Math 130

Practice with "Cut and Choose"

By Bronlyn Wassink

Review: Fractions, Decimals, and Percents

Before delving into the complexities of the Cut and Choose method, first I wish to talk to a little about percentages, fractions, decimals, and how these things are all related, and how knowledge of these concepts is the key to understanding this method (and many other things in life, too).

Most of you already have a notion of what a fraction, a decimal, and a percent is. If you don't see a TA in the math help room during office hours and he/she will explain it to you. The key to remember is that a fraction expresses a ratio. For example, suppose 3/4 of the math graduate students at BU are male. This means that it is possible to put all the math graduate students into groups of 4, and each of those groups will have 3 males (and 1 female) in them.

The next big step is knowing how to change between fractions, decimals, and percents.
  • Decimals to Percents: multiply your decimal by 100. (Example: .75 * 100 = 75%)
  • Percents to Decimals: divide your percent by 100. (Example: 82%/100 = .82)
  • Fractions to Decimals: use your calculator. (Example: 3/4 = .75)
  • Decimals to Fractions: First, take the decimal point out of your decimal to get a whole number. Put that whole number on the top of a fraction. Next, count the number of digits that are after the decimal point. A 1 followed by that many zeros goes on the bottom of the fraction.
    Examples: .3458 = 3,458/10,000
    .679215=679,215/1,000,000
    .85=85/100
  • Percents to Fractions: Change the percent to a decimal then change the decimal to a fraction.
  • Fractions to Percents: Change the fraction to a decimal then change the decimal to a percent.

    Using Fractions, Decimals, and Percents

    Percents and Decimals:

    A very common question that will arise several times in your life is "What is ___% of a certain number?".
    To find the answer to this, simply change the percent into a decimal, then multiply the decimal by the number.
    Examples:
    90% of 30 is .90 * 30 = 27
    50% of 59 is .50 * 59 = 29.5
    10% of 60 is .10 * 60 = 6
    A slightly more involved example: Suppose you have a cake that is half vanilla and half chocolate. Pretend for a minute that you know that Ann thinks the vanilla half is worth $15.00 and the chocolate half is worth $7.50. You are asked to find the value of a piece of the cake that has 20% of the vanilla half and 78% of the chocolate half.
  • 20% of the vanilla half is worth 20% of $15.00. This is .20 * 15.00 = $3.00.
  • 78% of the chocolate half is worth 78% of $7.50. This is .78 * 7.50 = $5.85.
  • So then this piece will be worth $3.00 + $5.85 = $8.85.
    You can also do so many fun things with this method, such as determining sale prices at a store and determining your overall course grades (using percentages given to you on the syllabus).

    Fractions and Ratios: Part 1

    Fractions are most helpful when determining the values for the different parts of the cake. For a person in a cake cutting problem (say Kate), you may be given certain preferences of hers. If she likes one flavor more than the other, then let "x" equal the value of the thing she likes the least. Then the other value is "x" times how ever much the problem tells you. Add these values up and you'll get the value of the whole cake. Then solve for x.
    For example: an $18 cake is half lemon, half chocolate. Kate likes chocolate 4 times as much as she likes lemon.

      x  =   Lemon
      +  4x  =   Chocolate
      5x  =   Whole Cake = $18

    Since (5x)/5 = x, and 5x = $18, we get x = (5x)/5 = ($18)/5 = $3.60. So the Lemon half is worth $3.60 to Kate.
    Once we know that x = $3.60, then 4x = 4 * $3.60 = $14.40. So to Kate, the Chocolate half is worth $14.40.

