A ratio can be defined as a comparison of two or more numbers that shows their sizes in relation to each other. The ratio is in the form [tex]\dfrac{\rm a}{\rm b}[/tex], where [tex]\rm b \neq 0[/tex].
When two ratios are set equal to each other, the equation is called proportion.
The solutions to the given ratios are:
[tex]\rm x = 9\\\rm x = 24\\\rm x = 24\\\rm x = 16[/tex]
Solution of the aforesaid will be obtained through forming quadratic equations:
(1.)
[tex]\begin{aligned} \dfrac{2}{3} &= \dfrac{6}{\rm x} \\\\\rm 2x &= 18 \end{aligned}[/tex]
[tex]\rm x = \dfrac{18}2} \\\\\rm x = 9[/tex]
(2.)
[tex]\begin{aligned} \dfrac{4}{7} &= \dfrac{\rm x}{42} \\\\4\times 42 &= 7\rm x \\\\168 &= 7\rm x\\\\\rm x &= \dfrac{168}{7}\\\\\rm x &= 24 \end[/tex]
(3.)
[tex]\begin {aligned}\dfrac{\rm 2x}{48} &= \dfrac{3}{12} \\\\24\rm x &= 3\times 48\\\\24\rm x &= 144\\\\\rm x &= \dfrac{144}{24} \\\\\rm x &= 6 \end[/tex]
(4.)
[tex]\begin{aligned} \dfrac{12}{15} &= \dfrac{\rm x}{20} \\\\12 \times 20 &= 15\rm x\\\\\rm 15x &= 240\\\\\rm x &= \dfrac{240}{15} \\\\\rm x &= 16 \end[/tex]
The value of x in all the cases is 9,24,6,16 respectively.
Learn more about Ratios here:
https://brainly.com/question/1504221
