Q: a : b = 2 : 3 , b : c = 4 : 5 , c : d = 6 : 7 . Find a : b : c : d.

Given \$a:b=2:3 , b:c=4:5 , c:d=6:7\$\$a:b=2\times 8:3\times 8=16:24\\ b:c=4\times 6:5\times 6=24:30\\ c:d=6\times 5:7\times 5=30:35\\ so\quad a:b:c:d=16:24:30:35\$

Q: Find the ratio of:\$9m\$ to \$27cm\$

9 m to 27 cm\$\Rightarrow 9\times100\ cm\ to\ 27\ cm\quad [\because 1\ m=100\ cm]\$\$\Rightarrow 900\ cm\ to\ 27\ cm\$Thus, ratio is \$=\dfrac{900}{27}=\dfrac{100}{3}=100:3\$

Q: If \$a, b, c\$ are all non-zero and \$a + b + c = 0\$, prove that \$\dfrac{a^2}{bc} + \dfrac{b^2}{ac} + \dfrac{c^2}{ab} = 3\$

Given, \$a+b+c=0\$or, \$a+b=-c\$cubing both sides we get,\$(a+b)^3=-c^3\$\$a^3+b^3+3ab(a+b)=-c^3\$ [\$\because(a+b)^3=a^3+3a^2b+3ab^2+b^3\$] \$a^3+b^3-3abc=-c^3\$ [Since \$a+b=-c\$] \$a^3+b^3+c^3=3abc\$.......(1).Now,\$\dfrac{a^2}{bc} + \dfrac{b^2}{ac} + \dfrac{c^2}{ab}\$\$=\dfrac{a^3+b^3+c^3}{abc} \$\$=\dfrac{3abc}{abc} \$ [Using (1)]\$=3\$.

Q: The length and breadth of a rectangular field are \$50\ m\$ and \$15\ m\$ respectively. Find the ratio of the breadth to the length of the field.

The length of the rectangular field is \$50 m\$ and breadth of the field is \$15 m\$. The ratio of breadth and length is, \${\rm{breadth}}:{\rm{length}} = 15:50=\dfrac{15}{50}=\dfrac{3}{10}\$ \$ = 3:10\$

Q: Find the ratio \$50\$ paise to Rs. \$5\$

\${\textbf{Step - 1: Converting rupees to paise}}\$ \${\text{We know that, 1 Rupee = 100 paise}}\$ \$\therefore {\text{ 5 Rupees = }}\dfrac{{{\text{5 }} \times {\text{ 100}}}}{{\text{1}}}{\text{ paise}}\$ \$ \Rightarrow {\text{ 5 Rupees = 500 paise}}\$ \$\quad \text{.........eqn(i)}\$ \${\textbf{Step - 2: Substitute above result and get required ratio}}\$ \${\text{Ratio of 50 paise to 5 Rupees = }}\$\$ {\text{ }}\dfrac{{{\text{50 paise}}}}{{{\text{5 Rupees}}}}\$ \$ ={\text{ }}\dfrac{{{\text{50 paise}}}}{{{\text{500 paise}}}}\$ \$\quad \textbf{[From eqn(i)]}\$ \$= {\text{ }}\dfrac{{\text{1}}}{{{\text{10}}}}\$ \${\textbf{Thus, the ratio of 50 paise to 5 Rupees is 1:10. Option C is correct.}}\$

Q: Find the ratio of the following in the simplest form.\$3\$m \$5\$cm to \$35\$cm.

We know that,1 meter = 100 cm.3 m 5 cm = 3 \$ \times \$ 100 cm + 5 cm. = 305 cm.Thus, 3 m 5 cm : 35 cm = 305 cm : 35 cm. = \$ \dfrac{305}{35}\$. = \$ \dfrac{61}{7}\$ = 61 : 7.

Q: If angles of a triangle are in the ratio \$2:3:4\$. Find the value of each angle.

Let common ratio be \$2x, 3x\$ and \$4x\$\$\rightarrow \$ let common ratio be \$x\$\$\therefore \ \$ Angle are \$2x,\ 3x\$ and \$4x\$\$\therefore \ 2x+3x4x=180^o\$\$\therefore \ 9x=180^o\$\$\therefore \ x=\dfrac {180^o}{9}=20^o\$\$\therefore \ \$ Angle are \$40^o,\ 60^o,\ 80^o\$

Q: The average age of three boys is \$25\$ years and their ages are in the ratio \$ 3: 5 : 7\$. The age of the youngest boy is

Total age of \$3\$ boys \$= (25 \$ \$\times\$ \$3)\$ years \$=75\$ years. Ratio of their ages \$ = 3 : 5 : 7\$.Age of the youngest \$=\$ \$\displaystyle {\left (75\, \times\, \frac{3}{15} \right )}\$ years \$= 15\$ years

Q: The ages of two persons are in the ratio of \$5:7\$. Eighteen years ago their ages were in the ratio \$8:13\$. Find their present ages.

