## Ratios

 4.1 Ratios

 Definition Compares two or three quantities of the same kind that are measured in the same unit.

•  The ratio of $$a$$ to $$b$$ is written as $$a:b$$.

 Example The ratio of $$5\,000\text{ g}$$ to $$9\text{ kg}$$ can be represented as, \begin{aligned}&\space5\,000\text{ g}:9\text{ kg} \\\\&=5\text{ kg}:9\text{ kg} \\\\&=5:9. \end{aligned} Thus, the ratio is $$5:9$$.

The ratio of three quantities:

• Represent the relation between three quantities in the form of $$a:b:c$$.

 Example Represent the ratio of $$0.02\text{ m}$$ to $$3\text{ cm}$$ to $$4.6\text{ cm}$$. \begin{aligned}&\space0.02\text{ m}:3\text{ cm}:4.6\text{ cm} \\\\&=2\text{ cm}:3\text{ cm}:4.6\text{ cm}\\\\&=2:3:4.6 \\\\&=20:30:46 \\\\&=10:15:23.\end{aligned}

Equivalent ratios:

 Definition Two or more ratios that have the same value.

• Equivalent ratios can be found by writing the ratios as equivalent fractions.

 Examples i) Multiplication Determine whether $$3:4$$ is the equivalent ratio of $$6:8$$. \begin{aligned} 3:4&=3\times2:4\times2 \\\\&=6:8. \end{aligned} Thus, $$3:4$$ is the equivalent ratio of $$6:8$$. ii) Division Determine whether $$7:28$$ is the equivalent ratio of $$1:4$$. \begin{aligned} 7:28&=7\div7:28\div7 \\\\&=1:4. \end{aligned} Thus, $$7:28$$ is the equivalent ratio of $$1:4$$.

Ratios in their simplest form:

• A ratio of $$a:b$$ is said to be in its simplest form if $$a$$ and $$b$$ are integers with no common factors other than $$1$$.

 Example State $$800\text{ g}:1.8\text{ kg}$$ in its simplest form. \begin{aligned}&\space800\text{ g}:1.8 \text{ kg} \\\\&=800\text{ g}:1\,800\text{ g} \\\\&=800\div200:1\,800\div200 \\\\&=4:9. \end{aligned} Thus, $$4:9$$ is the simplest form of $$800\text{ g}:1.8\text{ kg}$$.

 Examples i) Highest common factor (HCF) State the simplest form of $$32:24:20$$. Noted that $$4$$ is the HCF of $$32, 24$$ and $$20$$. Thus, \begin{aligned}&\space32:24:20 \\\\&= 32\div4:24\div4:20\div4 \\\\&=8:6:5. \end{aligned} ii) Lowest common multiple (LCM) State the simplest form of $$\dfrac{3}{5}:\dfrac{7}{10}$$. Noted that the LCM for $$5$$ and $$10$$ is $$10$$. Thus, \begin{aligned}&\space\dfrac{3}{5}:\dfrac{7}{10} \\\\&= \dfrac{3}{5} \times 10:\dfrac{7}{10}\times 10 \\\\&=6:7. \end{aligned}