Difference between revisions of "2007 AMC 8 Problems/Problem 20"
(→Solution 5) |
(→Solution 6) |
||
Line 9: | Line 9: | ||
<math>\frac{x}{y}=0.45</math> and <math>\frac{x+6}{y+8}=0.5.</math> | <math>\frac{x}{y}=0.45</math> and <math>\frac{x+6}{y+8}=0.5.</math> | ||
We first multiply both sides of the first equation by <math>y</math> to get <math>x=0.45y.</math> Then, we multiply both sides of the second equation by <math>(y+8)</math> to get <math>x+6=0.5(y+8).</math> Applying the Distributive Property gives yields <math>x+6=0.5y+4.</math> Now we substitute <math>0.45y</math> for <math>x</math> to get <math>0.45y+6=0.5y+4.</math> Solving gives us <math>y=40.</math> Since the problem asks for the total number of games, we add on the last 8 games to get the solution <math>\boxed{\textbf{(A)}\ 48}</math>. | We first multiply both sides of the first equation by <math>y</math> to get <math>x=0.45y.</math> Then, we multiply both sides of the second equation by <math>(y+8)</math> to get <math>x+6=0.5(y+8).</math> Applying the Distributive Property gives yields <math>x+6=0.5y+4.</math> Now we substitute <math>0.45y</math> for <math>x</math> to get <math>0.45y+6=0.5y+4.</math> Solving gives us <math>y=40.</math> Since the problem asks for the total number of games, we add on the last 8 games to get the solution <math>\boxed{\textbf{(A)}\ 48}</math>. | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
==Solution 7== | ==Solution 7== |
Revision as of 08:56, 18 June 2024
Contents
[hide]Problem
Before the district play, the Unicorns had won % of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
Solution 1
At the beginning of the problem, the Unicorns had played games and they had won of these games. From the information given in the problem, we can say that Next, the Unicorns win 6 more games and lose 2 more, for a total of games played during district play. We are told that they end the season having won half of their games, or of their games. We can write another equation: This gives us a system of equations: and We first multiply both sides of the first equation by to get Then, we multiply both sides of the second equation by to get Applying the Distributive Property gives yields Now we substitute for to get Solving gives us Since the problem asks for the total number of games, we add on the last 8 games to get the solution .
Solution 7
Let be the total number of games before the district play. The Unicorns have wins, therefore the rest are losses. But after the district play, they won 6 and lost 2 more games. We can solve for x by forming the equation . Subtracting and from both sides gives us , and from here we multiply both sides by 10 to get . We are not finished yet as the problem is asking for the total games (which includes the games after the district play), so we add 8 to our value of x to get our answer which is .
- LearnForEver
Solution 8 (Answer Choices)
We can check each answer choice from left to right to see which one is correct. Suppose the Unicorns played games in total. Then, after district play, they would have won games. Now, consider the situation before district play. The Unicorns would have won games out of . Converting to a percentage, %. Thus, the answer is .
Note: If A didn't work, we would have similarly tested the other choices until we found one that did.
Video Solution by OmegaLearn
https://youtu.be/rQUwNC0gqdg?t=1993
~pi_is_3.14
See Also
2007 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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.