1 Mark Question

Question 11 Mark

$a , b , c$ are in continued proportion. If $a =3$ and $c =27$, then find b .

Answer

$a , b , c$ are in continued proportion. ...[Given]
$\therefore b ^2= ac$
$\therefore b^2=3 \times 27 \ldots[\because a=3$ and $c =27]$
$\therefore b ^2=81$
$\therefore b =9$...[Taking square root of both sides]

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Question 21 Mark

Check whether the following numbers are in continued proportion : 3, 5, 8

Answer

If $a , b , c$ are in continued proportion then $b ^2= ac$.
$3,5,8$
Let, $a =3, b=5$ and $c =8$
Here, $b^2=5^2=25$
$a c=3 \times 8=24$
$\therefore b^2 \neq a c$
$\therefore 3,5,8$ are not in continued proportion.

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Question 31 Mark

Check whether the following numbers are in continued proportion: $9,12,16$

Answer

If $a , b , c$ are in continued proportion then $b ^2= ac$.
9, 12, 16
Let, $a =9, b=12$ and $c =16$
Here, $b^2=12^2=144$
$ac =9 \times 16=144$
$\therefore b ^2= ac$
$\therefore 9,12,16$ are in continued proportion.

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Question 41 Mark

Check whether the following numbers are in continued proportion: $1,2,3$

Answer

If $a, b, c$ are in continued proportion then $b^2=a c$.
$1,2,3$
Let, $a =1, b=2$ and $c =3$
Here, $b^2=2^2=4$
$ac =1 \times 3=3$
$\therefore b ^2 \neq ac$
$\therefore 1,2,3$ are not in continued proportion.

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Question 51 Mark

Check whether the following numbers are in continued proportion: $2,4,8$

Answer

If $a , b , c$ are in continued proportion then $b ^2= ac$.
i. $2,4,8$
Let, $a =2, b=4$ and $c =8$
Here, $b^2=4^2=16$
$ac =2 \times 8=16$
$\therefore b ^2= ac$
$\therefore 2,4,8$ are in continued proportion.

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Question 61 Mark

For the following numbers write the ratio of first number to second number in the reduced form : 114,133

Answer

$114,133 \\ \text { Ratio }=\frac{114}{113}=\frac{19 \times 6}{19 \times 7}=\frac{6}{7}=6: 7$

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Question 71 Mark

For the following numbers write the ratio of first number to second number in the reduced form : 138,161

Answer

$138,161 \\ \text { Ratio }=\frac{138}{161}=\frac{23 \times 6}{23 \times 7}=\frac{6}{7}=6: 7$

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Question 81 Mark

For the following numbers write the ratio of first number to second number in the reduced form : 65,117

Answer

$65,117 \\ \text { Ratio }=\frac{65}{117}=\frac{13 \times 5}{13 \times 9}=\frac{5}{9}=5: 9$

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Question 91 Mark

For the following numbers write the ratio of first number to second number in the reduced form : 36,90

Answer

$36,90 \\ \text { Ratio }=\frac{36}{90}=\frac{18 \times 2}{18 \times 5}=\frac{2}{5}=2: 5$

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Question 101 Mark

For the following numbers write the ratio of first number to second number in the reduced form : 21,48

Answer

$21,48 \\ \text { Ratio }=\frac{21}{48}=\frac{3 \times 7}{3 \times 16}=\frac{7}{16}=7: 16$

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Question 111 Mark

Five numbers are in continued proportion. The first term is 5 and the last term is 80 . Find these numbers.

Answer

Let the numbers in continued proportion be $a, a k, a k^2, a k^3, a k^4$.
Here $a=5$ and $a k^4=80$
$
\begin{array}{c}
\therefore 5 \times k^4=80 \\
\therefore k^4=16 \\
\therefore k=2 \quad \because 2^4=16 \\
a k=5 \times 2=10 \quad a k^2=5 \times 4=20 \\
a k^3=5 \times 8=40 \quad a k^4=5 \times 16=80
\end{array}
$
$\therefore$ the numbers are $5,10,20,40,80$.

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Question 121 Mark

If $4 a^2 b, 8 a b^2, p$ are in continued proportion then find the value of $p$.

Answer

From given information, $4 a^2 b, 8 a b^2, p$ are in continued proportion
$ \therefore \quad \frac{4 a^2 b}{8 a b^2}=\frac{8 a b^2}{p}$
$p=\frac{8 a b^2 \times 8 a b^2}{4 a^2 b}=16 b^3$

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Question 131 Mark

If $x$ is the geometric mean of 25 and 4 , then find the value of $x$.

Answer

$x$ is the geometric mean of 25 and 4
$
\begin{aligned}
\therefore x^2 & =25 \times 4 \\
\therefore x^2 & =100 \\
\therefore x & =10
\end{aligned}
$

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Question 141 Mark

Fill in the blanks in the following statements : $\frac{x}{3}=\frac{y}{5}=\frac{z}{4}=\frac{5 x-3 y+4 z}{\ldots \ldots \ldots \ldots \ldots}$

Answer

$
\begin{aligned}
\frac{x}{3}=\frac{y}{5}=\frac{z}{4} & =\frac{5 \times x}{5 \times 3}=\frac{-3 \times y}{-3 \times 5}=\frac{4 \times z}{4 \times 4} \\
\therefore \quad & =\frac{5 x}{15}=\frac{-3 y}{-15}=\frac{4 z}{16} \\
& =\frac{5 x-3 y+4 z}{15-15+16} \quad---(\text { by the theorem of equal ratio) } \\
& =\frac{5 x-3 y+4 z}{16}
\end{aligned}
$

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Question 151 Mark

Fill in the blanks in the following statements : $\frac{a}{3}=\frac{b}{7}=\frac{4 a+9 b}{\ldots \ldots \ldots . .}$

Answer

$\frac{a}{3}=\frac{\dddot{b}}{7}=\frac{4 a+9 b}{4 \times 3+9 \times 7}=\frac{4 a+9 b}{12+63}=\frac{4 a+9 b}{75}$

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Question 161 Mark

If $\frac{a}{b}=\frac{7}{3}$ then find the value of the ratio $\frac{a+2 b}{a-2 b}$.

