MSP Math Circle Summer Camp

Purple Haze and Purple Rain

June 15, 2016

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Purple Haze and Purple Rain

Anne decides to repaint her house in purple. She can buy twodifferent types of purple paint.

Paint A is made up from blue and red paint in the ratio 1 : 2.

Paint B is made up from blue and red paint in the ratio 1 : 8.

She can mix the paints to produce different shades of purple. IfPaint A and Paint B come in same size cans, what is the leastnumber she would need of each type in order to produce purplepaint containing blue and red in the ratio 1 : 3.

Source: NRICH enriching mathematicshttp://nrich.maths.org

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Things to do

Outline of Activity and Solution:

1. Ask each group to present their solution to the problem on aposter.

2. Sort the solutions (same answer, similar methods, type ofexposition etc.) and critique other groups solutions.

3. Compare different solutions and find the correct one.

4. See if we can find a solution if we use the ratios part to whole.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Picture

blue

blue

blue

red

red

red

red

red

red

Paint A

blue

red

red

red

red

red

red

red

red

Paint B

Purple Haze and Purple Rain MSP Math Circle Summer Camp

One Solution

From the picture above we can see that one can of Paint Acontains 3 parts of blue and 6 parts of red while one can of PaintB contains 1 part of blue and 8 parts of red. It might be a goodidea to imagine that one can contains 9 ounces of paint. Note thatcombining one can of Paint A with one can of Paint B yields amixture that contains 4 ounces of blue paint and 14 ounces of redpaint, ratio of 4 : 14. The table below lists the number of ouncesof blue and the number of ounces of red paint for thecorresponding mixtures of cans of A and B. Note that 5 cans of Aand 3 cans of B yield 18 ounces of blue and 54 ounces of red, aratio of 18:54, which equals the desired ratio of 1:3.

Paint A

Paint B

1 2 3 4 5

1 4:14 7:20 10:26 13:32 16:38

2 5:22 8:28 11:34 14:40 17:46

3 6:30 9:36 12:42 15:48 18:54

4 7:38 10:44 13:50 16:56 19:62

5 8:46 11:52 14:58 17:64 20:70Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Another Solution

Let’s start with one can of Paint A.

What we What we Too much blue or Correctionwant have too little blue?

1:3 3:6 too much blue add B

1:3 4:14 too little blue add A

1:3 7:20 too much blue add B

1:3 8:28 too little blue add A

1:3 11:34 too little blue add A

1:3 14:40 too much blue add B

1:3 15:48 too little blue add A

1:3 18:54 just right done!

We started with one can of A and added 4 more cans of A and 3cans of B. We used 5 cans of A and 3 cans of B.Does this algorithm always work?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Note that this argument uses the ratio of part to whole.1/3 of a can of Paint A is blue and 1/9 of one can of Paint B isblue.

Both cans are of the same size.We want to mix x cans of Paint A and y cans of paint B in orderto get the equivalent of x + y cans of a certain mixed paint, 1/4 ofwhich is blue.If we total the amount of red paint in x cans of A and y cans of B,we obtain

x · 1

3+ y · 1

9.

On the other hand x + y cans containing 1/4 of red paint eachwould give the following total of red paint

(x + y) · 1

4.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Note that this argument uses the ratio of part to whole.1/3 of a can of Paint A is blue and 1/9 of one can of Paint B isblue. Both cans are of the same size.

We want to mix x cans of Paint A and y cans of paint B in orderto get the equivalent of x + y cans of a certain mixed paint, 1/4 ofwhich is blue.If we total the amount of red paint in x cans of A and y cans of B,we obtain

x · 1

3+ y · 1

9.

On the other hand x + y cans containing 1/4 of red paint eachwould give the following total of red paint

(x + y) · 1

4.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Note that this argument uses the ratio of part to whole.1/3 of a can of Paint A is blue and 1/9 of one can of Paint B isblue. Both cans are of the same size.We want to mix x cans of Paint A and y cans of paint B in orderto get the equivalent of x + y cans of a certain mixed paint, 1/4 ofwhich is blue.

If we total the amount of red paint in x cans of A and y cans of B,we obtain

x · 1

3+ y · 1

9.

