Puzzle: JUZZ up

You have two bottles of orange juice mixed with lemonade, of two different strengths. The first one contains 3 litres of  lemonade and 1 liter of orange juice (25% concentration). The second one contains 2 litres of lemonade and 2 litres of orange juice (50% concentration). You take 1 liter from the 1st bottle and pour it into the 2nd bottle and mix. Then you take 1 liter of the 2nd, and pour it into the 1st. Now both bottle contain the same amount of liquid as they did at the start. What are the new concentrations?

 

Answer: Bottle 1: 30% orange juice, 70% lemonade

            Bottle 2: 45% orange juice and 55% lemonade.

BRAIN TEASER

On the planet Zogx, there are only four alphabets: Z,O,G and X. Luckily, almost every string formed by those letter is a word, for example XOGZX. Every word has at least one letter. No words have more than one Z, one O, or one G. No words have more than two Xs. Words with adjacent X’s are very rude! How many polite words are there on Planet Zogx?

 

Answer: 121 polite words

MULTIPLICATION BY NINE

1.Subtract the right –hand  figure of the long number from ten. This gives the right-hand figure of the answer.

2. Taking each of the following figures in turn, up to the last one, subtract it from nine and add the neighbor.

3. At the last step, when you are under the zero in front of the long number, subtract one from the neighbor and use that as the left-hand figure of the answer.

Of course, in all these steps it is understood that is there is a dot ( a carried one), you have to add it in as usual.

Here is an illustration of how it works out:8,769 times 9.

                    0 8 7 6 9    x     9

                    7 8 9 2 1

First: Take the 9 of 8,769 from 10, and we have the 1 of the answer.

Second:Take the 6 from 9 (we have 3) and add the neighbor, 9 ; the result is 12, so we write a dot and 2.

Third: 7 from 9 is 2, the neighbor (6) make it 8, and the dot makes  it 9.

Fourth: 8 from 9 is 1, and the neighbor makes it 8.

Fifth: This is the last step; we are under the left-hand zero. So we reduce left-hand figure of 8,769 by one, and 7 is the left –hand figure of the answer.

MULTIPLICATION BY FIVE

The rule for multiplying by 5 is like those for 6 and for 7, but simpler. Instead of adding in the “number” as we do for 6, or doubling it as we do for 7, we use the “number” only to look at. We look at it and see whether it is odd or even. It it is odd we add in the 5, as before:

Half the neighbor, plus 5 if the number is odd.

Suppose we wish to multiply 426 by 5:

0 4 2 6  /  0  x  5   look at 6, it is even; there is no 5 to add;                      there is no neighbor

0 4 2 6  / 30  x  5  look at 2, it is even, use half of 6

0 4 2 6  / 130 x  5  look at 4, it is even; use half of 2

0 4 2 6  / 2130 x 5 look at zero, it is even, use half of 4

Now if we had an odd digit in the multiplicand we would add 5:

0 4 3 6   /  0    x  5   as above

0 4 3 6  /   80  x  5   the 3 is odd; say 5 plus 3

0 4 3 6  / 2180  x 5 

This is easy to do. There is very little figuring involved. It seems rather odd at first, though, because of the little mental twist that you have to do: you use the neighbor, rather than the digit at the place where you are working. Actually it is good practice in keeping one’s place.  Later on, in multiplying one long number by another long number, we shall find that a certain amount of concentration is needed to remember where we are in the number we are multiplying. This multiplication method for 5 is a little preliminary practice.

What is a tessellation?

A tessellation is a repeated arrangement of plane figures, which fit together to fill a space with no overlaps or gaps between them. Another word for tessellation is tiling. The first tilings were made from square tiles. The word “tessellate” is derived from the Ionic version of the Greek word “tesseres”, which in English means “four”. Tessellations frequently appeared in the art of M.C.Escher.

Semi Regular tessellations

Patterns formed using two or more regular polygons are called semi regular tessellations. You can use a variety of regular polygons to make semi-regular tessellations. The beauty of a semi-regular tessellation is the identical arrangement of polygons at every vertex point.

Palindrome syndrome

A number that reads the same when read forwards and backwards is called a palindromic number.

eg: 121 is a palindromic number.

      134282431 is a palindromic number.

      12436683421 is not a palindromic number.

