比と比例

A ratio is simplified by dividing both parts by the same number; dividing by their highest common factor (HCF) gives the simplest form. For example $12:18$ divides by $6$ to give $2:3$. Multiplying both parts by the same number keeps the ratio equal, so $2:3=4:6=10:15$. When units differ, convert first (e.g. $1$ m $:50$ cm $=100:50=2:1$).

To share a quantity in the ratio $a:b$, first divide by the total number of parts $a+b$ to find the size of one part, then multiply. For example, dividing $60$ in the ratio $2:3$: one part is $\frac{60}{2+3}=12$, so the shares are $2\times 12=24$ and $3\times 12=36$. The same method works for three-part ratios $a:b:c$ (total parts $a+b+c$).

Saying $y$ is directly proportional to $x$ ($y\propto x$) means doubling $x$ doubles $y$; we write $y=kx$, where $k$ is the constant of proportionality. First find $k$ from a known pair of values, then use it for other values. For example, if $y=12$ when $x=3$, then $k=\frac{12}{3}=4$, so when $x=7$ we get $y=4\times 7=28$.

Saying $y$ is inversely proportional to $x$ ($y\propto \frac{1}{x}$) means doubling $x$ halves $y$; we write $y=\frac{k}{x}$, equivalently $xy=k$ (constant). Find $k$ from one pair of values and reuse it. For a best buy, compare the price per single unit. For example, $500$ g for $400$ versus $300$ g for $270$ gives $0.8$ per g and $0.9$ per g, so the first is better value.