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01. Fourth proportional to (a² - b²), (a² - ab), (a³ + b³) is

1(a - b)
2a⁴ + b⁴
3a(a² - ab + b²)
4a³ - a²b² + b²

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Answer:- 1

Explanation:-

Solution:

\begin{aligned} Let the fourth proportional to \\ (a^{2} - b^{2}), (a^{2} - a b), (a^{3} + b^{3}) be x . \\ Then, \\ = (a^{2} - b^{2}) : (a^{2} - a b) :: (a^{3} + b^{3}) : x \\ \Rightarrow (a^{2} - b^{2}) x = (a^{3} + b^{3}) (a^{2} - a b) \\ \Rightarrow x = \frac{(a^{3} + b^{3}) (a^{2} - a b)}{(a^{2} - b^{2})} \\ = \frac{(a + b) (a^{2} - a b + b^{2}) a (a - b)}{(a - b) (a + b)} \\ \Rightarrow x = a (a^{2} - a b + b) . \end{aligned}

$\eqalign{ & {\text{Let the fourth proportional to}} \cr & \left( {{a^2} - {b^2}} \right),\left( {{a^2} - ab} \right),\left( {{a^3} + {b^3}} \right)\,{\text{be}}\,x. \cr & {\text{Then,}} \cr & = \left( {{a^2} - {b^2}} \right):\left( {{a^2} - ab} \right)::\left( {{a^3} + {b^3}} \right):x \cr & \Rightarrow \left( {{a^2} - {b^2}} \right)x = \left( {{a^3} + {b^3}} \right)\left( {{a^2} - ab} \right) \cr & \Rightarrow x = \frac{{\left( {{a^3} + {b^3}} \right)\left( {{a^2} - ab} \right)}}{{\left( {{a^2} - {b^2}} \right)}} \cr & \,\,\,\,\,\,\, = \frac{{\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)a\left( {a - b} \right)}}{{\left( {a - b} \right)\left( {a + b} \right)}} \cr & \Rightarrow x = a\left( {{a^2} - ab + b} \right). \cr}$

02. The ratio of weekly income of A and B is 9 : 7 and the the ratio of their expenditure is 4 : 3. If each saves Rs. 200 per week, then the sum of their weekly income is = ?

1Rs. 3600
2Rs. 3200
3Rs. 4800
4Rs. 5600

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Answer:- 1

Explanation:-

Solution:

	A	:	B
Income	9	:	7
Expense	4	:	3

Income - Savings + Expenditure

\begin{aligned} ∴ \frac{9 x - 200}{7 x - 200} = \frac{4}{3} \\ \Rightarrow 27 x - 600 = 28 x - 800 \\ \Rightarrow x = 200 \\ Sum of weekly income \\ = 9 x + 7 x = 16 x \\ = 16 \times 200 = Rs . 3200 \end{aligned}

$\eqalign{ & \therefore \frac{{9x - 200}}{{7x - 200}} = \frac{4}{3} \cr & \Rightarrow 27x - 600 = 28x - 800 \cr & \Rightarrow x = 200 \cr & {\text{Sum of weekly income}} \cr & = 9x + 7x = 16x \cr & = 16 \times 200 = {\text{Rs}}.\,3200 \cr}$

03. In a mixture of 25 litres the ratio of acid to water is 4 : 1. Another 3 litres of water of water is added to the mixture. The ratio of acid to water in the new mixture is = ?