The ages of \$'2'\$ persons are in the ratio \$5:7\$1st person age = \$5x\$2nd person age = \$7x\$\$'18'\$ years ago their ratio \$8:13\$\$\dfrac{5x-18}{7x-18}=\dfrac{8}{13}\$\$13(5x-18)=8(7x-18)\$\$65x-234=56 x-144\$\$65x-56x=234-144\$\$9x=90\$\$x=10\$1st person age = \$5\times 10=50\,yrs\$2nd person age = \$7\times 10=70\,yrs\$

Q: Find the ratio of the given statement in the simplest form.\$3\$ weeks to \$3\$ days.

We know that,\$1 week = 7 days\$.So, \$3 weeks : 3 days\$ = \$3 \times 7 days : 3 days\$ = \$21 days : 3 days\$ = \$\dfrac{21}{3}\$ =\$\dfrac{7}{1}\$ = \$7 : 1\$

Question 1

a : b = 2 : 3 , b : c = 4 : 5 , c : d = 6 : 7 . Find a : b : c : d.

Accepted Answer

Given $a:b=2:3 , b:c=4:5 , c:d=6:7$$a:b=2\times 8:3\times 8=16:24\ b:c=4\times 6:5\times 6=24:30\ c:d=6\times 5:7\times 5=30:35\ so\quad a:b:c:d=16:24:30:35$

Question 2

Find the ratio of:$9m$ to $27cm$

Accepted Answer

9 m to 27 cm$\Rightarrow 9\times100\ cm\ to\ 27\ cm\quad [\because 1\ m=100\ cm]$$\Rightarrow 900\ cm\ to\ 27\ cm$Thus, ratio is $=\dfrac{900}{27}=\dfrac{100}{3}=100:3$

Question 3

If $a, b, c$ are all non-zero and $a + b + c = 0$, prove that $\dfrac{a^2}{bc} + \dfrac{b^2}{ac} + \dfrac{c^2}{ab} = 3$

Accepted Answer

Given, $a+b+c=0$or, $a+b=-c$cubing both sides we get,$(a+b)^3=-c^3$$a^3+b^3+3ab(a+b)=-c^3$     [$\because(a+b)^3=a^3+3a^2b+3ab^2+b^3$] $a^3+b^3-3abc=-c^3$ [Since $a+b=-c$] $a^3+b^3+c^3=3abc$.......(1).Now,$\dfrac{a^2}{bc} + \dfrac{b^2}{ac} + \dfrac{c^2}{ab}$$=\dfrac{a^3+b^3+c^3}{abc} $$=\dfrac{3abc}{abc} $ [Using (1)]$=3$.

Question 4

The length and breadth of a rectangular field are $50\ m$ and $15\ m$ respectively. Find the ratio of the breadth to the length of the field.

Accepted Answer

The length of the rectangular field is $50 m$ and&#10;breadth of the field is $15 m$.&#10;&#10;The ratio of breadth and length is,&#10;&#10;${\rm{breadth}}:{\rm{length}} = 15:50=\dfrac{15}{50}=\dfrac{3}{10}$&#10;&#10;$ = 3:10$

Question 5

Find the ratio $50$ paise to Rs. $5$

Accepted Answer

${\textbf{Step  - 1: Converting rupees to paise}}$&#10;&#10;                   ${\text{We&#10;know that, 1 Rupee  =  100 paise}}$&#10;&#10;                   $\therefore&#10;{\text{ 5 Rupees  =  }}\dfrac{{{\text{5 }} \times {\text{&#10;100}}}}{{\text{1}}}{\text{ paise}}$&#10;&#10;                   $&#10;\Rightarrow {\text{ 5 Rupees  =  500 paise}}$   $\quad \text{.........eqn(i)}$&#10;&#10;${\textbf{Step  - 2: Substitute above result and get required ratio}}$&#10;&#10;                   ${\text{Ratio&#10;of 50 paise to 5 Rupees = }}$$ {\text{ }}\dfrac{{{\text{50 paise}}}}{{{\text{5 Rupees}}}}$&#10;&#10;                                                                         $&#10;={\text{ }}\dfrac{{{\text{50 paise}}}}{{{\text{500 paise}}}}$  $\quad \textbf{[From eqn(i)]}$&#10;&#10;                                                                         $= {\text{ }}\dfrac{{\text{1}}}{{{\text{10}}}}$&#10;&#10;${\textbf{Thus, the ratio of 50 paise to 5 Rupees is 1:10. Option C is correct.}}$

Question 6

Find the ratio of the following in the simplest form.$3$m $5$cm to $35$cm.

Accepted Answer

We know that,1 meter = 100 cm.3 m 5 cm = 3 $ \times $ 100 cm + 5 cm.                 = 305 cm.Thus, 3 m 5 cm : 35 cm = 305 cm : 35 cm.                              = $ \dfrac{305}{35}$.                              = $ \dfrac{61}{7}$                              = 61 : 7.

Question 7

If angles of a triangle are in the ratio $2:3:4$. Find the value of each angle.

