In his time, Bob Hawke was regarded as one of Australia’s most popular political figures. Today he may be best remembered for some of his off-the cuff comments!

This leads us, perhaps unfairly, to the following puzzle:

BOB DID = .TALKTALKTALK ...

Can you solve this problem? As usual, different letters stand for different digits. The problem becomes somewhat easier when we realize that the expression on the right can be simplified to TALK / 9999 .

The solution is unique, if it is assumed that the fraction is in lowest terms.

Hint 1

Use the hint in the text to simplify the right hand side.

Hint 2

Cross multiply to obtain an equation, find the factors of 9999, and hence find some values for DID.

Solution

Replacing the repeating decimal by ^{TALK /} ₉₉₉₉ , and cross multiplying gives us

                  <sup>BOB x 9999 = DID x TALK.

Now 9999 = 3 x 3 x 11 x 101. We deduce that DID is 101 or 303 or 909.

Suppose first that DID = 101. Cancelling gives BOB x 99 = TALK. Clearly the product on the left has only four digits (TALK). But B ≠ 1, since different letters stand for different digits, and D = 1. Hence B = 2, BOB ≠ 200, and we have a contradiction. Hence DID ≠ 101.

Next suppose DID = 909. Cancelling, we obtain BOB x 11 = TALK. This is impossible, since for any value of B, the final digit of the product BOB x 11 is B (for example, 343 x 11 = 3773). But this implies that K and B stand for the same digit. Hence DID ≠ 909.

It follows that DID = 303, and BOB x 33 = TALK. Since the product has four digits, BOB ≠ 303. B cannot be 3, so B = 1 or B = 2. Now B ≠ 1, else on multiplying, K = 3 (= D). So B = 2. Testing in turn the values O = 1, 4, 5, 6, 7, 8, 9 gives TALK = 6996 (X), 7986 (possible), 8316 (X – contains a 3), 8646 (X), 8976 (X – contains a 7 = V), 9306 (X – contains a 3), 9636 (X).

Thus BOB = 242, DID = 303 and TALK = 7986.</sup>

Extensions

Use your calculator to
check the result.

Hint 1

Hint 2

Solution

Extensions