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Question

If the product of two consecutive odd integers is decreased by 1/3 of the smaller integer is 250, what are the two integers?

Show work and explain please

Answer 1

OK.

Let's say N is the smaller of the two consecutive odd integers. Then the other one is N+2.

Their product is N·(N+2).

Their product decreased by 1/3 of the smaller integer is N·(N+2) - 1/3·N.

So we have
N·(N+2) - 1/3·N = 250
N² + 2N - 1/3·N = 250
N² + 5/3·N - 250 = 0 {multiply by 3 to simplify}
3·N² + 5·N - 750 = 0

Use the quadratic formula
N = [-5 ± sqrt(5² - 4·3·(-750))] / 2·3
N = [-5 ± 95] / 6
N = [-5 - 95] / 6 = -16.67 (not an integer) or
N = [-5 + 95] / 6 = 15

Therefore the two odd integers in question are 15 and 17.

Hope this helps.

Answer 2

let the 2 integers = x + 1, x -1 (x = an even number)

product = (x + 1)(x - 1) = x^2 + x - x -1 = x^2 - 1

(x^2 - 1) - 1/3(x -1) = 250
x^2 -1/3x -2/3 = 250
3x^2 - x - 2 = 750
3x^2 - x -752 = 0
(3x + 47)(x -16) = 0

x = 16 or -47/3, want an integer so x = 16

the 2 numbers are 15 and 17

Answer 3

Breathe in - -
ab - a/3 = 250
b = a+2
a(a+2) - a/3 = 250
a^2 +2a -a/3 = 250
3a^2 +6a -a = 750
3a^2 +5a - 750 =0
a={ -5 +/- Sq rt (25- 4(3)(-750))}/6 quadratic formula
= {-5 - 95}/6 or {-5 + 95}/6

-100/6 is not an integer
90/6 =15 = a is an odd integer
a+2= 17= b is next highest odd integer
check by substitution
ab-a/3=250
15(17) -15/5 = 250
255 - 5 =250
Breathe out

Answer 4

x and x+2 are the integers

x(x+2) - 1/3 x = 250

x^2 + 2x - 1/3 x = 250

Mult by 3

3x^2 + 6x - x = 750

3x^2 + 5x - 750 = 0

(3x + 50)(x -15) = 0

x = 15 only

15, 17

Check: 15(17) = 255

255 - 1/3 ( 15) = 250

Answer 5

n(n+2) - n/3 =250
n^2 + (5/3) =250
(n+5/6)^2 = 250 + 25/36 = 9025/36 = (95/6)^2
n = 95/6 - 5/6 = 90/6 = 15
so the numbers are 15 and 17

Answer 6

Set up your equation...

x ( x + 2 ) - x/3 = 250

Expand out..

3x^2 + 5x - 750 = 0

Solve...

(3x + 50)(x - 15) = 0

x - 15 = 0

x = 15

The integers are 15 and 17.