    Fractions and Ratios: Part 2

    Another place where we use fractions and ratios in Cake Cutting is deciding how much a piece of the cake will be worth to one person when we know the values for another person. This is best explained by example.
    Example:
    Two friends, Molly and Nate, are splitting a half chocolate, half vanilla cake. Suppose we have already found the monetary value for the Choc. and Vanilla parts (using the method above). We found:
     Molly: Choc. = $9.75   Nate: Choc. = $3.50
      Vanilla = $2.25   Vanilla= $8.50
    Suppose that for a piece of cake that has some chocolate and some vanilla, Molly thinks that the chocolate part of that piece is worth $3.90 and the vanilla part of that piece is worth $0.45. So, to Molly, the whole piece is worth $3.90 + $0.45 = $4.35.
    How much is this same piece worth to Nate?

  • Chocolate part: We must set up a fraction based on ratios.
    To Molly: the chocolate in this piece is worth $3.90 and all the chocolate is worth $9.75.
    To Nate: the chocolate in this piece is worth "X" and all the chocolate is worth $3.50.
    The numbers will be different, but the ratios will be the same.
       Molly  Nate
      Part (Choc.) $3.90  =   X
      Whole (Choc.) $9.75  $3.50
    (Please forgive the poor alignment.)
    Cross multiply and solve for X:
    $3.90 * $3.50 = $9.75 * X
    $13.65 = $9.75 * X (Now divide both sides by 9.75)
    $1.40 = X
    So Nate thinks the chocolate part of this piece is worth $1.40

  • Vanilla part: We will follow the same method as above. Suppose Nate thinks the vanilla part of this piece is worth "Y".
    The ratios will be:
       Molly  Nate
      Part (Vanilla) $.45  =   Y
      Whole (Vanilla) $2.25  $8.50
    Again, we will cross multiply and solve for Y.
    .45 * 8.85 = 2.25 * Y
    3.825 = 2.25 * Y (Do not round yet!!)
    $1.70 = Y (I divided both sides by 2.25.)
    So Nate thinks the vanilla part of this piece is worth $1.70

    Combining the chocolate and vanilla parts together, we found that this piece is worth $1.40 + $1.70 = $3.10 to Nate!
    Recall this piece is worth $4.35 to Molly.

    A Cake Cutting Example

    Please note: all the math that I will do is all explained in an earlier part of this page. So please refer back for more clarification if needed.
    Also note: I do not know how to draw pictures and put them on a webpage. I suggest that you draw pictures as you go over this problem.

    Three friends, Ann, Bob, and Chris buy an $18 pizza. Half is Mushroom, half is Pickels.
    Ann likes Mushrooms and Pickels the same.
    Bob likes Mushrooms twice as much as he likes Pickels.
    Chris likes Pickels three times as much as he likes Mushrooms.
    Ann will make the first cut, and Bob will choose first.

    First find out how much each part of the pizza is worth to each of them.
  • Ann: Let X = mushrooms. So X = Pickels, too.
    So X + X = the whole pizza
    So 2X = $18.00 (divide both sides by 2)
    So X = $9.00 = Mushrooms = Pickels

  • Bob: Let X = Pickels. Then 2X = Mushrooms.
    So X + 2X = whole pizza
    So 3X = $18.00 (Divide both sides by 3)
    So X = $6.00 = Pickels
    and 2X = $12.00 = Mushrooms

  • Chris: Let X = Mushrooms. So 3X = Pickels.
    So X + 3X = whole pizza
    So 4X = $18.00 (now divide both sides by 4)
    So X = $4.50 = Mushrooms
    and 3X = $13.50 = Pickels
    Now would be a great time to draw a few pictures for yourself.

    Ann makes the first cut:
    Ann cuts the pizza into two pieces, which in her opinion are equal to her.
  • Piece 1 has 3/4 of the Mushrooms and 1/4 of the Pickels.
  • Piece 2 has 1/4 of the Mushrooms and 3/4 of the Pickels.
    This is the same information as:
  • Piece 1 has 75% of the Mushrooms and 25% of the Pickels.
  • Piece 2 has 25% of the Mushrooms and 75% of the Pickels.
    You will be given the information in only one way. It is up to you to know that these two things are the same.

