac{3}{2} = \frac{12}{2} = 6$$
Definition: A decimal fraction is a fraction whose denominator is a power of 10.
$$0.5 = \frac{5}{10} = \frac{1}{2}$$$$0.25 = \frac{25}{100} = \frac{1}{4}$$$$0.125 = \frac{125}{1000} = \frac{1}{8}$$
Method 1: Bring the denominator to a power of 10
Example:$$\frac{3}{4} = \frac{3 \cdot 25}{4 \cdot 25} = \frac{75}{100} = 0.75$$
Method 2: Divide the numerator by the denominator
Example:$$\frac{5}{8} = 5 : 8 = 0.625$$
#### Addition and subtractionAdd and subtract as you would ordinary numbers, aligning the decimal points.
Example:$$3.45 + 2.67 = 6.12$$$$5.2 - 1.85 = 3.35$$
#### MultiplicationMultiply the numbers without the decimal points, then count off as many digits after the decimal point as there were in both factors combined.
Example:$$2.5 \cdot 1.4 = ?$$
Compute: $25 \cdot 14 = 350$
There were $1 + 1 = 2$ digits after the decimal point, so:$$2.5 \cdot 1.4 = 3.50 = 3.5$$
#### DivisionConvert the divisor to an integer by multiplying both numbers by 10, 100, etc.
Example:$$4.2 : 0.6 = 42 : 6 = 7$$
A repeating decimal is a decimal in which one or more digits repeat infinitely.
$$0.333\ldots = 0.\overline{3}$$$$0.142857142857\ldots = 0.\overline{142857}$$$$0.1666\ldots = 0.1\overline{6}$$
Formula: If $x = 0.\overline{a_1a_2\ldots a_n}$, then$$x = \frac{a_1a_2\ldots a_n}{\underbrace{99\ldots9}_{n \text{ nines}}}$$
Example: Convert $0.\overline{3}$ to an ordinary fraction.
$$x = 0.\overline{3} = 0.333\ldots$$$$10x = 3.333\ldots$$$$10x - x = 3.333\ldots - 0.333\ldots$$$$9x = 3$$$$x = \frac{3}{9} = \frac{1}{3}$$
Example: Convert $0.\overline{27}$ to an ordinary fraction.
$$0.\overline{27} = \frac{27}{99} = \frac{3}{11}$$
Example: Convert $0.1\overline{6}$ to an ordinary fraction.
$$x = 0.1\overline{6} = 0.1666\ldots$$$$10x = 1.666\ldots$$$$100x = 16.666\ldots$$$$100x - 10x = 16.666\ldots - 1.666\ldots$$$$90x = 15$$$$x = \frac{15}{90} = \frac{1}{6}$$
Formula: If $x = 0.a_1a_2\ldots a_m\overline{b_1b_2\ldots b_n}$, then$$x = \frac{a_1a_2\ldots a_m b_1b_2\ldots b_n - a_1a_2\ldots a_m}{\underbrace{99\ldots9}_{n \text{ nines}}\underbrace{00\ldots0}_{m \text{ zeros}}}$$
$$\frac{3}{7} < \frac{5}{7} \text{ (since } 3 < 5\text{)}$$
$$\frac{3}{7} > \frac{3}{11} \text{ (since } 7 < 11\text{)}$$
(The larger the denominator, the smaller the fraction)
$$\frac{2}{3} \text{ and } \frac{3}{5}$$
LCM(3, 5) = 15
$$\frac{2}{3} = \frac{10}{15}, \quad \frac{3}{5} = \frac{9}{15}$$
$$\frac{10}{15} > \frac{9}{15}, \text{ so } \frac{2}{3} > \frac{3}{5}$$
$$\frac{a}{b} \text{ and } \frac{c}{d}$$
Compare $a \cdot d$ and $b \cdot c$
Example:$$\frac{3}{5} \text{ and } \frac{4}{7}$$
$3 \cdot 7 = 21$ and $5 \cdot 4 = 20$
$21 > 20$, so $\frac{3}{5} > \frac{4}{7}$
$$\frac{2}{3} = 0.666\ldots$$$$\frac{3}{5} = 0.6$$
$0.666\ldots > 0.6$, so $\frac{2}{3} > \frac{3}{5}$
Rule: The fraction bar means division. Evaluate the numerator and denominator, then divide.
$$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} : \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$$
Example:$$\frac{\frac{3}{4}}{\frac{5}{8}} = \frac{3}{4} : \frac{5}{8} = \frac{3}{4} \cdot \frac{8}{5} = \frac{24}{20} = \frac{6}{5}$$
Complex example:$$\frac{1 + \frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{3}{2}}{\frac{1}{2}} = \frac{3}{2} : \frac{1}{2} = \frac{3}{2} \cdot \frac{2}{1} = 3$$
Example:$$\frac{1 + \frac{1}{2 + \frac{1}{3}}}{4}$$
Starting with the innermost fraction:$$2 + \frac{1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$$
$$1 + \frac{1}{\frac{7}{3}} = 1 + \frac{3}{7} = \frac{7 + 3}{7} = \frac{10}{7}$$
$$\frac{\frac{10}{7}}{4} = \frac{10}{7} : 4 = \frac{10}{28} = \frac{5}{14}$$
A rational expression is an expression containing fractions with variables.
