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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?

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Answer:

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.

User Khundragpan
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