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Mathematics [2 Mark Question Answer]

Ratio and Proportion · ICSE STD 10 (ICSE - English Medium). 25 questions.

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[2 Mark Question Answer]

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25 questions · timed · auto-graded

Question 12 Marks
If (ma + nb): b :: (mc + nd) : d, prove that a, b, c, d are in proportion.
Answer

$
\begin{aligned}
& ( ma + nb ): b ::( mc + nd ): d \\
& \Rightarrow \frac{ ma + nb }{ b }=\frac{ mc + nd }{ d } \\
& \Rightarrow mad + nbd = mbc + nbd \\
& \Rightarrow mad = mbc \\
& \Rightarrow ad = bc \\
& \Rightarrow \frac{a}{b}=\frac{c}{d}
\end{aligned}
$
Hence a : b :: c : d.
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Question 22 Marks
If $a : b :: c : d$, prove that $\frac{2 a+5 b}{2 a-5 b}=\frac{2 c+5 d}{2 c-5 d}$
Answer

$
\begin{aligned}
& a : b :: c : d \\
& \text { then } \frac{a}{b}=\frac{c}{d} \\
& \Rightarrow \frac{2 a}{5 b}=\frac{2 c}{5 d} \ldots\left(\text { multiply by } \frac{2}{5}\right)
\end{aligned}
$
Applying componendo and dividendo,
$
\frac{2 a+5 b}{2 a-5 b}=\frac{2 c+5 d}{2 c-5 d}
$
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Question 32 Marks
Find the mean proportion of: 8.1 and 2.5
Answer
Let the $x$ be the mean proportion of 8.1 and 2.5
$
\begin{aligned}
& \therefore 8.1: x :: x : 2.5 \\
& \therefore x 2=8.1 x 2.5 \\
& \therefore x =\sqrt{8.1 \times 2.5} \\
& =\sqrt{20.25} \\
& =4.5 .
\end{aligned}
$
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Question 42 Marks
Find the mean proportion of: $\frac{1}{12}$ and $\frac{1}{75}$
Answer
Let $x$ be the mean proportion of $\frac{1}{12}$ and $\frac{1}{75}$ then $\frac{1}{12}: x:: x: \frac{1}{75}$
$\therefore x ^2=\frac{1}{12} \times \frac{1}{75}=\frac{1}{900}$
$\therefore x=\sqrt{\frac{1}{900}}=\frac{1}{30}$
Hence the mean proportion $=\frac{1}{30}$.
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Question 52 Marks
Find the mean proportion of: 5 and 80
Answer

$
\begin{aligned}
& \text { Let } x \text { be the mean proportion of } 5 \text { and } 80, \\
& \text { then } 5: x:: x: 80 \\
& x^2=5 x 80 \\
& \Rightarrow x=\sqrt{5 \times 80} \\
& =\sqrt{400} \\
& =20 \\
& x=20
\end{aligned}
$Hence mean proportion $=20$.
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Question 62 Marks
Find the third proportional to Rs. 3, Rs. 12
Answer
Let $x$ be the third proportional to Rs. 3 and Rs. 12 then Rs. 3 : Rs. 12 : : Rs. 12 : x
$
\begin{aligned}
& \therefore x =\frac{12 \times 12}{3} \\
& =48 \\
& \therefore \text { Third proportional = Rs. } 48 .
\end{aligned}
$
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Question 72 Marks
Find the third proportional to 0.24, 0.6
Answer
Let $x$ be the third proportional to $0.24,0.6$ then $0.24: 0.6:: 0.6: x$
$
\begin{aligned}
& \therefore 0.24 \times x =0.6 \times 0.6 \\
& x =\frac{0.6 \times 0.6}{0.24} \\
& =1.5 . \\
& \therefore \text { Third proportional }=1.5 .
\end{aligned}
$
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Question 82 Marks
Find the third proportional to 5, 10
Answer
Let $x$ be the third proportional to 5,10 , then $5: 10:: 10: x$
$
\begin{aligned}
& \therefore 5 \times x =10 \times 10 \\
& \Rightarrow x =\frac{10 \times 10}{5} \\
& =20 \\
& \therefore \text { Third proportional }=20 .
\end{aligned}
$
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Question 92 Marks
Find the fourth proportional to 9.6 kg, 7.2 kg, 28.8 kg
Answer

