Maths SolverA fraction is in lowest terms (or “simplified,” or “in simplest form” — same idea, three different phrasings) when its numerator and denominator have no common factor larger than $1$. So $\tfrac{6}{8}$ is not in lowest terms because $2$ divides both $6$ and $8$; $\tfrac{3}{4}$ is, because $3$ and $4$ have nothing in common.
There are two ways to simplify a fraction. The slow way is to spot a common factor, divide both top and bottom by it, and repeat. The fast way is to find the greatest common divisor (GCD) of the two numbers and divide by that once. They give the same answer; the second saves you several steps and the chance to lose a factor.
The one-step method
For $\tfrac{a}{b}$, the simplified form is
$$\frac{a}{b} = \frac{a / \gcd(a, b)}{b / \gcd(a, b)}.$$
$\gcd(6, 8) = 2$, so $\tfrac{6}{8} = \tfrac{6/2}{8/2} = \tfrac{3}{4}$. $\gcd(45, 60) = 15$, so $\tfrac{45}{60} = \tfrac{3}{4}$ as well. The whole question of simplifying a fraction reduces to computing the GCD of two integers, which the Euclidean algorithm does in a few subtractions.
Improper fractions and mixed numbers
A fraction whose numerator is bigger than its denominator (in absolute value) is called improper. $\tfrac{7}{3}$ is improper. You can either leave it as a fraction in lowest terms or convert it to a mixed number: $7 \div 3 = 2$ remainder $1$, so $\tfrac{7}{3} = 2\tfrac{1}{3}$.
Going the other way: a mixed number $a\tfrac{b}{c}$ converts to the improper fraction $\tfrac{ac + b}{c}$. So $2\tfrac{1}{3} = \tfrac{2 \cdot 3 + 1}{3} = \tfrac{7}{3}$. Calculators almost always want improper fractions, so do this conversion before plugging anything in.
Negative fractions and the sign
$\tfrac{-3}{4} = \tfrac{3}{-4} = -\tfrac{3}{4}$. The minus sign can sit on the numerator, on the denominator, or out front; all three mean the same thing. Pick one convention — usually the third — and stick with it. Calculators that put the minus sign on the denominator make easy errors when you then go to multiply or divide.
The mistakes I see most often
1. Stopping after one common factor
$\tfrac{12}{18}$ becomes $\tfrac{6}{9}$ if you divide by $2$, then $\tfrac{2}{3}$ if you divide by $3$. Both steps are right, but if you stop after the first you have not simplified all the way. Using the GCD ($\gcd(12, 18) = 6$) gets you to $\tfrac{2}{3}$ in one move and guarantees you have not stopped early.
2. Cancelling across an addition or subtraction
You can cancel a common factor from a single product, but not from a sum. $\tfrac{2x + 4}{2}$ is not $x + 4$; it is $\tfrac{2x}{2} + \tfrac{4}{2} = x + 2$. The $2$ has to divide every term, not just one of them. The same trap shows up in $\tfrac{6 + 9}{3}$, which is $5$, not $6 + 3 = 9$.
3. Forgetting the sign on a negative numerator
Simplifying $\tfrac{-6}{8}$ gives $-\tfrac{3}{4}$, not $\tfrac{3}{4}$. Sign loss is one of the most-marked errors on test scripts.
Where this comes up later
You simplify a fraction every time you finish an arithmetic problem with a fractional answer (probability, ratio, statistics) and every time you want to compare two fractions by sight. The same idea generalises into algebra: simplifying $\tfrac{x^2 - 1}{x + 1}$ to $x - 1$ is the exact same operation, just with polynomials instead of integers.
References
- Wikipedia: Fraction, Greatest common divisor.
- Khan Academy: Fraction arithmetic.