15 : 2
22 : 5
33 : 5
45 : 3

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Answer:- 1

Explanation:-

Solution:

\begin{aligned} Acid : Water \\ 4 : 1 \\ Let 4 x + x = 5 x \\ ∵ 4 x + x = 25 \\ \Rightarrow x = 5 \\ ∴ Acid = 5 \times 4 = 20 \\ Water = 5 \times 1 = 5 \\ ∴ 3 Litres water is added \\ So, new quantity of water \\ = 5 + 3 = 8 \\ ∴ Acid : Water \\ 20 : 8 \\ 5 : 2 \end{aligned}

$\eqalign{ & {\text{Acid}}:{\text{Water}} \cr & \,\,\,\,\,4\,\,\,\,\,:\,\,\,\,\,\,\,1 \cr & {\text{Let }}4x + x = 5x \cr & \because 4x + x = 25 \cr & \Rightarrow x = 5 \cr & \therefore {\text{Acid }} = {\text{ }}5 \times 4 = 20 \cr & {\text{Water }} = 5 \times 1 = 5 \cr & \therefore {\text{3 Litres water is added }} \cr & {\text{So, new quantity of water}} \cr & = 5 + 3 = 8 \cr & \therefore {\text{Acid}}:{\text{Water}} \cr & \,\,\,\,\,\,\,\,\,{\text{20}}\,\,\,:\,\,\,\,\,\,\,8 \cr & \,\,\,\,\,\,\,\,\,\,\,5\,\,\,\,:\,\,\,\,\,\,\,2 \cr}$

04. The fourth proportional to 0.12, 0.21 and 8 is-

18.9
214
317
456

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Answer:- 1

Explanation:-

Solution:

Let the fourth proportional to 0.12, 0.21 and 8 be x.

\begin{aligned} Then, \\ = 0 .12 : 0 .21 :: 8 : x \\ \Rightarrow 0 .12 x = 0.21 \times 8 \\ \Rightarrow x = \frac{0.21 \times 8}{0.12} \\ = \frac{21 \times 8}{12} \\ = 14. \end{aligned}

$\eqalign{ & {\text{Then,}} \cr & = {\text{0}}{\text{.12}}:{\text{0}}{\text{.21}}::{\text{8}}:x \cr & \Rightarrow {\text{0}}{\text{.12}}x = 0.21 \times 8 \cr & \Rightarrow x = \frac{{0.21 \times 8}}{{0.12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{21 \times 8}}{{12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 14. \cr}$

05. The fourth proportion to 5, 8, 15 is-

118
220
321
424

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Answer:- 1

Explanation:-

Solution:

Let the fourth proportional to 5, 8, 15 be x.
Then,

\begin{aligned} \Rightarrow 5 : 8 : 15 : x \\ \Rightarrow 5 x = 8 \times 15 \\ \Rightarrow x = \frac{8 \times 15}{5} = 24. \end{aligned}

$\eqalign{ & \Rightarrow 5:8:15:x \cr & \Rightarrow 5x = 8 \times 15 \cr & \Rightarrow x = \frac{{8 \times 15}}{5} = 24. \cr}$

06. Three numbers are in the ratio 3 : 4 : 5. The sum of the largest and the smallest equals to the sum of the second number and 52. The smallest number is = ?

120
227
39
452

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Answer:- 1

Explanation:-

Solution:

	A	:	B	:	C
	3	:	4	:	5
Let	3x	:	4x	:	5x

∴ Sum of (smallest + largest)
A + C = 8x
According of the question,
8x = 4x + 52
4x = 52
∴ Smallest number is = A
⇒ 13 × 3 = 39

07. If x : y = 3 : 4, then the value of (4x - y) : (2x + 3y) is = ?

14 : 9
28 : 9
34 : 3
48 : 3

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Answer:- 1

Explanation:-

Solution:

\begin{aligned} x : y = 3 : 4 \\ ∴ \frac{x}{y} = \frac{3}{4} \\ \Rightarrow \frac{4 x - y}{2 x + 3 y} \\ \Rightarrow \frac{y (4 \frac{x}{y} - 1)}{y (2 \frac{x}{y} + 3)} \\ \Rightarrow \frac{4 \times \frac{3}{4} - 1}{2 \times \frac{3}{4} + 3} \\ \Rightarrow \frac{(3 - 1) \times 2}{3 + 6} \\ \Rightarrow \frac{4}{9} \\ \Rightarrow 4 : 9 \end{aligned}