Accepted Answer

Let common ratio be $2x, 3x$ and $4x$$\rightarrow $ let common ratio be $x$$\therefore \ $ Angle are $2x,\ 3x$ and $4x$$\therefore \ 2x+3x4x=180^o$$\therefore \ 9x=180^o$$\therefore \ x=\dfrac {180^o}{9}=20^o$$\therefore \ $ Angle are $40^o,\ 60^o,\ 80^o$

Question 8

The average age of three boys is $25$ years and their ages are in the ratio $ 3: 5 : 7$. The age of the youngest boy is

Accepted Answer

Total age of $3$ boys $= (25 $ $\times$ $3)$ years $=75$ years. Ratio of their ages $ = 3 : 5 : 7$.Age of the youngest $=$ $\displaystyle {\left (75\, \times\, \frac{3}{15} \right )}$ years $= 15$ years

Question 9

The ages of two persons are in the ratio of $5:7$. Eighteen years ago their ages were in the ratio $8:13$. Find their present ages.

Accepted Answer

The ages of $'2'$ persons are in the ratio $5:7$1st person age = $5x$2nd person age = $7x$$'18'$ years ago their ratio $8:13$$\dfrac{5x-18}{7x-18}=\dfrac{8}{13}$$13(5x-18)=8(7x-18)$$65x-234=56 x-144$$65x-56x=234-144$$9x=90$$x=10$1st person age = $5\times 10=50\,yrs$2nd person age = $7\times 10=70\,yrs$

Question 10

Find the ratio of the given statement in the simplest form.$3$ weeks to $3$ days.

Accepted Answer

We know that,$1 week = 7 days$.So, $3 weeks : 3 days$ = $3 \times 7 days : 3 days$                              = $21 days : 3 days$                               = $\dfrac{21}{3}$                                =$\dfrac{7}{1}$                                = $7 : 1$

Revise with Concepts

Introduction to Ratios

Dividing a Quantity in a Given Ratio

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Quick Summary With Stories

Introduction to Ratios

Dividing a Quantity in a Given Ratio

a : b = 2 : 3 , b : c = 4 : 5 , c : d = 6 : 7 . Find a : b : c : d.

Find the ratio of:
$9 m$ to $27 c m$

If $a, b, c$ are all non-zero and $a + b + c = 0$ , prove that $\frac{a ^{2}}{b c} + \frac{b ^{2}}{a c} + \frac{c ^{2}}{a b} = 3$

The length and breadth of a rectangular field are $50 m$ and $15 m$ respectively. Find the ratio of the breadth to the length of the field.

Find the ratio $50$ paise to Rs. $5$

Find the ratio of the following in the simplest form.
$3$ m $5$ cm to $35$ cm.

If angles of a triangle are in the ratio $2 : 3 : 4$ . Find the value of each angle.

The average age of three boys is $25$ years and their ages are in the ratio $3 : 5 : 7$ . The age of the youngest boy is

The ages of two persons are in the ratio of $5 : 7$ . Eighteen years ago their ages were in the ratio $8 : 13$ . Find their present ages.

Find the ratio of the given statement in the simplest form.
$3$ weeks to $3$ days.

Revise with Concepts

Introduction to Ratios

Dividing a Quantity in a Given Ratio

Learn with Videos

Quick Summary With Stories

Introduction to Ratios

Dividing a Quantity in a Given Ratio

a : b = 2 : 3 , b : c = 4 : 5 , c : d = 6 : 7 . Find a : b : c : d.

Find the ratio of: 9m to 27cm

If a,b,c are all non-zero and a+b+c=0, prove that bca2​+acb2​+abc2​=3

The length and breadth of a rectangular field are 50 m and 15 m respectively. Find the ratio of the breadth to the length of the field.

Find the ratio 50 paise to Rs. 5

Find the ratio of the following in the simplest form. 3m 5cm to 35cm.

If angles of a triangle are in the ratio 2:3:4. Find the value of each angle.

The average age of three boys is 25 years and their ages are in the ratio 3:5:7. The age of the youngest boy is

The ages of two persons are in the ratio of 5:7. Eighteen years ago their ages were in the ratio 8:13. Find their present ages.

Find the ratio of the given statement in the simplest form. 3 weeks to 3 days.

Find the ratio of:
$9 m$ to $27 c m$

If $a, b, c$ are all non-zero and $a + b + c = 0$ , prove that $\frac{a ^{2}}{b c} + \frac{b ^{2}}{a c} + \frac{c ^{2}}{a b} = 3$

The length and breadth of a rectangular field are $50 m$ and $15 m$ respectively. Find the ratio of the breadth to the length of the field.

Find the ratio $50$ paise to Rs. $5$

Find the ratio of the following in the simplest form.
$3$ m $5$ cm to $35$ cm.

If angles of a triangle are in the ratio $2 : 3 : 4$ . Find the value of each angle.

The average age of three boys is $25$ years and their ages are in the ratio $3 : 5 : 7$ . The age of the youngest boy is

The ages of two persons are in the ratio of $5 : 7$ . Eighteen years ago their ages were in the ratio $8 : 13$ . Find their present ages.

Find the ratio of the given statement in the simplest form.
$3$ weeks to $3$ days.