    Which piece does Bob choose?
    Piece 1 is worth .75 * $12.00 + .25 * $6.00 = $9.00 + $1.50 = $10.50
    Piece 2 is worth .25 * $12.00 + .75 * $6.00 = $3.00 + $4.50 = $7.50
    So Bob will choose piece 1 because it is worth more to him.

    Now Bob cuts his piece into 3 equal pieces.
    Bob has a piece worth $10.50. If we want 3 equal pieces, we want 3 pieces each worth ($10.50)/3 = $3.50.
    Hopefully you have a picture to look at. If not, draw one. What you'd be looking at: for the piece that Bob has chosen, the Mushroom part is worth $9.00 to him, and the Pickels part is worth $1.50 to him.
    Cut this into three pieces - let's call the pieces I, II, and III (Roman Numerals).
    I: has all $1.50 of the Pickels part and an additional $2.00 of the Mushroom part.
    II: has $3.50 of the Mushroom part.
    III: also has $3.50 of the Mushroom part.

    How much are each of these pieces worth to Chris?
    Pieces II and III:
    Pieces II and III will be worth the same, so I will do them before Piece I.
    Suppose piece II is worth X to Chris.
       Bob  Chris
      Part (value of II) $3.50  =   X
      Whole (value of total Mushroom) $12.00  $4.50

    Now we solve for X.
    $3.50 * $4.50 = $12.00 * X
    $15.75 = $12.00 * X (Now divide both sides by $12.00)
    $1.3125 = X
    Round to $1.31. So Pieces II and III are each worth $1.31 to Chris.

    Piece I: We will break this into two parts: the mushroom part and the pickels part.
  • Mushroom part:
    Let X be the value to Chris of the Mushroom part of piece I.
       Bob  Chris
      Part (value of Mush. in I) $2.00  =   X
      Whole (value of total Mush.) $12.00  $4.50

    Again, we need to solve for X.
    $2.00 * $4.50 = $12.00 * X
    $.75 = X (Try doing the work yourself)

  • Pickel part:
    Let Y be the value to Chris of the Pickel part of Piece I.
       Bob  Chris
      Part (value of Pickels in I) $1.50  =   Y
      Whole (value of total Pickels) $6.00  $13.50

    Cross multiply and solve for Y: $1.50 * $13.50 = $6.00 * Y
    So $3.375 = Y. Round this to Y = $3.38

    So Piece I is worth X + Y = $.75 + $3.38 = $4.13 to Chris.

    Chris will choose Piece I from Bob's half because it is worth the most to Chris.

    Please note: this is not an entire cake cutting example!
    We have just done about half of the work.
    To have a complete example:
  • Ann still has half of the pizza. She needs to cut her half (the half that Bob did not choose) in to 3 pieces that she thinks are equal (worth the same amount of money to her). Call these pieces IV, V and VI (again Roman numerals).
    Questions for you to answer: How much money are they all worth to Ann? What percent of the mushroom and pickels part is contained in each of the three pieces that Ann just cut? What are the angles involved? (see below for percentages and angles).
  • Chris chooses one of the three pieces that Ann just cut.
    Questions for you to answer: How much money does Chris think the three pices that Ann just cut are worth (to him)? Which of the three pieces does Chris choose?
  • Final step: draw the pizza with all 6 pieces. Write the names of the owners of each piece (everyone should have 2 pieces).
    Recall that we already know that of the 6 pieces I, II, III, IV, V, VI, Bob has II and III. Chris has piece I and either IV, V, or VI, whichever he chooses. Ann will have two out of the 3 pieces IV, V, and VI, whichever Chris does not choose.
    Questions for you: How much does Ann think that her 2 pieces are worth (the total of the 2)? How much does Bob think his 2 pieces are worth to him? How much does Chris think his 2 pieces are worth to him? Is this an envy-free division? Do all parties think they got at least their fair share of the pizza?

    Suppose you are asked for percentages.


