1.2 Fractions

A fraction is in simplest form, or lowest terms, when numerator and denominator have no common factor greater than $1$. Cancel common factors until none remain: $\frac{18}{24}=\frac{3}{4}$ after dividing both by $6$.

Cross-products can check equivalence: $\frac{a}{b}=\frac{c}{d}$ when $ad=bc$, provided $b$ and $d$ are non-zero. For $\frac{2}{5}$ and $\frac{6}{15}$, both cross-products are $30$.

Both forms represent the same value. Since $\frac{11}{4}$ contains two complete groups of four quarters with three quarters left, it equals $2\frac34$.

Simplify the fractional part when needed. For example, $2\frac{6}{8}=2\frac34$.

Any common multiple gives a valid common denominator, so $42$ would also work. The smallest convenient choice usually keeps the arithmetic shorter.

Common denominators express fractions in equal-sized parts. Once the parts have the same size, their numerators can be compared, added or subtracted meaningfully.

When fractions have the same positive denominator, compare numerators. When they have the same positive numerator, the fraction with the smaller denominator is larger because each part is larger.

A proper fraction of a positive quantity must be smaller than the original quantity. An ordering can be checked by estimating against useful benchmarks such as $0$, $\frac12$ and $1$.

For a part of a group, denominator is the total group. If $19$ of $403$ people are crew, the fraction who are crew is $\frac{19}{403}$.

The result need not be proper. Expressing $12$ as a fraction of $8$ gives $\frac{12}{8}=\frac32$, because the first quantity is larger than the second.

For $\frac23+\frac57$, use denominator $21$: $\frac{14}{21}+\frac{15}{21}=\frac{29}{21}=1\frac{8}{21}$.

You may work with whole and fractional parts separately, but regroup one whole when the first fractional part is too small to subtract. Converting to improper fractions avoids that hidden borrowing step.

If an equivalent fraction has denominator $100$, its numerator is the percentage: $\frac{18}{100}=18\%=0.18$. Otherwise division always works.

Some decimals terminate; others repeat forever. Keep an ellipsis or recurring notation until the required rounding stage so the fraction's value is not silently changed.

Division by $n$ asks how much remains in one of $n$ equal groups. Multiplication by $\frac1n$ takes exactly that one equal share, so the operations have the same effect.

The inverse pair restores the starting value: $x\times n\times\frac1n=x$ for $n\ne0$.

Multiplying by the reciprocal works because $\frac83\times\frac38=1$; the reciprocal undoes multiplication by the divisor.

Estimate before calculating. Since $3\frac15$ is a little larger than $2\frac23$, their quotient should be a little larger than $1$, which agrees with $1\frac15$.