The ratio is used for comparing two quantities of the sane kind.
The ratio formula for two numbers says a and b is given by a:b or a/b. When two or more such ratios are equal, they are said to be in proportion.
The concept of ratio and proportion is majorly based on ratios and fractions.
Solution:
Given, $2/3:4/9$
$={{\left( 2/3 \right)}^{2}}:{{\left( 4/9 \right)}^{2}}$
$=4/9:16/81$
$=\left( 4/9 \right)\times \left( 81/16 \right)$
$=\left( 4\times 81 \right)/\left( 9\times 16 \right)$
$=\left( 1\times 9 \right)/\left( 1\times 4 \right)$
$=9/4$
Therefore, duplicate ratio = 9: 4
Solution:
Given,
$\left( a+b \right):\left( {{a}^{2}}-{{b}^{2}} \right)$
$={{\left( a+b \right)}^{2}}:{{\left( {{a}^{2}}-{{b}^{2}} \right)}^{2}}$
$={{\left( a+b \right)}^{2}}/\left( {{\left( a+b \right)}^{2}}{{\left( a-b \right)}^{2}} \right)$
$=1/{{\left( a-b \right)}^{2}}$
Therefore, duplicate ratio $=1:{{\left( a-b \right)}^{2}}$