Question 7

Find two consecutive natural numbers whose product is 30.

Solution:

Let x , (x + 1) are the two consecutive integers

whose product is 30

x x (x + 1) = 30

x² + 1 x = 30

x² + x - 30 = 0

x² + 6 x - 5 x - 30 = 0

x (x + 6) - 5 (x + 6) = 0

(x + 6) (x - 5) = 0

x + 6 = 0                 x - 5 = 0

x = -6                   x = 5

Since those are positive integer we should not take x = -6. So let us take the value 5 for x.

So the next consecutive integer is 6.

Therefore three consecutive integers are 5 and 6.

Verification:

Product of two positive integer is 30.

5 (6) = 30

30 =  30

Question 7

The sum of two numbers is 60 and their product is 576. Find the numbers.

Solution:

Let x and y are the two consecutive integers

The sum of two numbers is 60

x + y = 60  ---- (1)

their product is 576

x y = 576

y = 576/x  ----- (2)

Now we are going to apply this value in the first equation.

x + (576/x) = 60

(x² + 576)/x = 60

x² + 576 = 60 x

x² - 60 x + 576 = 0

x² - 48 x - 12 x + 576 = 0

x (x - 48) - 12 (x - 48) = 0

(x - 12) (x - 48) = 0

x - 12 = 0               x - 48 = 0

x = 12                       x = 48

Now we have to apply those values in the second equation inorder to get the value of y.

if x = 12                        if x = 48

then y = 576/12                        then y = 576/48

y = 48                                   y = 12

So the required integers are 18 and 48.

Verification:

The sum of two numbers is 60

48 + 18 = 60

60 = 60

their product is 576

48 (18) = 576

576 = 576