Difference between revisions of "1999 AMC 8 Problems/Problem 18"

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==Problem==
 
==Problem==
  
At Stinky Fat Papa Middle School the 108 students who take the Fat Papa Test meet in the evening to talk about food and eat an average of two full size, double chocolate, and creamy cakes apiece. Walter and Gretel are baking Bonnie's Smelliest Bar Cookies this year. Their recipe, which makes a pan of 15 cakes, lists this 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 pounds. They will make only full recipes, not partial recipes.
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At Central Middle School the 108 students who take the AMC8 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 Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this 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 only full recipes, not partial recipes.
  
They learn that a big concert is scheduled for the same night and attendance will be down 25%. How many recipes of cakes should they make for their smaller party?
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They learn that a big concert is scheduled for the same night and attendance will be down 25%. How many recipes of cookies should they make for their smaller party?
  
 
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11</math>
 
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11</math>
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===Solution 1===
 
===Solution 1===
  
If <math>108</math> students eat <math>2</math> cakes on average, there will need to be <math>108\cdot 2 = 216</math> cakes.  But with the smaller attendance, you will only need <math>100\% - 25\% = 75\%</math> of these cakes, or <math>75\% \cdot 216 = 0.75\cdot 216 = 162</math> cakes.
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If <math>108</math> students eat <math>2</math> cookies on average, there will need to be <math>108\cdot 2 = 216</math> cookies.  But with the smaller attendance, you will only need <math>100\% - 25\% = 75\%</math> of these cookies, or <math>75\% \cdot 216 = 0.75\cdot 216 = 162</math> cookies.
  
<math>162</math> cakes requires <math>\frac{162}{15} = 10.8</math> batches.  However, since half-batches are forbidden, we must round up to get <math>\lceil \frac{162}{15} \rceil = 11</math> batches, and the correct answer is <math>\boxed{E}</math>.
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<math>162</math> cookies requires <math>\frac{162}{15} = 10.8</math> batches.  However, since half-batches are forbidden, we must round up to get <math>\lceil \frac{162}{15} \rceil = 11</math> batches, and the correct answer is <math>\boxed{E}</math>.
  
===Solution papa ===
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===Solution 2===
  
If there were <math>108</math> students before, with the <math>25\%</math> of students missing, there will be <math>75\%</math> of <math>108</math> students left.  This is <math>75\% \cdot 108 =  0.75 \cdot 108 = 81</math> students.  These students eat <math>81 \cdot 2 = 162</math> cakes.  Follow the logic of the second paragraph above to find that there needs to be <math>11</math> batches, and the correct answer is <math>\boxed{E}</math>.
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If there were <math>108</math> students before, with the <math>25\%</math> of students missing, there will be <math>75\%</math> of <math>108</math> students left.  This is <math>75\% \cdot 108 =  0.75 \cdot 108 = 81</math> students.  These students eat <math>81 \cdot 2 = 162</math> cookies.  Follow the logic of the second paragraph above to find that there needs to be <math>11</math> batches, and the correct answer is <math>\boxed{E}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=1999|num-b=17|num-a=19}}
 
{{AMC8 box|year=1999|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:57, 31 October 2016

Problem

At Central Middle School the 108 students who take the AMC8 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 Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this 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 only full recipes, not partial recipes.

They learn that a big concert is scheduled for the same night and attendance will be down 25%. How many recipes of cookies should they make for their smaller party?

$\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$

Solution

Solution 1

If $108$ students eat $2$ cookies on average, there will need to be $108\cdot 2 = 216$ cookies. But with the smaller attendance, you will only need $100\% - 25\% = 75\%$ of these cookies, or $75\% \cdot 216 = 0.75\cdot 216 = 162$ cookies.

$162$ cookies requires $\frac{162}{15} = 10.8$ batches. However, since half-batches are forbidden, we must round up to get $\lceil \frac{162}{15} \rceil = 11$ batches, and the correct answer is $\boxed{E}$.

Solution 2

If there were $108$ students before, with the $25\%$ of students missing, there will be $75\%$ of $108$ students left. This is $75\% \cdot 108 =  0.75 \cdot 108 = 81$ students. These students eat $81 \cdot 2 = 162$ cookies. Follow the logic of the second paragraph above to find that there needs to be $11$ batches, and the correct answer is $\boxed{E}$.

See Also

1999 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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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