Half of the smaller of two consecutive even integers is equal to two more than the larger integer

Answer 1

Let't break down the words into mathematical terms where we can.

"Two consecutive EVEN integers". Since we are dealing with integers we know the answer can be either a positive or a negative whole number. Furthermore, we know consecutive means in order in increasing value. So if we pretend the number is 6, the next consecutive number would be 7 (the lower value integer plus 1, 6 + 1 = 7). But in this problem the consecutive numbers are EVEN. Going back to our pretend number of 6, the next integer that is even as we increase in value would be 8. So consecutive even integers are the lower value integer plus 2, 6 + 2 = 8 in our example.

But we don't know the integer. In that case we can use a variable to represent an unknown quantity, in this case an integer. Let's use the ever popular x to represent the integer of the lowest value. We know the next consecutive even integer after x is going to be the integer plus 2. Using our variable of x, we can write the next consecutive integer as x + 2

Going back to the problem, "Half of the smaller of two consecutive even integers" deals with the smaller integer, which we have defined as the variable x. "Half of this integer" means we can divide x by 2. We can write this as:

x/2

"is equal to"

We can write this as

=

So now we have

x/2 =

And the last part, "two more than the larger integer" refers to the consecutive integer after x. We have defined the large integer as x + 2. But we want "two more than" that integer. We can write this as 2 + (two more than) x + 2 (the larger integer) or

2 + x + 2

So the problem can now be written completely as:

x/2 = 2 + x + 2

We complete the addition of the constants on the right side of the equation (2 + 2), resulting in

x/2 = x + 4

We are now trying to isolate the variable, x. We can do this by first multiplying the left and right sides of the equation by 2, resulting in x on the left hand side.

2 * x/2 = 2(x + 4)

x = 2x + 8

We can now subtract x from the left and right sides of the equation,

x - x = 2x + 8 - x

0 = x + 8

And lastly we subtract 8 from both sides to isolate x.

0 - 8 = x + 8 - 8

-8 = x

Thus the smaller integer equals -8. The next consecutive even integer is x + 2, which is:

-8 + 2 = -6

So the consecutive even integers are -8 and -6.

We can check this by going back to the original equation

-8/2 = 2 + -8 + 2

-4 = 4 + -8

-4 = -4

Note, there are various ways to solve the equation once it has been defined and it is difficult and time consuming to explain all of that here. The beautiful part of this problem is in discovering the relationship between consecutive even integers and how we can express that without words.

Answer 2

S/2=2+ L

L=S+2

S/2=2+2+S

S/2=-4

S=-8= the Smaller integer

L=-6= the Larger integer

-8/2=-4 =2+(-6)