$\begin{aligned} & \text { Let fourth proportional to } 9.6 kg , 7.2 kg , 28.8 kg \text { be } x \\ & \text { then } 9.6: 7.2:: 28.8: x \\ & \Rightarrow 9.6 \times x=7.2 \times 28.8 \\ & x=\frac{7.2 \times 28.8}{9.6} \\ & =21.6\end{aligned}$
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Question 102 Marks
Find the fourth proportional to 1.5, 2.5, 4.5
Answer
Let fourth proportional to $1.5,2.5,4.5$ be $x$ then $1.5: 2.5:: 4.5: x$
$
\begin{aligned}
& \therefore 1.5 \times x=2.5 \times 4.5 \\
& x=\frac{2.5 \times 4.5}{1.5} \\
& =7.5
\end{aligned}
$
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Question 112 Marks
Find the fourth proportional to $\frac{1}{3}, \frac{1}{4}, \frac{1}{5}$
Answer
Let fourth proportional to $\frac{1}{3}, \frac{1}{4}, \frac{1}{5}$ be $x$
then $\frac{1}{3}: \frac{1}{4}:: \frac{1}{5}: x$
$\Rightarrow \frac{1}{3} \times x=\frac{1}{4} \times \frac{1}{5}$
$\Rightarrow x=\frac{1}{4} \times \frac{1}{5} \times \frac{3}{1}=\frac{3}{20}$.
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Question 122 Marks
Find the fourth proportional to 3, 12, 15
Answer

$\begin{aligned} & \text { Let fourth proportional to } \\ & 3,12,15 \text { be } x \\ & \text { then } 3: 12:: 15: x \\ & \Rightarrow 3 \times x=12 \times 15 \\ & x=\frac{12 \times 15}{3} \\ & =60\end{aligned}$
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Question 132 Marks
Find the value of x in the following proportions : x : 50 :: 3 : 2
Answer

$\begin{aligned} & x: 50:: 3: 2 \\ & \Rightarrow x \times  2=50 \times 3 \\ & x=\frac{50 \times 3}{2} \\ & =75\end{aligned}$
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Question 142 Marks
Find the value of x in the following proportions : 2.5 : 1.5 = x : 3
Answer

$\begin{aligned} & 2.5: 1.5=x: 3 \\ & \Rightarrow 1.5 \times x=2.5 \times 3 \\ & x=\frac{2.5 \times 3}{1.5} \\ & =5.0 .\end{aligned}$
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Question 152 Marks
Find the value of x in the following proportions : 3 : x = 24 : 2
Answer

$\begin{aligned} & 3: x=24: 2 \\ & \Rightarrow x \times  24=3 \times 2 \\ & \therefore x=\frac{3 \times 2}{24} \\ & =\frac{1}{4} .\end{aligned}$
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Question 162 Marks
Find the value of x in the following proportions : 10 : 35 = x : 42
Answer

