[2 Mark Question Answer]

Question 12 Marks

If $a: b:: c: d:: e: f$, then prove that $\frac{a e+b f}{a e-b f}=\frac{c e+d f}{c e-d f}$

Answer

$\frac{a}{b}=\frac{c}{d}=\frac{e}{f} $
$\frac{a}{b} \times \frac{e}{f}=\frac{c}{d} \times \frac{e}{f} $
$\Rightarrow \frac{a e}{b f}=\frac{c e}{d f}$
Applying componendo and dividendo
$\frac{a e+b f}{a e-b f}=\frac{c e+d f}{c e-d f}$
Hence, proved.

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Question 22 Marks

If $\frac{1}{12}, x$ and $\frac{1}{75}$ are in continued proportion, find $x$.

Answer

Since $\frac{1}{12}, x$ and $\frac{1}{75}$ are in continued proportion.
$\Rightarrow \frac{1}{12}: x :: x : \frac{1}{75}$
$\Rightarrow x ^2=\frac{1}{12} \times \frac{1}{75} $
$\Rightarrow x=\sqrt{\frac{1}{900}} $
$\Rightarrow x =\frac{1}{30} \frac{1}{30}$

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Question 32 Marks

Find the value of the unknown in the following proportion :
$3 : 4 : : p : 12$

Answer

$3: 4:: p: 12 $
$\Rightarrow 3 \times 12=4 \times p $
$ \Rightarrow p=\frac{3 \times 12}{4}$
$\Rightarrow p=9$
$ p=9$

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Question 42 Marks

Find the value of the unknown in the following proportion :
$5 : 12 :: 15 : x$

Answer

$5: 12:: 15: x $
$ \Rightarrow 5 \times X=12 \times 15$
$ \Rightarrow x=\frac{12 \times 15}{5} $
$ \Rightarrow x=36$

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Question 52 Marks

Find the duplicate ratio of the following:
$\frac{2}{3}: \frac{4}{9}$

Answer

$\frac{2}{3}: \frac{4}{9} $
$=\left(\frac{2}{3}\right)^2:\left(\frac{4}{9}\right)^2 $
$=\frac{4}{9}: \frac{16}{81} $
$=\frac{4}{9} \times \frac{81}{14} $
$=\frac{9}{4}$
Duplicate ratio $=9: 4$

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Question 62 Marks

Find the duplicate ratio of the following :
$3 \sqrt{2 a}: 2 \sqrt{3 a}$

Answer

$3 \sqrt{2 a}: 2 \sqrt{3 a} $
$=(3 \sqrt{2 a})^2:(2 \sqrt{3 a})^2 $
$=18 a: 12 a $
$=3: 2$
Duplicate ratio $=3: 2$

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Question 72 Marks

Find the duplicate ratio of the following :
$\sqrt{10}: \sqrt{14}$

Answer

$\sqrt{10}: \sqrt{14} $
$=(\sqrt{10})^2:(\sqrt{14})^2 $
$=10: 14 $
$=5: 7$
Duplicate ratio $=5: 7$

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Question 82 Marks

Find the compounded ratio of the following:
$3 : 5, 7 : 9$ and $15 : 28$

Answer

Compounded ratio of $3:5, 7:9$ and $15: 28$
$=\frac{3}{5} \times \frac{7}{9} \times \frac{15}{28} $
$=\frac{1}{4}$
Compounded ratio $=1: 4$

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Question 92 Marks

If $p : q = 2 : 5 , q : r = 4 : 3,$ then find $p : r$

Answer

$p: q=2: 5 \Rightarrow \frac{p}{q}=\frac{2}{5} $
$ q: r=4: 3 \Rightarrow \frac{q}{r}=\frac{4}{3} $
$ \Rightarrow \frac{p}{q} \times \frac{q}{r}=\frac{2}{5} \times \frac{4}{3}$
$\Rightarrow \frac{p}{q}=\frac{8}{15} $
$ p: r=8: 15$

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Question 102 Marks

If $a + b : a - b = 11 : 8 ;$ find the vaIue of $a : b$

Answer

$a+b: a-b=11: 8 $
$ \Rightarrow \frac{a+b}{a-b}=\frac{11}{8} $
$ \Rightarrow 8 a+8 b-11 a-11 b $
$ \Rightarrow 3 a-19 b $
$\Rightarrow \frac{a}{b}=\frac{19}{3} $
$a: b=19: 3$

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Question 112 Marks

If $m : n = 3 : 8,$ find the value of $(3m+2n) : (5m+n)$

Answer

$m: n=3: 8 \Rightarrow \frac{m}{n}=\frac{3}{8} $
$ (3 m+2 n):(5 m+n)=\frac{3 m+2 n}{5 m+n} $
$ =\frac{3 \times \frac{m}{n}+2}{5 \times \frac{m}{n}+1} $
$=\frac{3 \times \frac{3}{8}+2}{5 \times \frac{3}{8}+1} $
$ =\frac{\frac{9}{8}+2}{\frac{15}{8}+1}$
$ =\frac{25}{23}=25: 23$
$ (3 m+2 n):(5 m+n)=25: 23$

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Question 122 Marks

Divide a number into two parts in the ratio $5:7$ so that smaller part is $60.$ Find the number.

