Nana was baking cookies and talking to Sara

Nana was baking cookies. She does that quite often but this time she was making lots of them. There was a bake sale in their temple and her delicious cookies were always the first items to be sold out.  When Sara came home from school, she smelled the aroma and went straight to the kitchen to talk to her grandmother.  She sat down. They said hello to each other and started talking even though Nana was busy baking.

Sara: Nana, we spend a lot of time together because I love you.

Shanti: I love you too Sara.

Sara’s activities – proportion of time

Sara: I think on most days we spend about three hours talking to each other.  That’s about 3 hours out of 24.

Nana:  Yes, we do that but where are you going with this?

Sara: We learned about ratios and proportions in algebra.  We spend 3/24 hours together.  So if x was the number of hours in a day, this proportion will be simplified to x/8.

Nana: Yes, you leave for school at 8:30 and come back at 3:30.  So that’s 7 hours a day you spend at school.  Do you also want to call that 7x/24?  You also sleep about for about 8  hours at night.

Sara: Yes that would be 8x/24 or x/3.  I also spend time getting ready for school and brushing my teeth at night for another x/24 proportion of the day.  We also have dinner together for another x/24.

Nana:  You also spend x/6 of the time in your room doing your homework, watching TV, talking to your friends or whatever you do in there.

Sara:  Does that account for the whole day? I will add all of the activities together:

x/8 + 7x/24 + x/3 + x/24 + x/24 +x/6

= 3x/24 +7x/24 + 8/24 + x/24 + x/24 +4x/24 = 24 x/24 = x.

Yes, all these add up to x.

Why use x

Nana:  Why did you waste your time using the x?  You could have just checked if the number of hours for all the activities added up to 24.

Sara: Nana, because there is a beauty in these proportions.  There are 120 hours from Monday to Friday.  The beauty is that the proportion remains the same whether the total is 24 hours or 120. Now if were to say x = 120, we can still figure out the number of hours we spend together but this time by saying when x = 120 hours, and x/8 = 15 hours.  The same goes for the school, when x =120, 7x/24 = 35.

Nana:  I see where you are going.  You have school for about 36 weeks a year which would mean 180 school days or 4320 hours.

Sara: You got it Nana.  So when x = 4320 hours, we spend x/8 or 4320/8 or 540 hours together during that period.

Nana:  This ratio and proportion thing is nothing new to me.  I use it all the time.  I know that for making 400 grams of cookies, I need 150 grams of flour, 140 grams of sugar, 100 grams of butter, 5 grams of peanut crunch and 5 grams of chocolate chips. Today, I had to bake  10 kilograms of cookies. I adjusted it according to the amount of cookies I had to bake. The amount of 10 kilograms is 25 times 400 grams.  So I increased each ingredient to 25 times.  I used 3750 grams of flour, 3500 grams of sugar, 2500 grams of butter, 125 grams of peanut crunch and 125 grams of chocolate chips. That’s it.

Sara:  I already said that you know all this.  Only if you used x and y, you would call it algebra.  That’s all.  Nana, the math teacher said that starting with this, we can handle many more complicated problems more easily.  That’s why we are learning algebra.

Shanti hugged her goddess of knowledge and said, “I love you Sara”.

Challenge   

1. Sara’s parents went to an Indian wedding held in a Banquet Hall, and asked Sara to come along.  When Sara reached the Banquet Hall, she was amazed at the large number of people who had come there – men and women.  However, very soon she started to get bored. The only people she knew there were her parents, and they were busy chatting away with their old friends. Looking around Sara ran into Raji – a girl her own age, and they started chatting.  Raji told Sara that she loves fashion and goes to the Indian weddings with her parents just to see the types and colors of the women’s dresses. They are nothing like the colours she sees the girls wearing at her school. Raji started telling Sara the colours of the clothing of the women as she saw them.  Sara had not even heard of some of the colours. She also knew that she had many hours to kill and this could be one way.  So she tagged along with Raji. Never the less, she took out her smart phone and took notes.  Here are the number of women with the different coloured clothing: classic blue (130), scuba blue (140), marsala (80), toasted almond (70), aquamarine (20), tangerine (30), custard (120), strawberry ice (50), lucite green (100) and other colours (300).  Other colours were the ones that even Raji could not name. When they came home, Sara’s mom told her that there were 2000 people at the wedding – 1040 women and the rest were men, boys and girls.  If x was the total  number of people and y was the total number of women, determine the ratio y/x.  For each colour, determine the simplified proportion of women with respect to y.  Sara did the calculations and called Raji.  Raji was fascinated with the calculations and said that these proportions were similar at the 15 wedding parties she attended in the last two years.

Solution:  Given x = 2000, y =1040.  Therefore the ratio y/x = 1040/2000 =13/25.

Because y = 1040, based on the given values the proportions of the women with each colour dresses are:

classic blue: 130/1040 = y/8,  scuba blue: 140/1040 = 7y/52, marsala:  80/1040 = y/13,

toasted almond: 70/1040 = 7y/104, aquamarine: 20/1040 = y/52, tangerine: 30/1040 = 3y/104

custard: 120/1040 = 3y/26, strawberry ice: 50/1040 = 5y/104, lucite green: 100/1040 = 5y/52,

other colours: 300/1040 = 15y/52.

• Ratios and proportions are everywhere – even in sports. All sports have winning percentage. Baseball stats include batting average for batters and ERA for pitchers.  Soccer and hockey use shooting percentage, save percentage, and penalty kick percentage in soccer or penalty goal percentage in hockey.  In tennis, they talk of first serve percentage and ace percentage. Don’t forget the run rate in cricket.  All proportions can be written as x/y.  Discuss with your friends what the values of x and y are in the sport of your interest.  Check from the Internet if they are correct.   Note that a solution is not provided here.

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