Single digit numbers i.e., 0,1,2,3,4,5,6,7,8,9 are palindromic.

How may palindromic numbers are there between -

a) 0 and 100?       b) 100 and 200 ?

c) 200 and 300 ?   c) 0 and 1000?

answers:-

a) 19   b)  10   c) 10   d)  109

Multiplication by seven

The rule for multiplying by seven is very much like that for six:

Double the number and add half the neighbor; add 5 if the number is odd.

Suppose we wish to multiply 4,242 by 7. There are no odd digits in this number, so we shall not need to add the extra 5. For this example, the work goes the same as for six, except that now we double:

First step: 0 4 2 4 2 x 7 = _ _ _ _4 (double the number 2 )

Second step: 0 4 2 4 2 x 7 = _ _9 4 (double 4 plus half the neighbor)

Third step: 0 4 2 4 2 x 7 = _ 6 9 4 (double 2 plus half the neighbor)

Fourth step: 0 4 2 4 2 x 7 = 9 6 9 4

Last step: 0 4 2 4 2 x 7 = 2 9 6 9 4 (doubling zero we still have zero, but add half of neighbor)

Here is an example containing odd digits. The 3 and 1 are both odd:

First step: 0 3 4 1 2 x 7 = _ _ _ _ 4 (double 2; there is no neighbor)

Second step: 0 3 4 1 2 x 7 = _ _ 8 4 (double 1, plus 5 – because 1 is odd –is 7, and add half of 2)

Third step: 0 3 4 1 2 x 7 = _ _8 8 4 (not odd: double 4 and add half of 1)

Fourth step: 0 3 4 1 2 x 7 = _ 3 8 8 4

Last step: 0 3 4 1 2 x 7 = 2 3 8 8 4 (doubling zero is zero, but add half of 3, plus the dot)

The correct mental steps are:

1) Say “1” for the dot, if there is a carried 1.

2) Look at the next number to work on and notice whether it is odd. If it is, add 5 to the carried 1, saying “6”, or say “5” if there was no dot.

3) Looking at the number and doubling it mentally, we say the sum of the 5 and this doubled figure. If the figure is 3, for instance, we say “5”, then we say “11”, because doubling the 3 to get 6 and adding it to the 5 can be done in one step.

4) Looking at the neighbor, say 6 for example, we add half of it to what we have already. We were saying just now that we had 11. If the neighbor is 6, we next say “14”.

Let us take this process a little at a time. The mental training involved in doing this sort of thing is very valuable because it develops the ability to concentrate, and concentration is practically the whole secret of success.

Multiplication by five, by six and by seven

All these multiplications-5,6, and 7-make use of the idea of “half” a digit. We put “half” in quotation marks because it is a simplified half. We take half the easy way, by throwing away fractions if there are any. To take “half” of throwing away fractions if there are any. To take “half” of5, we say 2. It is really 2 1/2, but we won’t use the fractions. So “half” of 3 is 1, and “half” of 1 is zero. Of course “half” of 4 is still 2, and so with all even numbers.

This step is to be done instantly. We do not say to ourselves “half” of 4 is 2 or anything like that. We look at 4 and say 2. Try doing that now, on these digits:

2,6,4,5,8,7,2,9,4,3,0,7,6,8,5,9,3,6,1

The odd digits, 1,3,5,7 and 9, have this special feature of dropping the fractions. The even digits, 0.2.4.6 and 8 give the usual result anyway.

Multiplication by six

Now let us try out this business of “half”. Part of the rule for multiplying by 6 is:

To each number add “half” of the neighbor.

Let us assume for the moment that this is all we need to know about multiplying by 6 and work out this problem:

0 6 2 2 0 8 4 x 6

First step: 4 is the first “number” of the long number, and it has no neighbor so there is nothing to add:

0 6 2 2 0 8 4 x 6 = _ _ _ _ _ _ 4

Second step: the second number is the 8, and its neighbor is the 4, so we take the 8 and add half the 4 (2), and we get 10:

0 6 2 2 0 8 4 x 6 = _ _ _ _ _ 0 4

Third step: the next number is the zero. We add to it half its neighbor, and 8. Zero plus 4 is 4, and add the carry (1):

0 6 2 2 0 8 4 x 6 = _ _ _ _ 5 0 4

Repeat this last step with the 2, the 2, the 6, and the zero, in turn:

0 6 2 2 0 8 4 x 6 = 3 7 3 2 5 0 4

Would you like to see how easy it is ? Try it yourself on these two multiplications:

0 4 4 0 4 x 6

0 2 8 6 8 8 4 2 4 x 6

The answer to the first problem is 26,424. The answer to the second one is 172,130,544.

