The sum of two number a and b is 15, and the sum of their reciprocals and is . Find the numbers a and b.
Answers
Answered by
8
SOLUTION :
Given : Sum of two number a and b is 15 and Sum of their reciprocals is 3/10 .
a + b = 15
b = 15 - a …………………..(1)
1/a + 1/b = 3/10
(b + a) / ab = 3/10
[By taking LCM]
(a + b) / ab = 3/10
10(a + b) = 3(ab)
[By cross multiplication]
10a + 10b = 3ab …………..(2)
Put the value of b from eq 1 in eq 2
10a + 10(15 - a) = 3a(15 - a)
10a + 150 - 10a = 45a - 3a²
150 = 45a - 3a²
3a² - 45a + 150 = 0
3(a² - 15a + 50) = 0
a² - 15a + 50 = 0
a² - 10a - 5a + 50 = 0
[By middle term splitting]
a(a - 10) - 5(a - 10) = 0
(a - 5) (a - 10) = 0
(a - 5) = 0 or (a - 10) = 0
a = 5 or a = 10
Case 1 :
When a = 5 then b= (15 - a) = 15 - 5 = 10
Case 1 :
When a = 10 then b= (15 - a) = 15 - 10 = 5
Hence , the two numbers are (a = 5 ,b = 10) & (a = 10 , b = 5) .
HOPE THIS ANSWER WILL HELP YOU...
Answered by
3
The sum of two numbers a and b is 15.
( a + b ) = 15
a = 15 - b
Sum of their reciprocals =+ =
=
10 ( b + a ) = 3 ab
10 b + 10a = 3 ab
10b + 10( 15 - b ) = 3 ( 15 - b ) b
10b + 150 - 10b = 45b - 3b ^2
150 = - 3b^2 + 45b
3b^2 - 45b +150 = 0
3 ( b^2 - 15b + 50 ) = 0
b^2 - 15b + 50 = 0
b ^2 - 10b - 5b + 50 = 0
b ( b - 10 ) - 5 ( b - 10 ) = 0
( b - 5 ) ( b - 10 ) = 0
b = 5, 10
Putting value of b in a,
a = 15 - b = 15 - 5 = 10 or
a = 15 - 10 = 5
( a + b ) = 15
a = 15 - b
Sum of their reciprocals =+ =
=
10 ( b + a ) = 3 ab
10 b + 10a = 3 ab
10b + 10( 15 - b ) = 3 ( 15 - b ) b
10b + 150 - 10b = 45b - 3b ^2
150 = - 3b^2 + 45b
3b^2 - 45b +150 = 0
3 ( b^2 - 15b + 50 ) = 0
b^2 - 15b + 50 = 0
b ^2 - 10b - 5b + 50 = 0
b ( b - 10 ) - 5 ( b - 10 ) = 0
( b - 5 ) ( b - 10 ) = 0
b = 5, 10
Putting value of b in a,
a = 15 - b = 15 - 5 = 10 or
a = 15 - 10 = 5
Similar questions
Art,
6 months ago
Computer Science,
6 months ago
English,
11 months ago
Science,
11 months ago
Psychology,
1 year ago