$\begin{aligned} & 10: 35= x : 42 \\ & \Rightarrow 35 \times x =10 \times 42 \\ & \therefore x =\frac{10 \times 42}{35} \\ & =2 \times 6 \\ & =12 .\end{aligned}$
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Question 172 Marks
If x : y = 2 : 3, and y : z = 4 : 7, find x : y : z
Answer
LCM of y's terms 3 and $4=12$
Making equals of $y$ as 12
$\frac{x}{y}=\frac{2}{3}=\frac{2 \times 4}{3 \times 4}=\frac{8}{12}$ or $8: 12$
$\frac{y}{z}=\frac{4}{7} \times \frac{3}{3}=\frac{12}{21}$ or $12: 21$
Then $x: y: z=8: 12: 21$.
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Question 182 Marks
Find the sub-triplicate ratio of $\frac{1}{8}: \frac{1}{125}$
Answer
Sub-triplicate ratio of $\frac{1}{8}: \frac{1}{125}$
$
\begin{aligned}
& =\left(\frac{1}{8}\right)^{\frac{1}{3}}:\left(\frac{1}{125}\right)^{\frac{1}{3}} \\
& =\left[\left(\frac{1}{2}\right)^3\right]^{\frac{1}{3}}:\left[\left(\frac{1}{5}\right)^3\right]^{\frac{1}{3}} \\
& =\frac{1}{2}: \frac{1}{5} \\
& =5: 2 .
\end{aligned}
$
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Question 192 Marks
Find the sub-triplicate ratio of 1 : 216
Answer

$\begin{aligned} & \text { Sub-triplicate ratio of } 1: 216 \\ & =\sqrt[3]{1}: \sqrt[3]{216} \\ & =\left(1^3\right) \frac{1}{3}:\left(6^3\right) \frac{1}{3} \\ & =1: 6\end{aligned}$
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Question 202 Marks
Find the sub-duplicate ratio of $\frac{1}{4}: \frac{1}{9}$
Answer
Sub-duplicate ratio of $\frac{1}{4}: \frac{1}{9}$
$
\begin{aligned}
& =\sqrt{\frac{1}{4}}: \sqrt{\frac{1}{9}} \\
& =\frac{1}{2}: \frac{1}{3} \\
& =3: 2 .
\end{aligned}
$
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Question 212 Marks
Find the triplicate ratio of $1^3 : 2^3$
Answer
Triplicate ratio of $1^3 : 2^3$
$= (1^3)^3 : (2^3)^3$
$= (1)^3 : (8)^3$
$= 1 : 512.$
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Question 222 Marks
Find the triplicate ratio of $\frac{1}{2}: \frac{1}{3}$
Answer
Triplicate ratio of $\frac{1}{2}: \frac{1}{3}$
$
\begin{aligned}
& =\left(\frac{1}{2}\right)^3:\left(\frac{1}{3}\right)^3 \\
& =\frac{1}{8}: \frac{1}{27} \\
& =27: 8 .
\end{aligned}
$
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Question 232 Marks
Find the compounded ratio of $: (a – b) : (a + b), (a + b)^2 : (a^2 + b^2)$ and $(a^4 – b^4) : (a^2 – b^2)^2$
Answer
$(a-b):(a+b),(a+b)^2:\left(a^2+b^2\right)$ and $\left(a^4-b^4\right):\left(a^2-b^2\right)^2$
Compound ratio
$=\frac{ a - b }{ a + b } \times \frac{(a+b)^2}{a^2+b^2} \times \frac{a^4-b^4}{\left(a^2-b^2\right)^2}$
$ =\frac{ a - b }{ a + b } \times \frac{(a+b)(a+b)}{a^2+b^2} \times \frac{\left(a^2+b^2\right)(a+b)(a-b)}{(a+b)^2(a-b)^2}$
$ =\frac{1}{1}$
or
$1: 1$.
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Question 242 Marks
Find the compounded ratio of: 4 : 5, 5 : 7 and 9 : 11
Answer
$4: 5,5: 7$ and $9: 11$Compound ratio
$
\begin{aligned}
& =\frac{4}{5} \times \frac{5}{7} \times \frac{9}{11} \\
& =\frac{36}{77}
\end{aligned}
$
or
36: 77.
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Question 252 Marks
Find the compounded ratio of: 2 : 3 and 4 : 9
Answer
$2: 3$ and $4: 9$ Compound ratio
$
\begin{aligned}
& =\frac{2}{3} \times \frac{4}{9} \\
& =\frac{8}{27} \text { or } 8: 27 .
\end{aligned}
$
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