On the other hand x + y cans containing 1/4 of red paint eachwould give the following total of red paint

(x + y) · 1

4.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Note that this argument uses the ratio of part to whole.1/3 of a can of Paint A is blue and 1/9 of one can of Paint B isblue. Both cans are of the same size.We want to mix x cans of Paint A and y cans of paint B in orderto get the equivalent of x + y cans of a certain mixed paint, 1/4 ofwhich is blue.If we total the amount of red paint in x cans of A and y cans of B,we obtain

x · 1

3+ y · 1

9.

On the other hand x + y cans containing 1/4 of red paint eachwould give the following total of red paint

(x + y) · 1

4.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Note that this argument uses the ratio of part to whole.1/3 of a can of Paint A is blue and 1/9 of one can of Paint B isblue. Both cans are of the same size.We want to mix x cans of Paint A and y cans of paint B in orderto get the equivalent of x + y cans of a certain mixed paint, 1/4 ofwhich is blue.If we total the amount of red paint in x cans of A and y cans of B,we obtain

x · 1

3+ y · 1

9.

On the other hand x + y cans containing 1/4 of red paint eachwould give the following total of red paint

(x + y) · 1

4.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Setting the two sides equal to each other gives us

x · 1

3+ y · 1

9= (x + y) · 1

4.

Multiplying both sides with the common denominator of 36 yields

12x + 4y = 9(x + y) = 9x + 9y

Collecting terms results in

3x = 5y

orx

y=

5

3

Hence the ratio of number of cans of Paint A to the number ofcans of Paint B should be 5 : 3 in order to obtain the desiredmixture. We need at least 5 cans of A and 3 cans of B.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Setting the two sides equal to each other gives us

x · 1

3+ y · 1

9= (x + y) · 1

4.

Multiplying both sides with the common denominator of 36 yields

12x + 4y = 9(x + y) = 9x + 9y

Collecting terms results in

3x = 5y

orx

y=

5

3

Hence the ratio of number of cans of Paint A to the number ofcans of Paint B should be 5 : 3 in order to obtain the desiredmixture. We need at least 5 cans of A and 3 cans of B.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Setting the two sides equal to each other gives us

x · 1

3+ y · 1

9= (x + y) · 1

4.

Multiplying both sides with the common denominator of 36 yields

12x + 4y = 9(x + y) = 9x + 9y

Collecting terms results in

3x = 5y

orx

y=

5

3

Hence the ratio of number of cans of Paint A to the number ofcans of Paint B should be 5 : 3 in order to obtain the desiredmixture. We need at least 5 cans of A and 3 cans of B.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Yet Another Solution

Setting the two sides equal to each other gives us

x · 1

3+ y · 1

9= (x + y) · 1

4.

Multiplying both sides with the common denominator of 36 yields

12x + 4y = 9(x + y) = 9x + 9y

Collecting terms results in

3x = 5y

orx

y=

5

3

Hence the ratio of number of cans of Paint A to the number ofcans of Paint B should be 5 : 3 in order to obtain the desiredmixture. We need at least 5 cans of A and 3 cans of B.

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Poker

Three players, Peggy, Jason and Susan, participated in SouthAlabama Poker Championship. At the beginning of the night theamount of money each had was in the ratios 7 : 6 : 5. At the endof the night the ratio was 6 : 5 : 4. One of the players won $1, 200.How much money did the players have at the beginning of theevening?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Crunchy

A farmer is supplying a mix of seeds, nuts and dried fruits to amanufacturer of crunchy cereal bars. The ingredients cost:

dried fruits $7 per pound

nuts $6 per pound

seeds $4 per pound

He has been asked to supply a mix which costs $5 per pound.What combination of ingredients could he supply?

Purple Haze and Purple Rain MSP Math Circle Summer Camp

Buying Shoes

Serena and LeBron went shopping for shoes at the Shoe Store,which was having a sale. The sign in the window said: “Buy onepair, get the second pair (of equal or lesser value) for half price.Discount taken at the register.” Serena found a pair of shoes sheliked marked $60, and LeBron found a pair he liked marked $40.When they got to the register, the cashier rang up the sale andsaid “That will be $80.” How much of the $80 should each ofthem pay?

Purple Haze and Purple Rain MSP Math Circle Summer Camp