Answer

$\frac{5}{7}=\frac{60}{x}$
$\Rightarrow 5 x=420 $
$\Rightarrow x=84$
Number $=84+60=144$

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Question 132 Marks

A sum of money Is divided in the ratio $2 : 3.$ If the larger portion is $Rs.7,47,300.$ find the sum distributed.

Answer

$\frac{2}{3}=\frac{x}{747300} $
$\Rightarrow 14,94,600=3 x $
$\Rightarrow x=4,98,200$
Sum distributed $=\operatorname{Rs}(7,47,300+4,98,200)=\operatorname{Rs} 12,45.500$

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Question 142 Marks

If $a: b=4: 7$, find the following:
$\frac{5 a-4 b}{2 a-3 b}$

Answer

$a: b=4: 7 \Rightarrow \frac{a}{b}=\frac{4}{7} $
$ \frac{5 a-4 b}{2 a-3 b}=\frac{5 \times \frac{a}{b}-4}{2 \times \frac{a}{b}-3} $
$ =\frac{5 \times \frac{4}{7}-4}{2 \times \frac{4}{7}-3} $
$=\frac{\frac{20}{7}-4}{\frac{8}{7}-3} $
$ =\frac{-8}{-13}=\frac{8}{13}$

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Question 152 Marks

If $a: b=4: 7$, find the following:
$\frac{6 a-b}{a+3 b}$

Answer

$a: b=4: 7 \Rightarrow \frac{a}{b}=\frac{4}{7} $
$\frac{6 a-b}{a+3 b}=\frac{6 \frac{a}{b}-1}{\frac{a}{b}+3}$
$ =\frac{6 \times \frac{4}{7}-1}{\frac{4}{7}+3} $
$=\frac{\frac{24}{7}-1}{\frac{4}{7}+3} $
$ =\frac{17}{25}$

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Question 162 Marks

If $a: b=4: 7$, find the following:
$\frac{5 a+2 b}{5 a-2 b}$

Answer

$a: b=4: 7 \Rightarrow \frac{a}{b}=\frac{4}{7} $
$ \frac{5 a+2 b}{5 a-2 b}=\frac{5 \frac{a}{b}+2}{5 \frac{a}{b}-2}$
$ =\frac{5 \times \frac{4}{7}+2}{5 \times \frac{4}{7}-2} $
$ =\frac{\frac{20}{7}+2}{\frac{20}{7}-2}$
$=\frac{34}{6}$
$=\frac{17}{3}$

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Question 172 Marks

Find the ratio of the following in the simplest fcrm:
$x^2 + 4x +4$ and $x^2 - x - 6$

Answer

$x^2+4 x+4$ and $x^2-x-6$
$\frac{x^2+4 x+4}{x^2-x-6}=\frac{(x+2)^2}{(x-3)(x+2)}=\frac{x+2}{x-3}$
$=(x+2):(x-3)$

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Question 182 Marks

Find the ratio of the following in the simplest fcrm:
$a^4 + b^4$ and $a^3 - b^3$

Answer

$\frac{a^4+b^4}{a^3-b^3} $
$ =\frac{(a-b)\left(a^3+a b^2+a^2 b+b^3\right)}{(a-b)\left(a^2+a b+b^2\right)} $
$ =\frac{a^3+a b^2+a^2 b+b^3}{a^2+a b+b^2} $
$=a^3+a b^2+a^2 b+b^3: a^2+a b+b^2$

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Question 192 Marks

Find the reciprocal ratio of e following:
$\frac{1}{45}: \frac{1}{54}$

Answer

$\frac{1}{45}: \frac{1}{54} $
$=45: 54$
$=5: 6$
Reciprocal ratio $=5: 6$

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Question 202 Marks

Find the sub-triplicate ratio of the following :
$125a^3 : 343b^6$

Answer

$125 a^3: 343 b^6 $
$=\left(125 a^3\right)^{\frac{1}{3}}:\left(343 b^6\right)^{\frac{1}{3}} $
$=5 a: 7 b^2$
Sub-triplicate ratio $=5 a: 7 b^2$

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Question 212 Marks