Theorem on Equal Ratios

Fill in the blanks of the following

 $\frac{a}{3}=\frac{b}{4}=\frac{c}{7}=\frac{a-2b+3c}{......}=\frac{......}{6-8+14}$

Solve $\frac{5y^2+40y-12}{5y^2+10y-4}=\frac{y+8}{1+2y}=\frac{5y\left(y+8\right)}{5y\left(1+2y\right)}$ and find the value of $y$.

$5m-n=3m+4n,$ then find the values of the following expressions.

$\frac{3m+4n}{3m-4n}$

If $5m-n=3m+4n,$ then find the value of the following expression:

$\frac{m^2+n^2}{m^2-n^2}$

Solve:

$\frac{12x^2+18x+42}{18x^2+12x+58}=\frac{2x+3}{3x+2}$

If a, b, c, d are in proportion, then prove that

$\frac{a^2+ab+b^2}{a^2-ab+b^2}=\frac{c^2+cd+d^2}{c^2-cd+d^2}$

If a, b, c, d are in proportion, then prove that,

$\sqrt{\frac{a^2+5c^2}{b^2+5d^2}}=\frac{a}{b}$

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{2}{3}$ then find the value of the following expression.

$\frac{7\mathrm{b}-4\mathrm{a}}{7\mathrm{b}+4\mathrm{a}}$

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{2}{3}$ then find the value of the following expression.

$\frac{{\mathrm{a}}^3+{\mathrm{b}}^3}{{\mathrm{b}}^3}$

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{2}{3}$ then find the values of the following expression.

$\frac{5{\mathrm{a}}^2+2{\mathrm{b}}^2}{5{\mathrm{a}}^2-2{\mathrm{b}}^2}$

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{2}{3}$ then find the value of the following expression. Write your answer in fraction form.

$\frac{4\mathrm{a}+3\mathrm{b}}{3\mathrm{b}}$

Write the ratio of first quantity to second quantity in the reduced form.

$1.5 \mathrm{kg}, 2500 \mathrm{gm}$

Write the ratio of first quantity to second quantity in the reduced form.

$4 \mathrm{sq}.\mathrm{m}, 800 \mathrm{sq}.\mathrm{cm}$

Write the ratio of first quantity to second quantity in the reduced form: $5$ dozens, $120$ units

Write the ratio of first quantity to second quantity in the reduced form.

$₹17$, $₹25$ and $60$ paise

Write the ratio of first quantity to second quantity in the reduced form.

$1024 \mathrm{MB}, 1.2 \mathrm{GB} \left[\left(1024 \mathrm{MB}=1 \mathrm{GB}\right)\right]$

If a, b, c, d are in proportion, then prove that,

$\frac{11a^2+9ac}{11b^2+9bd}=\frac{a^2+3ac}{b^2+3bd}$

If $\frac{by+cz}{b^2+c^2}=\frac{cz+ax}{c^2+a^2}=\frac{ax+by}{a^2+b^2}$ then prove that $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$.

If $\frac{2x-3y}{3z+y}=\frac{z-y}{z-x}=\frac{x+3z}{2y-3x}$, then prove that every ratio $=\frac{x}{y}$.

Solve.

$\frac{16x^2-20x+9}{8x^2+12x+21}=\frac{4x-5}{2x+3}$