Difference between revisions of "1999 AMC 8 Problems/Problem 17"

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==Problem==
 
==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: <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.
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At Fat Papa Middle School the 108 students who take the Papa meet in the evening to talk about food and eat an average of two full size, double chocalate, creamy cream 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 cakes. They will make only full recipes, not partial recipes.
  
Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)
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Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cakes? (Some eggs and some cakes may be left over.)
  
 
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 15</math>
 
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 15</math>
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==Solution==
 
==Solution==
  
If <math>108</math> students eat <math>2</math> cookies on average, there will need to be <math>108\cdot 2 = 216</math> cookies.  There are <math>15</math> cookies per pan, meaning there needs to be <math>\frac{216}{15} = 14.4</math> pans.  However, since half-recipes are forbidden, we need to round up and make <math>\lceil \frac{216}{15}\rceil = 15</math> pans.
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If <math>108</math> students eat <math>2</math> cakes on average, there will need to be <math>108\cdot 2 = 216</math> cakes.  There are <math>15</math> cakes per pan, meaning there needs to be <math>\frac{216}{15} = 14.4</math> pans.  However, since half-recipes are forbidden, we need to round up and make <math>\lceil \frac{216}{15}\rceil = 15</math> pans.
  
 
<math>1</math> pan requires <math>2</math> eggs, so <math>15</math> pans require <math>2\cdot 15 = 30</math> eggs.  Since there are <math>6</math> eggs in a half dozen, we need <math>\frac{30}{6} = 5</math> half-dozens of eggs, and the answer is <math>\boxed{C}</math>
 
<math>1</math> pan requires <math>2</math> eggs, so <math>15</math> pans require <math>2\cdot 15 = 30</math> eggs.  Since there are <math>6</math> eggs in a half dozen, we need <math>\frac{30}{6} = 5</math> half-dozens of eggs, and the answer is <math>\boxed{C}</math>

Revision as of 23:27, 28 October 2016

Problem

At Fat Papa Middle School the 108 students who take the Papa meet in the evening to talk about food and eat an average of two full size, double chocalate, creamy cream 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: $1\frac{1}{2}$ cups flour, $2$ eggs, $3$ tablespoons butter, $\frac{3}{4}$ cups sugar, and $1$ package of chocolate cakes. They will make only full recipes, not partial recipes.

Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cakes? (Some eggs and some cakes may be left over.)

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 15$

Solution

If $108$ students eat $2$ cakes on average, there will need to be $108\cdot 2 = 216$ cakes. There are $15$ cakes per pan, meaning there needs to be $\frac{216}{15} = 14.4$ pans. However, since half-recipes are forbidden, we need to round up and make $\lceil \frac{216}{15}\rceil = 15$ pans.

$1$ pan requires $2$ eggs, so $15$ pans require $2\cdot 15 = 30$ eggs. Since there are $6$ eggs in a half dozen, we need $\frac{30}{6} = 5$ half-dozens of eggs, and the answer is $\boxed{C}$


See Also

1999 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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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