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Difference between revisions of "2009 Grade 8 CEMC Gauss Problems/Problem 7"

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==Problem==
 
==Problem==
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The number of faces (<math>F</math>), vertices (<math>V</math>), and edges (<math>E</math>) of a polyhedron are related by the equation <math>F + V - E = 2</math>. If a polyhedron has <math>6</math> faces and <math>8</math> vertices, how many edges does it have?
+
Kayla went to the fair with <math>\$100</math>. She spent <math>\frac14</math> of her <math>\$100</math> on rides, and <math>\frac{1}{10}</math> of her <math>\$100</math> on food. How much money did she spend?
  
−
<math> \text{ (A) }\ 12 \qquad\text{ (B) }\ 14 \qquad\text{ (C) }\ 16 \qquad\text{ (D) }\ 18 \qquad\text{ (E) }\ 10 </math>
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<math> \text{ (A) }\ \$65 \qquad\text{ (B) }\ \$32.50 \qquad\text{ (C) }\ \$2.50 \qquad\text{ (D) }\ \$50 \qquad\text{ (E) }\ \$35 </math>
 
==Solution 1==
 
==Solution 1==
−
We can use the equation provided in the problem, and plug in <math>6</math> for <math>F</math>, and <math>8</math> for <math>V</math>:
+
We can calculate how much she spent on her rides, then the amount she spent on food, and then add them together.
  
−
<math>6 + 8 - E = 2</math>
+
For the rides, she spent:
  
−
We can combine <math>6</math> and <math>8</math> to get:
+
<math>\frac14 \times \$100 = \$25</math>
  
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<math>14 - E = 2</math>
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For the food, she spent:
  
−
Adding <math>E</math> to both sides, we get:
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<math>\frac{1}{10} \times \$100 = \$10</math>
  
−
<math>E + 2 = 14</math>
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Thus, altogether, she spent:
  
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Subtracting <math>2</math> from both sides of the equation, we get:
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<math>\$25 + \$10 = \boxed {\textbf {(E) } \$35}</math>
−
 
 
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<math>E = \boxed {\textbf {(A) } 12}</math>
 
  
 
~anabel.disher
 
~anabel.disher
 
==Solution 2==
 
==Solution 2==
−
We can remember that a rectangular prism has <math>6</math> faces, <math>8</math> vertices, and <math>\boxed {\textbf {(A) } 12}</math> edges, without doing any calculation.
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We can combine the fractions to see what fraction of the <math>\$100</math> she spent altogether:
 +
 
 +
<math>\frac14 + \frac{1}{10} = \frac{1 \times 5}{4 \times 5} + \frac{1 \times 2}{10 \times 2} = \frac{5}{20} + \frac{2}{20} = \frac{7}{20}</math>
 +
 
 +
We can now multiply this by the <math>\$100</math> she was given to see how much she spent altogether:
 +
 
 +
<math>\frac{7}{20} \times \$100 = \boxed {\textbf {(E) } \$35}</math>
  
 
~anabel.disher
 
~anabel.disher

Latest revision as of 23:09, 18 June 2025

Problem

Kayla went to the fair with $$100$. She spent $\frac14$ of her $$100$ on rides, and $\frac{1}{10}$ of her $$100$ on food. How much money did she spend?

$\text{ (A) }\ $65 \qquad\text{ (B) }\ $32.50 \qquad\text{ (C) }\ $2.50 \qquad\text{ (D) }\ $50 \qquad\text{ (E) }\ $35$

Solution 1

We can calculate how much she spent on her rides, then the amount she spent on food, and then add them together.

For the rides, she spent:

$\frac14 \times $100 = $25$

For the food, she spent:

$\frac{1}{10} \times $100 = $10$

Thus, altogether, she spent:

$$25 + $10 = \boxed {\textbf {(E) } $35}$

~anabel.disher

Solution 2

We can combine the fractions to see what fraction of the $$100$ she spent altogether:

$\frac14 + \frac{1}{10} = \frac{1 \times 5}{4 \times 5} + \frac{1 \times 2}{10 \times 2} = \frac{5}{20} + \frac{2}{20} = \frac{7}{20}$

We can now multiply this by the $$100$ she was given to see how much she spent altogether:

$\frac{7}{20} \times $100 = \boxed {\textbf {(E) } $35}$

~anabel.disher

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