$\eqalign{ & x:y = 3:4 \cr & \therefore \frac{x}{y} = \frac{3}{4} \cr & \Rightarrow \frac{{4x - y}}{{2x + 3y}} \cr & \Rightarrow \frac{{y\left( {4\frac{x}{y} - 1} \right)}}{{y\left( {2\frac{x}{y} + 3} \right)}} \cr & \Rightarrow \frac{{4 \times \frac{3}{4} - 1}}{{2 \times \frac{3}{4} + 3}} \cr & \Rightarrow \frac{{\left( {3 - 1} \right) \times 2}}{{3 + 6}} \cr & \Rightarrow \frac{4}{9} \cr & \Rightarrow 4:9 \cr}$

$\begin{aligned} If \frac{a + b}{c} = \frac{b + c}{a} = \frac{c + a}{b} = k, \\ then k is equal to - \end{aligned}$ $\eqalign{ & {\text{If }}\frac{{{\text{a}} + {\text{b}}}}{{\text{c}}} = \frac{{{\text{b}} + {\text{c}}}}{{\text{a}}} = \frac{{{\text{c}} + {\text{a}}}}{{\text{b}}} = {\text{k}}, \cr & {\text{then k is equal to - }} \cr}$

10
21
32
4a + b + c

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Answer:- 1

Explanation:-

Solution:

a + b = ck,
b + c = ak,
c + a = bk
Adding, we get : 2(a + b + c) = ak + bk + ck
⇒ k(a + b + c) or k = 2

$\begin{aligned} Let \frac{a}{b} : - \frac{b}{a} = x : y . \\ If (x - y) = {\frac{a}{b} + \frac{b}{a}}, \\ then x is equal to - \end{aligned}$ $\eqalign{ & {\text{Let }}\frac{{\text{a}}}{{\text{b}}}: - \frac{{\text{b}}}{{\text{a}}} = {\text{x}}:{\text{y}}{\text{.}} \cr & {\text{If}}\left( {{\text{x - y}}} \right) = \left\{ {\frac{{\text{a}}}{{\text{b}}}{\text{ + }}\frac{{\text{b}}}{{\text{a}}}} \right\}{\text{,}} \cr & {\text{then x is equal to - }} \cr}$

1^{(a - b)}/_a
2^{(a + b)}/_a
3^{(a + b)}/_b
4None of these

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Answer:- 1

Explanation:-

Solution:

\begin{aligned} \Rightarrow \frac{x}{y} = \frac{(\frac{a}{b})}{(- \frac{b}{a})} = - \frac{a^{2}}{b^{2}} \\ \Rightarrow y = (- \frac{b^{2}}{a^{2}}) x . \\ ∴ x - y = \frac{a}{b} + \frac{b}{a} \\ \Rightarrow x + \frac{b^{2}}{a^{2}} x = \frac{a^{2} + b^{2}}{a b} \\ \Rightarrow x (\frac{a^{2} + b^{2}}{a^{2}}) = \frac{a^{2} + b^{2}}{a b} \\ \Rightarrow x = \frac{a^{2}}{a b} = \frac{a}{b} . \end{aligned}

$\eqalign{ & \Rightarrow \frac{x}{y} = \frac{{\left( {\frac{a}{b}} \right)}}{{\left( { - \frac{b}{a}} \right)}} = - \frac{{{a^2}}}{{{b^2}}} \cr & \Rightarrow y = \left( { - \frac{{{b^2}}}{{{a^2}}}} \right)x. \cr & \therefore x - y = \frac{{\text{a}}}{{\text{b}}}{\text{ + }}\frac{{\text{b}}}{{\text{a}}} \cr & \Rightarrow x + \frac{{{b^2}}}{{{a^2}}}x = \frac{{{a^2} + {b^2}}}{{ab}} \cr & \Rightarrow x\left( {\frac{{{a^2} + {b^2}}}{{{a^2}}}} \right) = \frac{{{a^2} + {b^2}}}{{ab}} \cr & \Rightarrow x = \frac{{{a^2}}}{{ab}} = \frac{a}{b}. \cr}$

10. The ratio of the numbers of boys and girls in a school was 5 : 3. Some new boys and girls were admitted to the school, in the ratio 5 : 7. At this, the total number of students in the school became 1200 and the ratio of boys to girls changed to 7 : 5. The number of students in the school before new admissions was = ?

