Olivia is scared of the word algebra
Sara goes over to her friend Olivia’s house to play but finds that the friend is in a really bad mood. So here is their conversation.
Sara: Olivia, let us go play.
Olivia: What do you want to play?
Sara: Anything you want to play Olivia, but why the long face?
Olivia: The Algebra lesson this week! It just went over my head and the teacher said that this was just an introduction. We will do the real stuff in high school. It is scaring me.
Sara: So what should we play then?
Olivia: How will playing help me with my problem? I don’t feel like playing anything.
Sara: Sometimes, playing refreshes the mind.
Olivia: I don’t know.
Sara: Let’s look at your new dresses then. I promise you that by the time I leave, you will not feel so bad. What new dresses did you get last month?
Olivia has this friendly relationship with Sara. She thinks that Sara may do what she says. They start going through the clothes. Olivia shows her the new colourful silk gown her mom bought her last month. They also go through some of her other clothes. She has a blouse with a knee high skirt, a shirt pant suit and even one miniskirt. They appreciate and talk about the colours and designs of all the clothes. Suddenly, out of nowhere, Olivia comes up with something.
Cost of Perry and Gomez posters
Olivia: I don’t have any posters of Katy Perry or Selena Gomez. I want both. I want a Selena poster and a Perry poster. One of my friends told me that she bought these posters to give them as gifts to other girls for Christmas. She said that she bought three Gomez posters and two Perry posters together for $200.
Sara: From where did she buy them?
Olivia tells her the place and then Sara phones them. She tells them that she wants to buy three Gomez posters and two Perry posters for gifts to friends. The storekeeper says that they do not have that many Gomez posters left, but that they can give her only two Gomez posters plus three Perry posters for $180. She gives all this information to Olivia.
Olivia really wanted to know the prices of the posters So she started figuring this out by writing everything. Here is what she did:
3 Gomez posters + 2 Perry posters come for $200, and
2 Gomez posters + 3 Perry posters come for $180. That means
6 Gomez posters + 4 Perry posters come for $200 x 2 or $400, and
6 Gomez posters + 9 Perry posters for $180 x 3 or $540.
That means the difference in price between the two sets is $140. Because the two sets have the same number of Gomez posters, the difference in price must be because of the 5 Perry posters. That means that 5 Perry posters are worth $140 or one of them is worth $ 28.
Now that she knew the price of one Perry poster to be $28, two of them would cost $56. She went back to 3 Gomez posters and 2 Perry posters being $200. The three Gomez posters must be worth $144 and one of these posters would be worth $48.
Sara: Olivia, I watched what you just did. You already know algebra.
Olivia: What do you mean?
P is for Perry and G is for Gomez
Sara: Here, I will show you. I don’t like too much writing. Instead of writing Gomez poster for full, I can just write G, and P for Perry poster. So what you just did was:
3G + 2P = 200…..equation 1.
2G + 3P =180….. equation 2.
Multiplying both sides of equation 1 by 2 gives 6G +4P=400…equation 3.
Multiplying both sides of equation 2 by 3 gives 6G +9P=540…equation 4.
Subtract equation 3 from 4: 6G + 9P – 6S-4P=540 – 400.
This way you eliminated G to get 5P=140 or P = 28.
Then you modified equation 1 to get 3S +56 = 200 or S = 48.
Olivia: You did the same thing as I did except that you used short hand for writing the names.
Sara: Exactly. I could have written x instead of G for Gomez and y for Perry and done the same thing. Why would that scare you? You already know what to do but you are putting a mental block just because somebody used the word algebra.
Tinku suddenly walked in.
Why write x and y?
Olivia: I know what you just did but why use x and y, and why write the equations?
Sara: Three reasons. One, you know that it took me a very short time to get the answer. Two, I can quickly put the values of x and y back into the equation to see if my answer is right. Three, Tinku here loves cars. The letters x and y could have been for his different toy cars or for that matter for the decorations his mom wanted to buy. It would be the same idea and the same way of doing everything.
Olivia: That makes sense. Thanks Sara. You kept your promise. You had said, ” I promise you that by the time I leave, you will not feel so bad.”
Tinku: I don’t care for toy cars anymore. I like drones now. Oh, Sara I see that you showed her how to solve the equations by elimination. Did you show her how to do them by substitution or should I show it to her?
Sara: Tinku, you don’t know how smart Olivia is. Olivia, show him how you can solve the equations by substitution. You don’t need to use x and y. For now, just use G for Gomez and P for Perry as we did before.
Ice cream and chocolates
Tinku: No, it is not fair if you solve the same problem again. Here, I give you a new one. You go to a store which sells ice cream and chocolates. They say 7 ice creams plus 2 chocolates will cost you $29 but 2 ice creams and 5 chocolates will cost you $26. If you do this right, I will take you out for an ice cream. What is the price of an ice cream?
Olivia: Tinku. If you want to take me out for an ice cream, just say so. Tell you what, if I get it right, you have to take both me and Sara out for an ice cream. Agreed?
Olivia: I will write x for an ice cream and y for a chocolate. Then, I will get
7x + 2y = 29…..equation 1, and 2x + 5y = 26….. equation 2.
From equation 1, I will subtract 2y from both sides
7x – 2y + 2y = 29 – 2y or 7x = 29 – 2y or x = (29-2y)/7.
Now equation 2 is 2x + 5y = 1740.
Substituting (29-2y)/7 for x, it becomes (29-2y) 2/7 + 5y =26.
Multiplying both sides by 7, it becomes (29-2y) 2 + 35 y =182 or 58 -4y+35y=182
or 31 y= 182-58 = 124 or y = 4.
Now back to equation 1, 7x + 2y = 29 or 7x + 8 = 29 or 7x = 21 or x =3.
Tinku: OMG, you got it right.
Always verify your answer
Olivia: There is one more thing that Olivia taught me. You should learn it too. Use these values of x and y to make sure that they fit into equations 1 and 2. Now, when are you buying us the ice cream?
Sara: Olivia, see I told you that you can use the same way of solving for anything – not just posters Now, you just showed Tinku that algebra works for ice cream and chocolates too. Tinku, I am glad that you agreed to take both of us for ice cream.
A week later Olivia called Sara over. She showed her the two posters in her room – Gomez and Perry. She thanked Sara for them. Sara said that she did not buy the posters and should not be thanked. Olivia then explained that she told her parents about her learning algebra and then they bought her the posters.
Sara and Tinku find new ways for rivalry every day. Today’s rivalry is finishing a 10 kilometer race. Now, neither of them is a good athlete who can run for 10 kilometers. They both decide that they would walk if they cannot run. Sara knows that she walks fast at about 5 kilometers per hour and she thinks that she will run 5 km at 9 km per hour and walk the rest. Tinku thinks that he will run 5 km at 10 km per hour and walk the rest. How fast will Tinku have to walk so that Sara does not beat him.
Solution: Time taken by Tanya to run 5 km = 5/9 hours
Time for her 5 km walk = 5/5 hours. So her total time = 1+5/9 or 14/9 hours.
Time for Tinku’s 5 km run would be 5/10 or 1/2 hour.
Let us say Tinku will have to walk can walk only for x hours.
Then x +1/2 is less than equal to Tanya’s time of 14/9 hours. Let us solve for
x +1/2 = 14/9
Subtracting 1/2 from both sides of the equation x = 14/9 – 1/2 = (28 – 9)/18 = 19/18 hours.
If he walks 5 km in 19/18 hours, his walking speed = 5/(19/18) =90/19 = 4.74 km/hour.
So for Tanya not to beat him, Tinku has to walk the 5 km for no slower than 4.74 km/hour.