If A+B = 76 and A-B = 38, what is the value of A/B = ?

*Extra time? *

*Which four-digit number multiplied by 4 gives the same number in reverse?*

**Answer(s)**. The correct answer for the main question was 3 and for the extra one was 2178 x 4 = 8712. (Please see a small selection of the many fine replies just below).

Very well done to: **Michael Easton**, director, Globus Group, Trafford Park, Manchester, UK; **Flemming Vorbeck**, purchasing manager commercial, NDI Group A/S, Brørup, Denmark; **David Mann**, key account manager, SPC Rubber Compounding, UK; **Mehmet Koral**, C&C Endüstriyel Danışmanlık, Eğitim ve Mümessillik Ltd, Göztepe, Istanbul, Turkey; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **France Veillette**, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **T N Dahal**, Kankai Rubber industries, Nepal; **Stefano Carletti**, R&D Indorama Mobility Group, Office address: Glanzstoff Sicrem Spa, Pizzighettone (CR), Italy; **Giorgio Marzari**, sales director, RDC srl, Bareggio (MI), Italy; **Alexandra Poschlep**, head of product management sheeting, Semperit Sheetings, Semperit Technische Produkte GmbH, Wimpassing, Austria.

*Solutions*

*Flemming Vorbeck *

*If A+B = 76 and A-B = 38*

76-38 = 38

38 / 2 = 19 (B)

76 - 19 (B) = 57 (A)

38 + 19 (B) = 57 (A)

So…

A (57) + B (19) = 76

A (57) - B (19) = 38

Result:

A (57) / B (19) = 3

*Mehmet Koral*

*If A+B = 76 and A-B = 38, what is the value of A/B = ?*

We start by adding the left and right sides of the 2 given equations:

A+B = 76

A-B = 38

2A+B-B=114. --->2A=114

A=57.

Then,

A+B=76

57+B=76 ------> B=76-57=19.

Now, we are ready to calculate the A/B=?

57/19=3

The answer is 3.

*Andrew Knox*

*Extra time*

Bit of trial and error required here, but by process of elimination one can narrow down the possibilities fairly quickly.

I.e. the 1000 series x 4 can never end in a 1, anything above the 2000 series will result in a 5 digit number, so it has to be in the 2000 series.

So to be reversible it has to end in an 8.

Of these possibilities, only the 21X8 and 22X8 series will work, as 23X8 and above will yield a “9“ as the last digit. So, 2178 is found to work after only 7 “trial an error” calculations.

*John Bowen*

*Extra time*

1. Lets say the number is written as ABCD (1000a+100b+10c+d).

2. Multiplying it by 4, gives you DCBA (1000d+100c+10b+a)

3. By divisibility of 4, BA is a two-digit number divisible by 4.

4. Now, the largest possible value of ABCD is 2499, because anything more will result in a five digit number when multiplied by 4.

5. Out of the two possibilities 1 and 2 for A, [3] follows that A is even, so 2.

6. From [5], D is either 8 or 9. However, since D times 4 would end in 2, D is 8.

7. Using [2], 4000a+400b+40c+4d = 1000d+100c+10b+a, which would simplify to 2c-13b = 1 after substituting A = 2 and D = 8.

8. The only possible value of ABCD is 2178

*Alexandra Poschlep*

*Extra time:*

It must be something below 2500

ABCD = DCBA

From the logic it must be at least A*4 and found out 2 and 8 (as 8:2 = 4) and it must be starting with the low and and with the high fit the best >2000 but <2500.

A = 2

D = 8

2BC8

Then I tried several possibilities and found out that the inner part B and C must be reduced by -1 from the left and from the right

Therefore, I came to 2178.