1700
2720
3900
4960

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Answer:- 1

Explanation:-

Solution:

\begin{aligned} A : B \\ 5 : 3 \\ Let 5 x : 3 x = 8 x \\ \Rightarrow New comers \\ 5 y : 7 y = 12 y \\ ∴ 8 x + 12 = 1200 \\ \Rightarrow 2 x + 3 y = 300...... (i) \\ Again, \frac{5 x + 5 y}{3 x + 7 y} = \frac{7}{5} \\ 25 x + 25 y = 21 x + 49 y \\ \Rightarrow 4 x - 24 y = 0 \\ \Rightarrow 4 x = 24 y \\ \Rightarrow x = 6 y . . . . . . . . . . . (i i) \\ ∴ From equation . . . . . . . (i) \\ 12 y + 3 y = 300 \\ y = 20 \\ ∴ x = 6 \times 20 = 120 \end{aligned}

$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,{\text{A}}:{\text{B}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,5:3 \cr & {\text{Let }}5x:3x = 8x \cr & \Rightarrow {\text{New comers}} \cr & 5y:7y = 12y \cr & \therefore 8x + 12 = 1200 \cr & \Rightarrow 2x + 3y = 300......(i) \cr & {\text{Again,}}\frac{{5x + 5y}}{{3x + 7y}} = \frac{7}{5} \cr & 25x + 25y = 21x + 49y \cr & \Rightarrow 4x - 24y = 0 \cr & \Rightarrow 4x = 24y \cr & \Rightarrow x = 6y...........(ii) \cr & \therefore {\text{From equation }}.......(i) \cr & 12y + 3y = 300 \cr & y = 20 \cr & \therefore x = 6 \times 20 = 120 \cr}$ The number of students initially
8x = 8 × 120 = 960

Hello Guest

Ratio and Proportion - 05

01. Fourth proportional to (a² - b²), (a² - ab), (a³ + b³) is

02. The ratio of weekly income of A and B is 9 : 7 and the the ratio of their expenditure is 4 : 3. If each saves Rs. 200 per week, then the sum of their weekly income is = ?

03. In a mixture of 25 litres the ratio of acid to water is 4 : 1. Another 3 litres of water of water is added to the mixture. The ratio of acid to water in the new mixture is = ?

04. The fourth proportional to 0.12, 0.21 and 8 is-

05. The fourth proportion to 5, 8, 15 is-

06. Three numbers are in the ratio 3 : 4 : 5. The sum of the largest and the smallest equals to the sum of the second number and 52. The smallest number is = ?

07. If x : y = 3 : 4, then the value of (4x - y) : (2x + 3y) is = ?

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NCERT
6 to 12

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Hello Guest

Ratio and Proportion - 05

01. Fourth proportional to (a2 - b2), (a2 - ab), (a3 + b3) is

02. The ratio of weekly income of A and B is 9 : 7 and the the ratio of their expenditure is 4 : 3. If each saves Rs. 200 per week, then the sum of their weekly income is = ?

03. In a mixture of 25 litres the ratio of acid to water is 4 : 1. Another 3 litres of water of water is added to the mixture. The ratio of acid to water in the new mixture is = ?

04. The fourth proportional to 0.12, 0.21 and 8 is-

05. The fourth proportion to 5, 8, 15 is-

06. Three numbers are in the ratio 3 : 4 : 5. The sum of the largest and the smallest equals to the sum of the second number and 52. The smallest number is = ?

07. If x : y = 3 : 4, then the value of (4x - y) : (2x + 3y) is = ?

Subject Wise

Exam Wise

NCERT 6 to 12

Paid Course

Practice Batch

01. Fourth proportional to (a² - b²), (a² - ab), (a³ + b³) is

NCERT
6 to 12