Question

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.​

Answer 1

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