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Difference between revisions of "1999 AMC 8 Problems/Problem 19"

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==Problem==
 
==Problem==
  
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At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookeis this year. Their recipe, which makes a pan of 15 cookies, lists these items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops They will only make full recipes, not partial recipes.
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At Central Middle School, the 108 students who take the AMC 8 meet in the evening to talk about food and eat an average of two cookies apiece. Hansel and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists these items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make full recipes, not partial recipes.
  
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The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.)
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Hansel and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be leftover, of course.)
  
 
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math>
 
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math>
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==Solution==
 
==Solution==
  
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For <math>216</math> cakes, you need to make <math>\frac{216}{15} = 14.4</math> pans.  Since fractional pans are forbidden, round up to make <math>\lceil \frac{216}{15} \rceil = 15</math> pans.
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For <math>216</math> cookies, you need to make <math>\frac{216}{15} = 14.4</math> pans.  Since fractional pans are forbidden, round up to make <math>\lceil \frac{216}{15} \rceil = 15</math> pans.
  
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There are <math>3</math> tablespoons of butter per pan, meaning <math>3 \cdot 15 = 45</math> tablespoons of butter is required for <math>15</math> pans.
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There are <math>3</math> tablespoons of butter per pan, meaning <math>3 \cdot 15 = 45</math> tablespoons of butter are required for <math>15</math> pans.
  
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Each stick of butter has <math>8</math> tablespoons, so we need <math>\frac{45}{8} = 5.625</math> sticks of butter.  However, we must round up again because partial sticks of butter are forbidden since they will be eaten in one gulp. Thus, we need <math>\lceil \frac{45}{8} \rceil = 6</math> sticks of butter, and the answer is <math>\boxed{B}</math>.
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Each stick of butter has <math>8</math> tablespoons, so we need <math>\frac{45}{8} = 5.625</math> sticks of butter.  However, we must round up again because partial sticks of butter are forbidden! Thus, we need <math>\lceil \frac{45}{8} \rceil = 6</math> sticks of butter, and the answer is <math>\boxed{B}</math>.
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==Video Solution==
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https://youtu.be/jg2xcwR-y6o Soo, DRMS, NM
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== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
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 +
https://www.youtube.com/watch?v=WU0hVEyevgE
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=1999|num-b=18|num-a=20}}
 
{{AMC8 box|year=1999|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 21:26, 24 January 2024

Problem

At Central Middle School, the 108 students who take the AMC 8 meet in the evening to talk about food and eat an average of two cookies apiece. Hansel and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists these items: $1\frac{1}{2}$ cups flour, $2$ eggs, $3$ tablespoons butter, $\frac{3}{4}$ cups sugar, and $1$ package of chocolate drops. They will make full recipes, not partial recipes.

Hansel and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be leftover, of course.)

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

Solution

For $216$ cookies, you need to make $\frac{216}{15} = 14.4$ pans. Since fractional pans are forbidden, round up to make $\lceil \frac{216}{15} \rceil = 15$ pans.

There are $3$ tablespoons of butter per pan, meaning $3 \cdot 15 = 45$ tablespoons of butter are required for $15$ pans.

Each stick of butter has $8$ tablespoons, so we need $\frac{45}{8} = 5.625$ sticks of butter. However, we must round up again because partial sticks of butter are forbidden! Thus, we need $\lceil \frac{45}{8} \rceil = 6$ sticks of butter, and the answer is $\boxed{B}$.

Video Solution

https://youtu.be/jg2xcwR-y6o Soo, DRMS, NM

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=WU0hVEyevgE

See Also

1999 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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