    Percentages: We can use a ratio just like in everything else. It's slightly different, but with the same idea behind it as before.

  • Pieces II and III:
    Recall that pieces II and III are the identical.
    Suppose Piece II has X% of the total mushrooms. We need to find X.

       Money (to Bob)  Percent
      Part (Mushrooms in Piece II) $3.50  =   X%
      Whole (Mushrooms in whole pizza) $12.00  100%

    Now we cross multiply to solve for X.
    3.50 * 100 = 12.00 * X
    350 = 12.00 * X
    29.167 = X (Always round at least 3 decimal places)
    So Pieces II and III contain 29.167% of the mushrooms in the pizza.

    Note, if we did this part first, we could find that Chris thinks that Piece II is worth 29.167% of $4.50. This is .29167 * 4.50 = $1.3125, or $1.31. This is the same value we found before.
    If you solve this way, you do not need to solve the other way.


  • Piece I:
    Suppose Piece I has X% of Mushrooms and Y% of Pickels.
    First, let us work with the mushroom part and solve for X.

       Money (to Bob)  Percent
      Part (Mushrooms in Piece I) $2.00  =   X%
      Whole (Mushrooms in whole pizza) $12.00  100%

    Again, we cross multiply to find X.
    2.00 * 100 = 12.00 * X
    200 = 12.00 * X
    16.667 = X
    So Piece I has 16.667% of the Mushrooms.

    Now let's look at the Pickels in Piece I.

       Money (to Bob)  Percent
      Part (Pickels in Piece I) $1.50  =   Y%
      Whole (Pickels in whole pizza) $6.00  100%

    Cross multiply to find Y.
    1.50 * 100 = 6.00 * Y
    150 = 6.00 * Y
    25 = Y
    So Piece I has 25% of the Pickels and 16.667% of the Mushrooms.

    Note: this means that to Chris, Piece I is worth .25 * $13.50 + .16667 * $4.50 = $3.375 + $0.750015 = $4.125015 or $4.13. This is the same answer we found before.

    Suppose you are asked for angles.


    Angles: Angles can be computed using ratios, just like the previous methods.
    Remember that 180 degrees is a straight line - so the entire mushroom part of the pizza has 180 degrees, and the entire pickel part of the pizza also has 180 degrees.

  • Pieces II and III:
    Suppose the angle of the Mushrooms in Piece II is X degrees.
         Money (to Bob)    Angle (in degrees)
       Part (Mushrooms in Piece II)  $3.50   =   X
       Whole (Mushrooms in whole pizza)  $12.00    180

    You may have guessed by now that we cross multiply to find X.
    3.50 * 180 = 12.00 * X
    630 = 12.00 * X
    52.5 = X
    So Piece II (and also Piece III) has an angle of 52.5 degrees, which is all inside the Mushroom part.

  • Piece I:
    There are 2 angles in Piece I, the angle of the mushroom part and the angle of the pickel part. Let's do the mushroom part first.
         Money (to Bob)    Angle (in degrees)
       Part (Mushrooms in Piece I)  $2.00   =   X
       Whole (Mushrooms in whole pizza)  $12.00    180

    Hopefully you know to cross multiply by now.
    2.00 * 180 = 12.00 * X
    360 = 12.00 * X
    30 = X
    So the angle of the Mushroom part of Piece I is 30 degrees.

    Now Pickels:
       Money (to Bob)  Angle (in degrees)
      Part (Pickels in Piece I) $1.50  =   Y
      Whole (Pickels in whole pizza) $6.00  180

    1.50 * 180 = 6.00 * Y
    270 = 6.00 * Y
    45 = Y
    So the angle of the Pickel part of Piece I is 45 degrees.

    Aren't ratios fun? :-)

    Remember that we only looked at half of the pizza - that half that Bob chose after Ann did the first cut. To have a complete example, see the 3 steps listed before the "Percentages" began to remind yourself how to complete the example (and do the previous things to the other 3 pieces of the pizza).