Difference between revisions of "2024 AIME I Problems/Problem 1"
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Multiplying by <math>(s)(s+2)</math>, we get | Multiplying by <math>(s)(s+2)</math>, we get | ||
| − | <math>9s+18-9s=1.6s^{2} + 3.2s</math> | + | <math>9s+18-9s=18=1.6s^{2} + 3.2s</math> |
| + | |||
| + | Multiplying by 5/2 on both sides, we get | ||
| + | |||
| + | <math>0 = 4s^{2} + 8s - 45</math> | ||
| + | |||
| + | Factoring gives us | ||
| + | |||
| + | <math>(2x-5)(2x+9) = 0</math>, of which the solution we want is <math>x=2.5</math>. | ||
| + | |||
| + | Substituting this back to the first equation, we can find that <math>t = 0.4</math> hours. | ||
| + | |||
| + | Lastly, <math>x + \frac{1}{2} = 3</math> kilometers per hour, so | ||
| + | |||
| + | <math>\frac{9}{3} + 0.4 = 3.4</math> hours, or <math>204</math> minutes | ||
| + | |||
| + | -Failure.net | ||
Revision as of 13:21, 2 February 2024
Problem
Every morning Aya goes for a 
-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of 
kilometers per hour, the walk takes her 4 hours, including 
minutes spent in the coffee shop. When she walks 
kilometers per hour, the walk takes her 2 hours and 24 minutes, including minutes spent in the coffee shop. Suppose Aya walks at 
kilometers per hour. Find the number of minutes the walk takes her, including the minutes spent in the coffee shop.
Solution 1
in hours and 
in hours.
Subtracting the second equation from the first, we get,
Multiplying by 
, we get
Multiplying by 5/2 on both sides, we get
Factoring gives us
, of which the solution we want is 
.
Substituting this back to the first equation, we can find that 
hours.
Lastly, 
kilometers per hour, so
hours, or 
minutes
-Failure.net
