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Math - One-Half versus One-Third of Pizza











































Choice of Pizza

How to answer: you are offered the choice of a third of a pizza or half a pizza. Because you are hungry and love pizza, you choose the half. Your friend gets a third of a pizza but ends up with more than you!!! How can this be?

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Dec 12, 2012
Choice of Pizza
by: Staff



Answer

Part I

Obviously your friend’s pizza is larger than your pizza.

We assume the pizzas are formed in the traditional round shape, so the radius of your friend’s pizza is larger than the radius of your pizza.

The area of a pizza is:

APizza = πr²


The area of your pizza


Ayou = area of pizza eaten by you

ryou = radius of pizza eaten by you

Ayou = πr²you


The area of your friend’s pizza

Afriend = area of pizza eaten by friend

rfriend = radius of pizza eaten by friend

Afriend = πr²friend


At what radius will the one-third the area of the friend’s pizza exceed one-half the area of your pizza?

⅓ * πr²friend > ½ * πr²you


Solve for rfriend


Multiply each side of the inequality by 3

3 * (⅓) * πr²friend > 3 * (½) * πr²you

1 * πr²friend > 3/2 * πr²you

πr²friend > 3/2 * πr²you


Divide each side of the inequality by π

πr²friend / π > 3/2 * πr²you / π

friend * (π/ π) > 3/2 * r²you * (π/ π)

friend * 1 > 3/2 * r²you * 1

friend > 3/2 * r²you

Take the square root of each side of the inequality

√(r²friend) > √(3/2 * r²you)

rfriend > √(3/2) * √(r²you)

rfriend > 1.2247448713916 * ryou

rfriend > 1.225 * ryou

If the radius of your friend’s pizza is over 22.5% larger than your pizza, then ⅓ of your friend’s pizza will be greater than ½ of your pizza.

For example, if your pizza has a radius of 5 inches, and your friend’s pizza has a radius of 6.2 inches, then ⅓ of your friend’s pizza will be greater than ½ of your pizza.

⅓ * πr²friend > ½ * πr²you

⅓ * π * (6.2)² > ½ * π * (5)²

⅓ * π * (38.44) > ½ * π * (25)

40.2542738679972 in² > 39.2699081698724 in²


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Dec 12, 2012
Choice of Pizza
by: Staff


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Part II

Math – Example - ⅓ of a larger pizza is greater than ½ of a smaller pizza.







Thanks for writing.

Staff
www.solving-math-problems.com


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