Letters A, B, C, and D represent four different digits selected from 0, 1, 2, …, 9. If (A+B)/(C+D) is an integer that is as large as possible, what is the value of A+B?

1998 AMC 12 Problem 2

Attempt Problem 2 of the 1998 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1998 AMC 12 solutions, or check the answer key.

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Letters

A,

B,

C,

and

D

represent four different digits selected from

0,

1,

2,

\ldots,

9.

\frac{A+B}{C+D}

is an integer that is as large as possible, what is the value of

A+B?

13

14

15

16

17

Make the denominator as small as possible using distinct digits

The two smallest digits and the two largest digits are disjoint pairs

The least positive denominator is

0+1=1,

and the greatest numerator using two other digits is

8+9=17.

Their ratio is the integer

17,

which is plainly maximal. Thus

A+B=17,

and E is correct.