Theorems on Equal Ratios

Q.

Geometric mean of $7,7^2,7^3,...,7^n$ is

1. ${\left(7\right)}^{\frac{(n+1)}{2}}$

2. $7$

3. ${\left(7\right)}^{\frac{n}{2}}$

4. $7^n$

Q.

Identify the equivalent ratios among the following:

$1:4,3:9,1:3,4:12,2:6$

Q.

Express each of the following ratios in the simplest form:-

$15:25$

Q.

Solve using appropriate proper properties.

$\frac{2}{5}\times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7}\times \frac{3}{5}$

Q. If $\frac{y}{b+c-a}=\frac{z}{c+a-b}=\frac{x}{a+b-c}$ and
$x+y+z\ne 0$, then which of the following hold true?

1. $\frac{a}{z+x}=\frac{b}{x+y}=\frac{c}{y+z}$
2. $\frac{a}{z-x}=\frac{b}{x-y}=\frac{c}{y-z}$
3. $\frac{z-x}{a}=\frac{x-y}{b}=\frac{y-z}{c}$
4. $\frac{z+x}{a}=\frac{b}{x+y}=\frac{c}{y+z}$

Q. If $\frac{x}{3}=\frac{y}{5}=\frac{z}{4}=\frac{5x-3y+4z}{P}$. Find the value of $P$?

Q. Which of the following option is correct according to theorem of equal ratios?

1. If $\frac{a}{b}=\frac{c}{d}$, then $\frac{a}{c}=\frac{b}{d}$
2. If $\frac{a}{b}=\frac{c}{d}$, then $\frac{a}{d}=\frac{c}{b}$
3. If $\frac{a}{b}=\frac{c}{d}$, then $\frac{a}{b}=\frac{a+c}{b+d}=\frac{c}{d}$
4. If $\frac{a}{b}=\frac{c}{d}$, then $\frac{a}{b}=\frac{a+b}{c+d}=\frac{c}{d}$

Q.

Which of the following is equivalent to $\frac{2}{5}$?

1. $4\%$

2. $2.5$

3. $0.25$

4. $2\times \frac{1}{5}$

Q.

Fill in the following blanks: $\frac{15}{18}=\frac{\boxed{}}{6}=\frac{10}{\boxed{}}=\frac{\boxed{}}{30}$. Are these equivalent ratios?

Q. If $\frac{27}{36}=\frac{9}{12}$ $=k$, then value of $k$ is .

1. $\frac{38}{73}$
2. $\frac{36}{48}$
3. $\frac{48}{36}$
4. $\frac{4}{8}$

Q. Find the value of $x$ in the following equation using theorems on equal ratio.

$\frac{14x^2-6x+8}{10x^2+4x+7}=\frac{7x-3}{5x+2}$

1. $\frac{37}{3}$
2. $\frac{37}{9}$
3. $\frac{37}{5}$
4. $\frac{37}{4}$

Q. If $\frac{a}{10}$ = $\frac{b}{4}$ = $\frac{c}{6}$ = $\frac{4a-3b+9c}{x}$. The value of x is .

1. 41
2. 82
3. 84
4. 42

Q. If $\frac{a}{10}$ = $\frac{b}{4}$ = $\frac{c}{6}$ = $\frac{4a-3b+9c}{x}$. Find the value of $x.$

1. 82

Q. If $\frac{a}{3}=\frac{b}{7}=\frac{4a+5b}{Q}$. Find the value of Q?

1. 37
2. 47
3. 35
4. 49

Q. If $\frac{x}{3}=\frac{y}{5}=\frac{z}{4}=\frac{5x-3y+4z}{P}$. Find the value of $P$?

1. $14$
2. $10$
3. $12$
4. $16$

Q. Show that
$\frac{2}{1-}\frac{3}{5-}\frac{8}{7-}\cdots \frac{n^2-1}{2n+1}=\frac{n(n+3)}{2}$.

Q. If $\frac{a}{b+c}$ = $\frac{b}{c+a}$ = $\frac{c}{a+b}$ and $a+b+c\ne$ 0, value of $a$ is .

1. $\frac{b+c}{2}$
2. $\frac{1}{2}$
3. $2\times (b+c)$
4. $2\times \left(b\right)$

Q. If 6 men build a house in a week, how many men will be required to build the house in 2 days?

1. 12
2. 21
3. 18
4. 30

Q. Burst out the ratios equivalent to $1:2$.

1. 20:40
2. 13:39
3. 4:8
4. 7:21
5. 9:18
6. 8:16
7. 6:12
8. 7:14

Q.

Suppose $x$, $y$, and $z$ form a geometric sequence. if you know that $x+y+z=18$ and $x^2+y^2+z^2=612$, find the value of $y$.