


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





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,
\(\begin{aligned}&\space\dfrac{3}{5}:\dfrac{7}{10} \\\\&= \dfrac{3}{5} \times 10:\dfrac{7}{10}\times 10 \\\\&=6:7. \end{aligned}\)

