Ratio and Proportion - 05

Explanation:-

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

Explanation:-

	A	:	B
Income	9	:	7
Expense	4	:	3

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

Explanation:-

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

Explanation:-

Let the fourth proportional to 0.12, 0.21 and 8 be x. $$\eqalign{ & {\text{Then,}} \cr & = {\text{0}}{\text{.12}}:{\text{0}}{\text{.21}}::{\text{8}}:x \cr & \Rightarrow {\text{0}}{\text{.12}}x = 0.21 \times 8 \cr & \Rightarrow x = \frac{{0.21 \times 8}}{{0.12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{21 \times 8}}{{12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 14. \cr}$$

Explanation:-

Let the fourth proportional to 5, 8, 15 be x.
Then, $$\eqalign{ & \Rightarrow 5:8:15:x \cr & \Rightarrow 5x = 8 \times 15 \cr & \Rightarrow x = \frac{{8 \times 15}}{5} = 24. \cr}$$

Explanation:-

	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

Explanation:-

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

Explanation:-

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

Explanation:-

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

Explanation:-

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