Word Problem

Question

A student gave "2,4,6" as the answer to the problem "Find three consecutive even integers such that 10 times the largest value exceeds the product of the other two by more than 40."  Explain why the student's answer is not entirely correct. 

Answer 1

It is correct, but there is more than one set of answers.
6 x 10 = 60
2 x 4 = 8

60 - 8 = 52 > 40

 

4,6, 8 is also correct as 80 - 24 = 56 > 40

 

 

x is the smallest even integer

x+2 is the middle even integer

x+4 is the largest even integer

 

 

 

10(x+4) > x(x+2)

 

10x + 40 > x^2 + 2x

 

0 > x^2 - 8x - 40

 

8 +- sqrt(  64 + 4*1*40)  /   2

8 +- sqrt ( 224)     /   2

8 +- sqrt( 2 x 2 x 2 x 2x2x7)  / 2

8 +- 4*sqrt(14)  /   2

 

4 +- 2 sqrt(14) 

 

 

4+2*sqrt(14) = 11.483

 

4 - 2*sqrt(14) = -3.4833

 

So the smallest even integer is between those values:

-4,-2, 0   ---->   0x10 - -4*-2 = 0 - 8 = -8 <---- doesn't work because it is outside the range of those two values.   

 

-2,0,2   ---> 2 x 10 - -2*0 = 20 - 0 = 20 <--- doesn't work because it is outside the range of those two values

 

  0,2,4 --->  4 * 10 - 0 x 2 = 40 - 0  = 40

 

 2,4,6  ---> 6 x 10 - 2 x 4 = 60 - 8 = 52 > 40   yes it works

 

4,6,8  ---> 8 x 10 - 4 x 6 = 80 - 24 = 56 > 40 yes it works

 

6,8,10 --> 10 x 10 - 6 x 8 = 100 - 48 = 52 > 40  yes it works

 

8,10,12  --->  12 x 10 - 8 x 10 = 120 - 80 = 40

 

10,12,14 ---> 14 x 10 - 10 x 12 = 140 - 120 = 20

 

12,14,16 ---> 16 x 10 - 12 x 14 = 160 - 168 = -8  <--- doesn't work ; outside the range of values

 

 

Answer 2

lets consider

consecutive even numbers are n,n+2,n+4

10(n+4) =n(n+2)+40

 

10n+40 =n² +2n +40

 

 

8n = n²

 

8 = n

 

consecutive even numbers are n,n+2,n+4  = 8,10,12

 

IN case of 2,4,6

 

10(6) ≠ (2*4 ) +40 

 

 

 

 

Answer 3

Well, let's check the answer against the conditions.

(10)(6) - (2)(4) > 40

60 - 8 > 40

52 > 40

True.

Unless I misread, the student is correct.