Question

$\rm If\: \: \dfrac{3 {a}^{2} +2 {b}^{2} }{2 {b}^{2} - {a}^{2} } = \dfrac{15}{7} \: \: and \: \: {a}^{2} : {b}^{2} = m : n. \: Then \: m - n = ..$​

Answer 1

$\sf :\implies \dfrac{3 {a}^{2} +2 {b}^{2} }{2 {b}^{2} - {a}^{2} } = \dfrac{15}{7}$

By cross multiplication:-

$\\ \implies \sf 7(3 {a}^{2} +2 {b}^{2}) = 15(2 {b}^{2} - {a}^{2} )$

$:\implies \sf 21 {a}^{2} +14 {b}^{2} = 30 {b}^{2} - 15{a}^{2}$

$:\implies \sf 21 {a}^{2} +15{a}^{2} = 30 {b}^{2} - 14{b}^{2}$

$:\implies \sf 36{a}^{2} = 16 {b}^{2}$

$:\implies \sf \dfrac{ {a}^{2} }{ {b}^{2} } = \dfrac{16}{36}$

$:\implies \sf \dfrac{ {a}^{2} }{ {b}^{2} } = \dfrac{4}{9}$

$\underline{ \boxed{ \red{\sf :\implies { {a}^{2} } : { {b}^{2} } = {4} : {9}}}}$

$\sf \green { Given , \: { {a}^{2} } : { {b}^{2} } = m :n}$

$\sf \green { Comparing \: { {a}^{2} } : { {b}^{2} } = m :n \: (with) \: { {a}^{2} } : { {b}^{2} } = {4} : {9}}$

$\sf \purple{ : \implies m = 4 \: \: and \: \: n = 9}$

Find the Value of m - n

$\sf : \implies m - n \: = \: 4 - 9 \: = \: - 5$

$\underline{ \boxed{ \purple{\sf \therefore m - n \: = \: - 5}}}$

Answer 2

-5

Step-by-step explanation:

(3a² +2b²)/(2b² − a²) = 15/7

7(3a² +2b²) = 15(2b² − a²)

21a² + 14b² = 30b² − 15a²

21a² +15a² = 30b² − 14b²

36a² = 16b²

a²/b² = 16/36

a²/b² = 4/9

a² : b² = 4 : 9

GIVEN

• a² : b² = m : n
• m : n = 4 : 9
• m = 4
• n = 9

NOW

m - n = 4 - 9 = -5