What we have done gave the correct answer in these problems. However, it was not quite the full rule for multiplying by 6. The full rule is:

To each “number” add half the neighbor; plus 5 if “number” is odd.

“If odd” means if the “number”is odd, it makes no difference whether the “neighbor” is odd. We look at the “number” and see whether is is odd or even. If it is even we merely add to it half of the neighbor. If it is odd, we add 5 to it and “half” of the neighbor, as we did just above. For instance:

0 4 4 3 0 5 2 x 6

The figures 3 and 5 are odd. We see that as soon as we look at the multiplicand. When we come to work on the 3 and the 5 we shall have to add and extra 5, simply because of their oddness. It goes like this:

First step: 0 4 4 3 0 5 2 x 6 = 2 (2 is even and has no neighbor; carry it down)

Second step: 0 4 4 3 0 5 2 x 6 = 1 2 ( Five is odd! 5 plus 5, plus “half” of 2 is 11)

Third step: 0 4 4 3 0 5 2 x 6 = 3 1 2 ( “half” of 5 is 2; then add the carry)

Fourth step:0 4 4 3 0 5 2 x 6 = 8 3 1 2 (3 is odd! 3 plus 5 is 8)

Fifth step: 0 4 4 3 0 5 2 x 6 = 5 8 3 1 2 (4 plus “half” of 3)

Sixth step: 0 4 4 3 0 5 2 x 6 = 6 5 8 3 1 2 (4 plus “half” of 4)

Last step: 0 4 4 3 0 5 2 x 6 = 2 6 5 8 3 1 2 (zero plus “half” of 4)

The answer is 2,658,312. Of course, all this explanation is only for the sake of the greatest possible clarify in showing the method for the first time. In actual practice, it goes fast because the step of adding half the neighbor is a very simple one. With only a reasonable amount of practice it becomes more automatic than conscious.

You will see this more clearly, perhaps, if you go through these two for yourself:

0 8 2 3 4 x 6

0 6 2 5 0 1 8 8 x 6

The answer to the first problem is 49,404. The answer to the second one is 37,501,128.

MULTIPLICATION BY TWELVE

To multiply any number by 12, you do this:

Double each number in turn and add its neighbor.

This is the same as multiplying by 11 except that now we double the “NUMBER” before we add its “neighbor”. If we wish to multiply 413 by 12, it goes like this:

First step:          0 4 1 3  x  1 2 = _ _ _ 6     double the right-hand figure and carry it down (there is no neighbor)

Second step:      0 4 1 3  x  1 2  = _ _ 56      double the 1 and add 3

Third step:         0 4 1 3  x  1 2  =  _ 956      double the 4, add the 1

Last step:          0 4 1 3  x  1 2   =  4956       zero doubled is zero; add the 4

The answer is 4,956. If you go through it yourself you will find that the calculation goes very fast and is very easy.

Try one yourself:63,247 times 12.  Write it out with the figures spaced apart, and put each figure of the answer directly under the figure of the 63,247 that it came from. This is not only a good habit because of neatness, it also is worth its weight in gold as a protection against errors. In the particular case of speed system of multiplication, we mention it because it will identify the “number” and the “neighbor”. The next blank space in the answer, where the next figure of the answer will go, is directly below the “number” (in this example the figure that you must double). The figure to its right is the “neighbor” that must be added. The example works out in this way:

0 6 3 2 4 7  x  1 2   = _ _ _ _ _ 4     double 7, 14; carry 1

0 6 3 2 4 7  x  1 2   = _ _ _ _ 6 4  double 4, plus 7, plus 1 is 16, carry 1

0 6 3 2 4 7  x  1 2   = _ _ _ 9 6 4 double 2, plus 4, plus 1 is 9

until you end up with: 0 6 3 2 4 7  x  1 2  =  7 5 8 9 6 4