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Difference between revisions of "2009 AMC 10B Problems/Problem 1"

m (→‎Solution 1 (Algebra))
(Added in Sol 2.)
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~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
== Solution 2 (Answer Choices) ==
+
== Solution 2 (Arithmetic) ==
 +
In this solution, all amounts are in the unit of cents.
 +
 
 +
The amount spent on muffins must end in either <math>00</math> or <math>50,</math> and the amount spent on bagels must end in one of <math>00,25,50,</math> or <math>75.</math>
 +
 
 +
We have two possible cases:
 +
<ol style="margin-left: 1.5em;">
 +
  <li>The amounts spent on muffins and bagels both end in <math>00.</math></li><p>
 +
The number of muffins bought must be even, and the number of bagels bought must be a multiple of <math>4.</math> <p>
 +
In this case, there are no solutions.
 +
  <li>The amounts spent on muffins and bagels both end in <math>50.</math></li><p>
 +
The number of muffins bought must be odd, and the number of bagels bought must be <math>2</math> more than a multiple of <math>4.</math> <p>
 +
In this case, the only solution is <math>3</math> muffins and <math>2</math> bagels, from which the answer is <math>b=\boxed{\text{(B) } 2}.</math>
 +
</ol>
 +
 
 +
== Solution 3 (Answer Choices) ==
 
* If Jane bought <math>1</math> bagel, then she bought <math>4</math> muffins. Her total cost for the week would be <math>75\cdot1+50\cdot4=275</math> cents, or <math>2.75</math> dollars. So, <math>\text{(A)}</math> is incorrect.
 
* If Jane bought <math>1</math> bagel, then she bought <math>4</math> muffins. Her total cost for the week would be <math>75\cdot1+50\cdot4=275</math> cents, or <math>2.75</math> dollars. So, <math>\text{(A)}</math> is incorrect.
  

Revision as of 22:43, 9 June 2021

The following problem is from both the 2009 AMC 10B #1 and 2009 AMC 12B #1, so both problems redirect to this page.

Problem

Each morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?

$\text{(A) } 1\qquad\text{(B) } 2\qquad\text{(C) } 3\qquad\text{(D) } 4\qquad\text{(E) } 5$

Solution 1 (Algebra)

A muffin costed $2$ quarters, and a bagel costed $3$ quarters.

Suppose that Jane bought $m$ muffins and $b$ bagels, where $m$ and $b$ are nonnegative integers. We need:

  1. $m+b=5.$

  2. $2m+3b$ is divisible by $4.$

From Condition 2, it is clear that $b$ must be even, so we narrow down the choices to either $\text{(B)}$ or $\text{(D)}.$ By a quick inspection, the only solution is $(m,b)=(3,2),$ from which the answer is $b=\boxed{\text{(B) } 2}.$

~MRENTHUSIASM

Solution 2 (Arithmetic)

In this solution, all amounts are in the unit of cents.

The amount spent on muffins must end in either $00$ or $50,$ and the amount spent on bagels must end in one of $00,25,50,$ or $75.$

We have two possible cases:

  1. The amounts spent on muffins and bagels both end in $00.$

    2. The number of muffins bought must be even, and the number of bagels bought must be a multiple of $4.$

    In this case, there are no solutions.

  2. The amounts spent on muffins and bagels both end in $50.$

    3. The number of muffins bought must be odd, and the number of bagels bought must be $2$ more than a multiple of $4.$

    In this case, the only solution is $3$ muffins and $2$ bagels, from which the answer is $b=\boxed{\text{(B) } 2}.$

Solution 3 (Answer Choices)

  • If Jane bought $1$ bagel, then she bought $4$ muffins. Her total cost for the week would be $75\cdot1+50\cdot4=275$ cents, or $2.75$ dollars. So, $\text{(A)}$ is incorrect.
  • If Jane bought $2$ bagels, then she bought $3$ muffins. Her total cost for the week would be $75\cdot2+50\cdot3=300$ cents, or $3.00$ dollars. So, $\boxed{\text{(B) } 2}$ is correct.

For completeness, we will check $\text{(C)},\text{(D)},$ and $\text{(E)}$ too. If you decide to use this solution on the real test, then you will not need to do that, as you want to save more time.

  • If Jane bought $3$ bagels, then she bought $2$ muffins. Her total cost for the week would be $75\cdot3+50\cdot2=325$ cents, or $3.25$ dollars. So, $\text{(C)}$ is incorrect.
  • If Jane bought $4$ bagels, then she bought $1$ muffin. Her total cost for the week would be $75\cdot4+50\cdot1=350$ cents, or $3.50$ dollars. So, $\text{(D)}$ is incorrect.
  • If Jane bought $5$ bagels, then she bought $0$ muffins. Her total cost for the week would be $75\cdot5+50\cdot0=375$ cents, or $3.75$ dollars. So, $\text{(E)}$ is incorrect.

~MRENTHUSIASM

See also

2009 AMC 10B (Problems • Answer Key • Resources)
Preceded by
First Question 		Followed by
Problem 2
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 AMC 10 Problems and Solutions
2009 AMC 12B (Problems • Answer Key • Resources)
Preceded by
First Question 		Followed by
Problem 2
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 AMC 12 Problems and Solutions

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