Answer

Method I :
Let $a=7 m, b=3 m$
$ \therefore \frac{a+2 b}{a-2 b} =\frac{7 m+2 \times 3 m}{7 m-2 \times 3 m}$
$ =\frac{7 m+6 m}{7 m-6 m}$
$ =\frac{13 m}{m}=\frac{13}{1} $
Method II :
$\therefore \frac{a}{b}=\frac{7}{3}$
$ \therefore \quad \frac{a}{2 b}=\frac{7}{6} \quad \ldots\left(\text { multiplying both sides by } \frac{1}{2}\right. \text { ) }$
$\therefore \frac{a+2 b}{a-2 b}=\frac{7+6}{7-6} \text { (using componendo dividendo )}$
$\therefore \frac{a+2 b}{a-2 b}=\frac{13}{1} $

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Question 171 Mark

Compare The Folowing (=,<,>): $\frac{\sqrt{13}}{\sqrt{8}}, \frac{\sqrt{7}}{\sqrt{5}}$

Answer

$\frac{\sqrt{13}}{\sqrt{8}}, \frac{\sqrt{7}}{\sqrt{5}}$
$\sqrt{13} \times \sqrt{5} \quad ? \quad \sqrt{8} \times \sqrt{7}$
$\sqrt{65} \quad ?\quad \sqrt{56}$
$\sqrt{65}>\sqrt{56}$
$\therefore \quad \frac{\sqrt{13}}{\sqrt{8}}>\frac{\sqrt{7}}{\sqrt{5}}$

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Question 181 Mark

Compare The Folowing (=,<,>): $\frac{4}{9}, \frac{7}{8}$

Answer

$\frac{4}{9}, \frac{7}{8}$
$4 \times 8\quad ?\quad 7 \times 9$
$
\begin{aligned}
& 32<63 \\
\therefore \quad & \frac{4}{9}<\frac{7}{8}
\end{aligned}
$

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Question 191 Mark

Convert the following ratios into percentages : $\frac{7}{16}$

Answer

$\text { Let } \frac{7}{16}=x \%$
$\therefore \quad \frac{7}{16}=\frac{x}{100}$
$\therefore \quad x=\frac{7}{16} \times 100=\frac{7}{4} \times 25=\frac{175}{4}=43.75 \%$
$\therefore \quad \frac{7}{16}=43.75 \%$

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Question 201 Mark

Convert the following ratios into percentages : $\frac{546}{600}$

Answer

Let $\frac{546}{600}=x \%$
$\begin{array}{ll}
\therefore & \frac{546}{600}=\frac{x}{100} \\
\therefore & x=\frac{546}{600} \times 100=\frac{546}{6}=91 \% \\
\therefore & \frac{546}{600}=91 \%
\end{array}$

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Question 211 Mark

Convert the following ratios into percentages : $\frac{7}{10}$

Answer

$\text {Let } \frac{7}{10}= x \%$
$\therefore \quad \frac{7}{10}=\frac{x}{100}$
$\therefore \quad x=\frac{7}{10} \times 100=7 \times 10=70 \%$
$\therefore \quad \frac{7}{10}=70 \%$

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Question 221 Mark

Convert the following ratios into percentages $: 47:50$

Answer

Let $47 : 50 = x\%$
$\therefore \quad \frac{47}{50}=\frac{x}{100}$
$\therefore \quad x=\frac{47}{50} \times 100=47 \times 2=94 \%$
$\therefore 47: 50=94 \%$

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Question 231 Mark

Convert the following ratios into percentages $: 15:25$

Answer

Let $15 : 25 = x \%$
$\therefore \quad \frac{15}{25}=\frac{x}{100}$
$\therefore \quad x=\frac{15}{25} \times 100=15 \times 4=60 \%$
$\therefore 15: 25=60 \%$

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Question 241 Mark

Express the following percentages as ratios : 0.64%

Answer

$\begin{aligned} \text { Ratio } & =0.64 \%=\frac{0.64}{100}=\frac{64}{100 \times 100} \\ & =\frac{4 \times 4 \times 4}{25 \times 4 \times 25 \times 4}=\frac{4}{25 \times 25} \\ & =\frac{4}{625}=4: 625\end{aligned}$

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Question 251 Mark

Express the following percentages as ratios : 52:100

Answer

$\begin{aligned} \text { Ratio } & =52: 100=\frac{52}{100}=\frac{4 \times 13}{4 \times 25} \\ & =\frac{13}{25}=13: 25\end{aligned}$

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Question 261 Mark

Express the following percentages as ratios : 6.25%

Answer

Ratio $\begin{aligned} & =6.25 \%=\frac{6.25}{100}=\frac{625}{100 \times 100} \\ & =\frac{25 \times 25}{25 \times 4 \times 25 \times 4}=\frac{1}{16}=1: 16\end{aligned}$

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Question 271 Mark

Express the following percentages as ratios $: 44:100$

Answer

$\text {Ratio }=44: 100=\frac{44}{100}=\frac{4 \times 11}{4 \times 25}$
$=\frac{11}{25}=11: 25$

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Question 281 Mark

Express the following percentages as ratios $: 75:100$

Answer

$\text {Ratio }=75: 100=\frac{75}{100}=\frac{25 \times 3}{25 \times 4}$
$=\frac{3}{4}=3: 4$

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