Question

If a > b > c > d, then which is larger, a+c or b+d ? Can we tell from a > b > c > d which of a+d and b+c is larger?

Answer 1

1. a+c is larger than b+d

2. No way to tell whether a+d or b+c is larger.

Explanation:

1. Which is larger, a+c or b+d?

Let a, b, c, and d be any numbers such that
`a > b > c > d`.

Specifically, note that
`a > b`, and subtracting b from both sides of the inequality, observe that
`a-b > 0`.

Similarly,
`c > d`, and subtracting d from both sides of the inequality, observe that
`c-d > 0`.

From this, add "a-b" (a positive number, as proven above) to both sides of the inequality.

`(a-b)+(c-d) > (a-b)+0`

Addition by zero (the additive identity) doesn't change anything, so the right side remains "a-b"...

`(a-b)+(c-d) > a-b`

... and "a-b" is positive...

`(a-b)+(c-d) > a-b > 0`

... so, by the transitive property of inequality...

`(a-b)+(c-d) > 0`

Recall that subtraction is addition by a negative number...

`a+(-b)+c+(-d) > 0`

...and that addition is associative and commutative, so things can be added in any order, so the middle two terms on the left side can be rearranged...

`a+c+(-b)+(-d) > 0`

Adding b + d to both sides of the inequality

`(a+c+(-b)+(-d))+(b+d) > 0+(b+d)`

... and simplifying

`a+c > b+d`

So, a+c is larger than b+d.

2. Which is larger, a+d or b+c?

Consider the following two examples:

Example 1

Suppose a=10; b=3; c=2; d=1.

Note that
`a > b > c > d` (
`10 > 3 > 2 > 1`) and, also observe that
`a+d=(10)+(1)=11`, and
`b+c=(3)+(2)=5`, so a+d is larger than b+c.

Example 2

However, suppose a=10; b=9; c=8; d=1.

Note that
`a > b > c > d` (
`10 > 9 > 8 > 1`) but that
`a+d=(10)+(1)=11`, and
`b+c=(9)+(8)=17`, so a+d is smaller than b+c.

So, in one example, a+d is bigger, and in the other, a+d is smaller. Therefore, there is no way to tell which of a+d or b+c is larger from only the given information.