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Difference between revisions of "2019 AMC 8 Problems/Problem 14"

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==Solution 1==
 
==Solution 1==
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Let <math>Day 1</math> to <math>Day 2</math> denote a day where one coupon is redeemed and the day when the second coupon is redeemed.
  
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If she starts on a <math>Monday</math> she redeems her next coupon on <math>Thursday</math>.
 +
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<math>Thursday</math> to <math>Sunday</math>.
 +
 +
Thus <math>\boxed{\textbf{(A)}\ Monday}</math> is incorrect.
 +
 +
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If she starts on a <math>Tuesday</math> she redeems her next coupon on <math>Friday</math>.
 +
 +
<math>Friday</math> to <math>Monday</math>.
 +
 +
<math>Monday</math> to <math>Thursday</math>.
 +
 +
<math>Thursday</math> to <math>Sunday</math>.
 +
 +
Thus <math>\boxed{\textbf{(B)}\ Tuesday}</math> is incorrect.
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If she starts on a <math>Wednesday</math> she redeems her next coupon on <math>Saturday</math>.
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 +
<math>Saturday</math> to <math>Tuesday</math>.
 +
 +
<math>Tuesday</math> to <math>Friday</math>.
 +
 +
<math>Friday</math> to <math>Monday</math>.
 +
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<math>Monday</math> to <math>Thursday</math>.
 +
 +
And on <math>Thursday</math> she redeems her last coupon.
 +
 +
No sunday occured thus <math>\boxed{\textbf{(C)}\ Wednesday}</math> is correct.
 +
 +
Checking for the other options,
 +
 +
If she starts on a <math>Thursday</math> she redeems her next coupon on <math>Sunday</math>.
 +
 +
Thus <math>\boxed{\textbf{(D)}\ Thursday}</math> is incorrect.
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If she starts on a <math>Friday</math> she redeems her next coupon on <math>Monday</math>.
 +
 +
<math>Monday</math> to <math>Thursday</math>.
 +
 +
<math>Thursday</math> to <math>Sunday</math>.
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Checking for the other options gave us negative results, thus the answer is <math>\boxed{\textbf{(C)}\ Wednesday}</math>
  
 
==See Also==
 
==See Also==

Revision as of 14:12, 20 November 2019

Problem 14

Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every $10$ days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the $6$ dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?

$\textbf{(A) }$Monday$\qquad\textbf{(B) }$Tuesday$\qquad\textbf{(C) }$Wednesday$\qquad\textbf{(D) }$Thursday$\qquad\textbf{(E) }$Friday

Solution 1

Let $Day 1$ to $Day 2$ denote a day where one coupon is redeemed and the day when the second coupon is redeemed.

If she starts on a $Monday$ she redeems her next coupon on $Thursday$.

$Thursday$ to $Sunday$.

Thus $\boxed{\textbf{(A)}\ Monday}$ is incorrect.


If she starts on a $Tuesday$ she redeems her next coupon on $Friday$.

$Friday$ to $Monday$.

$Monday$ to $Thursday$.

$Thursday$ to $Sunday$.

Thus $\boxed{\textbf{(B)}\ Tuesday}$ is incorrect.

If she starts on a $Wednesday$ she redeems her next coupon on $Saturday$.

$Saturday$ to $Tuesday$.

$Tuesday$ to $Friday$.

$Friday$ to $Monday$.

$Monday$ to $Thursday$.

And on $Thursday$ she redeems her last coupon.

No sunday occured thus $\boxed{\textbf{(C)}\ Wednesday}$ is correct.

Checking for the other options,

If she starts on a $Thursday$ she redeems her next coupon on $Sunday$.

Thus $\boxed{\textbf{(D)}\ Thursday}$ is incorrect.

If she starts on a $Friday$ she redeems her next coupon on $Monday$.

$Monday$ to $Thursday$.

$Thursday$ to $Sunday$.

Checking for the other options gave us negative results, thus the answer is $\boxed{\textbf{(C)}\ Wednesday}$

See Also

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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