Find three numbers, the second of which is as much greater than the first as the third is greater than the second, if the product of the two smaller numbers is 85 and the product of the two larger numbers is 115. About the author Kennedy
Given : Second number – First number = Third number- second number Product of two smaller number = 85 Product of two larger number = 115 To find : All three numbers Solution : Let – First number (x) = a – d Second number (y) = a Third number (z) = a + d According to question, Second number – First number = Third number- second number ➝ y – x = z – y ➩ a – ( a-d) = a+d – a ➩ a – a + d = d ➩ d = d [ Note – The above calculation is done to show that how I had taken three numbers as above ] ________________________________ According to question, Product of two smaller number = 85 ➝ xy = 85 Put value of x and y ➩ (a-d)a = 85 ➩ a² – ad = 85 equation 1 ________________________________ According to question, Product of two larger number = 115 ➝ yz = 115 Put value of y & z ➩ a(a+d) = 115 ➩ a² + ad = 115 equation 2 ________________________________ Add equation 1 and 2 [ LHS of equation 1 will be added to LHS of equation 2 and similarly RHS of equation 1 will be added to RHS of equation 2 ] ➝ (a² – ad) + (a² + ad) = (85 + 115) ➩ a² – ad + a² + ad = 200 ➩ 2a² = 200 ➩ a² = 200/2 ➩ a² = 100 ➩ a = √100 ➩ a = ±10 ________________________________ When a = +10 Put value of a in equation 1 ➝ (10)² – (10)d = 85 ➩ 100 – 10d = 85 ➩ 100 – 85 = 10d ➩ 15 = 10d ➩ d = 15/10 ➩ d = 1.5 Therefore, ➝ x = a – d ➩ x = 10 – 1.5 ➩ x = 8.5 ➝ y = a ➩ y = 10 ➝ z = a + d ➩ z = 10 + 1.5 ➩ z = 11.5 This gives the required three numbers are [ 8.5 , 10 , 11.5 ] ________________________________ When a = -10 Put value of a in equation 1 ➝ (-10)² – (-10)d = 85 ➩ 100 + 10d = 85 ➩ 100 – 85 = -10d ➩ 15 = -10d ➩ d = -15/10 ➩ d = -1.5 Therefore, ➝ x = a – d ➩ x = (-10) – (-1.5) ➩ x = -10 + 1.5 ➩ x = -8.5 ➝ y = a ➩ y = -10 ➝ z = a + d ➩ z = (-10) + (-1.5) ➩ z = -10 – 1.5 ➩ z = –11.5 This gives the required three numbers are [ -11.5 , -10 , -8.5 ] Note – Product of two smaller number = (-11.5) × (-10) = 115 Product of two larger number = (-10)×(-8.5) = 85 Therefore, these pair does not fulfill the given conditions ________________________________ ANSWER : The required three numbers are x = 8.5 y = 10 z = 11.5 Reply
Given :
To find :
All three numbers
Solution :
Let –
According to question, Second number – First number = Third number- second number
➝ y – x = z – y
➩ a – ( a-d) = a+d – a
➩ a – a + d = d
➩ d = d
[ Note – The above calculation is done to show that how I had taken three numbers as above ]
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According to question, Product of two smaller number = 85
➝ xy = 85
Put value of x and y
➩ (a-d)a = 85
➩ a² – ad = 85 equation 1
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According to question, Product of two larger number = 115
➝ yz = 115
Put value of y & z
➩ a(a+d) = 115
➩ a² + ad = 115 equation 2
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Add equation 1 and 2 [ LHS of equation 1 will be added to LHS of equation 2 and similarly RHS of equation 1 will be added to RHS of equation 2 ]
➝ (a² – ad) + (a² + ad) = (85 + 115)
➩ a² – ad + a² + ad = 200
➩ 2a² = 200
➩ a² = 200/2
➩ a² = 100
➩ a = √100
➩ a = ±10
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When a = +10
Put value of a in equation 1
➝ (10)² – (10)d = 85
➩ 100 – 10d = 85
➩ 100 – 85 = 10d
➩ 15 = 10d
➩ d = 15/10
➩ d = 1.5
Therefore,
➝ x = a – d
➩ x = 10 – 1.5
➩ x = 8.5
➝ y = a
➩ y = 10
➝ z = a + d
➩ z = 10 + 1.5
➩ z = 11.5
This gives the required three numbers are [ 8.5 , 10 , 11.5 ]
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When a = -10
Put value of a in equation 1
➝ (-10)² – (-10)d = 85
➩ 100 + 10d = 85
➩ 100 – 85 = -10d
➩ 15 = -10d
➩ d = -15/10
➩ d = -1.5
Therefore,
➝ x = a – d
➩ x = (-10) – (-1.5)
➩ x = -10 + 1.5
➩ x = -8.5
➝ y = a
➩ y = -10
➝ z = a + d
➩ z = (-10) + (-1.5)
➩ z = -10 – 1.5
➩ z = –11.5
This gives the required three numbers are [ -11.5 , -10 , -8.5 ]
Note –
Therefore, these pair does not fulfill the given conditions
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ANSWER :
The required three numbers are