Example: $\frac{x^2 - 4}{x - 2}$
Method: Factor the numerator and denominator, then cancel common factors.
Example 1:$$\frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2 \quad (x \neq 2)$$
Example 2:$$\frac{x^2 + 5x + 6}{x + 2} = \frac{(x+2)(x+3)}{x+2} = x + 3 \quad (x \neq -2)$$
#### Addition$$\frac{x}{x+1} + \frac{2}{x+1} = \frac{x + 2}{x+1}$$
#### Subtraction$$\frac{3}{x} - \frac{2}{x} = \frac{1}{x}$$
#### Multiplication$$\frac{x}{x+1} \cdot \frac{x+1}{x^2} = \frac{x(x+1)}{(x+1)x^2} = \frac{1}{x}$$
#### Division$$\frac{x}{x+1} : \frac{x}{x^2} = \frac{x}{x+1} \cdot \frac{x^2}{x} = \frac{x \cdot x^2}{(x+1) \cdot x} = \frac{x}{x+1} \cdot x = \frac{x^2}{x+1}$$
Statement: Which fraction is larger: $\frac{7}{11}$ or $\frac{9}{14}$?
Solution by cross-multiplication:
$7 \cdot 14 = 98$$11 \cdot 9 = 99$
$98 < 99$, so $\frac{7}{11} < \frac{9}{14}$
Answer: $\frac{9}{14}$ is larger
Statement: Convert $0.\overline{45}$ to an ordinary fraction.
Solution:
$$x = 0.\overline{45} = 0.454545\ldots$$$$100x = 45.454545\ldots$$$$100x - x = 45$$$$99x = 45$$$$x = \frac{45}{99} = \frac{5}{11}$$
Answer: $\frac{5}{11}$
Statement: Evaluate $\frac{1}{1 + \frac{1}{2 + \frac{1}{3}}}$
Start with the inner part:$$2 + \frac{1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$$
$$\frac{1}{\frac{10}{7}} = \frac{7}{10}$$
Answer: $\frac{7}{10}$
Statement: Find the value of $\frac{2}{3} + \frac{3}{4} - \frac{1}{6}$
LCM(3, 4, 6) = 12
$$\frac{2}{3} = \frac{8}{12}$$$$\frac{3}{4} = \frac{9}{12}$$$$\frac{1}{6} = \frac{2}{12}$$
$$\frac{8}{12} + \frac{9}{12} - \frac{2}{12} = \frac{8 + 9 - 2}{12} = \frac{15}{12} = \frac{5}{4}$$
Answer: $\frac{5}{4}$ or $1\frac{1}{4}$
Statement: Simplify $\frac{x^2 - 1}{x^2 + 2x + 1}$
Factor the numerator and denominator:$$x^2 - 1 = (x-1)(x+1)$$$$x^2 + 2x + 1 = (x+1)^2$$
$$\frac{(x-1)(x+1)}{(x+1)^2} = \frac{x-1}{x+1} \quad (x \neq -1)$$
Answer: $\frac{x-1}{x+1}$
Wrong: $\frac{1}{2} + \frac{1}{3} = \frac{1+1}{2+3} = \frac{2}{5}$
Right: Find the LCM and bring to a common denominator:$$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$$
Wrong: $\frac{2}{3} : \frac{4}{5} = \frac{2:4}{3:5} = \frac{1/2}{3/5}$
Right: $\frac{2}{3} : \frac{4}{5} = \frac{2}{3} \cdot \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$
Wrong: $0.\overline{3} = \frac{3}{9}$ left as-is without reducing
Right: $0.\overline{3} = \frac{3}{9} = \frac{1}{3}$ (always reduce)
Wrong: $\frac{x+1}{x^2+2x+1}$ reduced as $\frac{1}{x+1}$ (parentheses missed)
Right: $\frac{x+1}{(x+1)^2} = \frac{1}{x+1}$
When simplifying $\frac{x^2-1}{x-1} = x+1$ people forget to state $(x \neq 1)$
Right: Always state the restrictions inherited from the original expression.
### Tip 1: Memorize the key decimal equivalentsKnow by heart:- $\frac{1}{2} = 0.5$- $\frac{1}{3} = 0.\overline{3}$- $\frac{1}{4} = 0.25$- $\frac{1}{5} = 0.2$- $\frac{1}{8} = 0.125$
### Tip 2: Use cross-multiplication to compareIt's faster than finding a common denominator.
### Tip 3: Cancel first, then multiplyThis simplifies the arithmetic: $\frac{12}{15} \cdot \frac{10}{18} = \frac{4}{5} \cdot \frac{5}{9} = \frac{20}{45} = \frac{4}{9}$
### Tip 4: In compound fractions, start from the innermost partWork layer by layer, simplifying gradually.
### Tip 5: Verify the answer by substituting small numbersIf you got $\frac{x-1}{x+1}$, substitute $x=3$: $\frac{2}{4} = \frac{1}{2}$ and check against the original expression.
### Tip 6: Remember the exceptionsWhen simplifying rational expressions, always state for which values of the variable the expression is undefined.
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