⚖️ Ratio Basics

• What a ratio is: in $a:b$, picture the whole cut into $a+b$ equal parts — one quantity takes $a$ of them, the other takes $b$. The numbers are part-counts, not values.

• Simplify by the GCD: divide both sides by their highest common factor. For example $\dfrac{12}{18}=\dfrac{12\div 6}{18\div 6}=\dfrac{2}{3}$, so $12:18=2:3$. Always cancel the common factor in fraction form first.

• The k-method: a ratio hides the real numbers, so write them as equal multiples of one unit $k$: $a:b$ means the actual quantities are $ak$ and $bk$. Find $k$ from whatever the question gives you, then every quantity is just its parts $\times k$.

• Splitting a total: to share a total in $a:b:c$, the number of parts is $a+b+c$, so one part $=\dfrac{\text{total}}{a+b+c}$. Each share is its own parts multiplied by that one-part value.

• From a difference: the gap between two shares is also made of parts. In $a:b$ the difference is $(b-a)$ parts; set $(b-a)k=\text{given difference}$, solve for $k$, then read off whatever is asked.