# Playing at Tanya’s Place

During the summer holidays Tanya, Tinku, Kate and Peter played together at each other’s place.  They had to play indoors because it was very hot outside.  Wherever they played, they also got something to eat and drink.

Today they were at Tanya’s place and her mom gave them a large pizza to share equally.  The large pizza had 8 slices.  So it was easy, first each of them ate one slice of the pizza, and then the second one. The pizza was finished.  Peter was full and started to belch.  Kate started laughing but Tinku had a different idea. He wanted to do something with the numbers.

# 1/8 + 1/8 = 1/4

Tinku:  The pizza had 8 slices, first I took one. Tanya, would you say that I took 1/8 fraction of the pizza at that time?

Tanya: Yes, you took 1/8 of the pizza two times and that made it a total of 1/8 + 1/8 which is 2/8 which is equal to 1/4 or a quarter.

Tinku:  Thanks Tanya. I didn’t know that you could divide the numerator and the denominator by the same number and yet get the same value for the fraction.  That way the four of us got 1/4 +1/4 +1/4 +1/4 = 4/4 = 1 or the whole pizza.

# Four persons share a 6-slice pizza

The next day, the kids played at Peter’s place.  Peter’s mom  also gave them pizza but she thought that the kids should not eat only pizza.  So she got them a smaller pizza which had 6 slices but afterwards she also gave them two English cucumbers.  She told them to share everything equally amongst themselves.

First they ate the pizza.  All four took one slice each out of the six, and 2 slices were left.  That meant each one of them got 1/6  fraction of the pizza, and that there are 2/6 left behind.  They asked Peter’s mom to cut each of the remaining two slices into equal halves as shown with a black line in the picture.  Thus each of the remaining smaller pieces were half of the 1/6 of the pizza.  This came out to 1/12 of the pizza for each of them (see picture of 6 slice pizza).

Tinku:  We divided one pizza equally amongst ourselves.  So, in total I should have got 1/4 of the whole pizza.  Tanya, did I?

Tanya:  Yes, you did.  Remember,  you took 1/6 of the pizza (0ne slice out of 6) the first round and 1/12 the second round.  So together in both rounds you took 1/6 + 1/12 of the pizza.  One sixth of the pizza is equal to 2/12, and  2/12 + 1/12 = 3/12 which is equivalent to 1/4.

Tinku:  Yes, I did.  Thanks Tanya.  I like the way you explained it.

# Challenge

You have a large party pizza with 24 slices.  You and your friends are 10 people to share it equally.  That means each person will get 10/24 of this pizza.  Explain, how this can be done most fairly to everyone.  Your mom is ready to help by cutting any piece into two halves.

Solution: First everybody gets two slices thus 20 slices will give each 2/24 of a pizza.  Remaining are 4 slices – each 1/24 of pizza.  They can be cut into halves to get 8 pieces and then again into halves to get 16 pieces – each piece would be 1/96 of the pizza. Everyone can have a piece and 6 pieces will be left.  Each of the six pieces can be cut into two halves, to get what we will call them tiny pieces 1/192 of the pizza each.  Everyone takes one so that two tiny pieces will be left behind. This way each person will get 2/24 + 1/96 + 1/192 or 16/192 + 2/192 + 1/192 or 19/192 of one pizza. The exact share should be 1/10 or 19.2/192 per person. One can be happy with this or keep cutting again and again.  It is this way because both 2 and 5 are factors of 10 but the cuts are only by the factor 2.

### Fractional Dice

Normal dice are six sided with the numbers 1,2,3,4,5 or 6 printed on the different sides or that many dots on each side.  You may have used them while playing Ludo, monopoly or other games.

Tanya’s uncle brought her fractional dice instead of the normal ones.  Fractions were printed on their sides.  She rolled the dice and got the following picture.  Determine the sum of all the fractions she rolled.

Answer: The denominators of the fractions showing are 2,3,4,5,6,8.  Find their LCM (lowest common multiple).

First factor them all:  2 = 2. 3= 3, 4 =2×2, 5=5, 6= 2×3, 8= 2x2x2.

The number 2 x 3 x 2 x 5 x 2 = 120 contains all the factors shown above.  Therefore it is the LCM.

Now write all the equivalent of all the fractions in which LCM is the denominator.

1/2 = 60/120, 1/3 = 40/120, 1/4 = 30/120, 1/5 =24 /120,  1/6 = 20/120, 1/8 = 15/120.

The sum of all the numerators divided by the LCM will be your answer

(60+40+30+24+20+15)/120 = 189/120