Applying componendo and dividendo: \[\begin{array}{*{35}{l}} 8a/18b\text{ }=\text{ }8c/18d  \\ a/b\text{ }=\text{ }c/d  \\ \end{array}\]

fig 1: SOLUTION: \[\begin{array}{*{35}{l}} x\text{ }=\text{ }2ab/\text{ }\left( a\text{ }+\text{ }b \right)  \\ x/a\text{ }=\text{ }2b/\left( a\text{ }+\text{ }b \right)  \\ \end{array}\] Applying...

\[\begin{array}{*{35}{l}} x\left( {{y}^{2}}~+\text{ }{{z}^{2}}~+\text{ }2yz \right)\text{ }=\text{ }y\left( {{x}^{2}}~+\text{ }{{z}^{2}}~+\text{ }2xz \right)  \\ x{{y}^{2}}~+\text{...

Let  the required numbers be a and b. Given, 14 is the mean proportional between a and b. \[\begin{array}{*{35}{l}} a:\text{ }14\text{ }=\text{ }14:\text{ }b  \\ ab\text{ }=\text{ }196  \\ a\text{...

(iii) Let the mean proportional to \[\left( x\text{ }-\text{ }y \right)\text{ }and\text{ }\left( {{x}^{3}}~\text{ }-{{x}^{2}}y \right)\] be n. \[\left( x\text{ }-\text{ }y \right)\text{ }and\text{...

(i) Let the fourth proportional to 2xy, x2 and y2 be n. \[\begin{array}{*{35}{l}} 2xy:\text{ }{{x}^{2}}~=\text{ }{{y}^{2}}:\text{ }n  \\ 2xy~\times ~n\text{ }=\text{ }{{x}^{2}}~\times ~{{y}^{2}}  \\...

15(2x2 – y2) = 7xy Let the substitution as \[\begin{array}{*{35}{l}} x/y\text{ }=\text{ }a  \\ 2a\text{ }-\text{ }1/a\text{ }=\text{ }7/15  \\ \left( 2{{a}^{2}}~-\text{ }1 \right)/\text{ }a\text{...

Let the woman’s reduced weight as x. the original weight = 84 kg So, we have \[\begin{array}{*{35}{l}} 84:\text{ }x\text{ }=\text{ }7:\text{ }5  \\ 84/x\text{ }=\text{ }7/5  \\ 84\text{ * }5\text{...

Let the required quantity which has to be added be p. \[\begin{array}{*{35}{l}} dx\text{ }+\text{ }pd\text{ }=\text{ }cy\text{ }+\text{ }cp  \\ pd\text{ }-\text{ }cp\text{ }=\text{ }cy-\text{...

(iii) (3x – 7): (4x + 3) is the sub-triplicate ratio of 8: 27 And the sub-triplicate ratio of \[\begin{array}{*{35}{l}} 8:\text{ }27\text{ }=\text{ }2:\text{ }3  \\ \left( 3x\text{ }-\text{ }7...

(i) (2x + 3): (5x – 38) is the duplicate ratio of √5: √6 And, the duplicate ratio of √5: √6 = 5: 6, \[\begin{array}{*{35}{l}} \left( 2x\text{ }+\text{ }3 \right)/\text{ }\left( 5x\text{ }-\text{ }38...

(v) Reciprocal ratio of \[3:\text{ }5\text{ }=\text{ }5:\text{ }3\] (vi) Duplicate ratio of \[5:\text{ }6\text{ }=\text{ }25:\text{ }36\] Reciprocal ratio of \[25:\text{ }42\text{ }=\text{...

(iv) Sub-triplicate ratio of \[216:\text{ }343\text{ }=~{{\left( 216 \right)}^{1/3}}:\text{ }{{\left( 343 \right)}^{1/3}}~=\text{ }6:\text{ }7\] (v) Reciprocal ratio of \[3:\text{ }5\text{ }=\text{...

(i) Duplicate ratio of \[2\surd 2:\text{ }3\surd 5\text{ }=\text{ }{{\left( 2\surd 2 \right)}^{2}}:\text{ }{{\left( 3\surd 5 \right)}^{2}}~=\text{ }8:\text{ }45\] (ii) Triplicate ratio of...

Since, \[\left( 3x\text{ }-\text{ }4y \right):\text{ }\left( 2x\text{ }-\text{ }3y \right)\text{ }=\text{ }\left( 5x\text{ }-\text{ }6y \right):\text{ }\left( 4x\text{ }-\text{ }5y \right)\] This...

Given, On cross multiplying, we get \[\begin{array}{*{35}{l}} 45x\text{ }+\text{ }54y\text{ }=\text{ }64x\text{ }+\text{ }40y  \\ 14y\text{ }=\text{ }19x  \\ {}  \\ x/y\text{ }=\text{ }14/19  \\...

\[\begin{array}{*{35}{l}} a/b\text{ }=\text{ }3/5  \\ \left( 10a\text{ }+\text{ }3b \right)/\text{ }\left( 5a\text{ }+\text{ }2b \right)  \\ ~  \\ \end{array}\]