- Three hockey teams, the Crabs, the Eels, and the Seals, played each other once with no shutouts ( every team scored at least 1 goal in each game). A team gets 10 points for a win, 5 point for a tie and one point for each goal scored. If the Crabs received 20 points, the Eels 19 points and the Seals 5 points,what was the score of the game between the Crabs and the Eels?
- X is defined as one more than the number of digits in the integer x. For example , 1000 = 5 . If x has 900 digiits , find the value of x
what is the answer?
What is the answer to the value of X
To determine the score of the game between the Crabs and the Eels, we can use the given information about the points each team received.
The Crabs received 20 points, which can be broken down into wins, ties, and goals. Let's assume they won x games, tied y games, and scored z goals. The total number of points they received can then be expressed as:
10x + 5y + z = 20 (equation 1)
Similarly, the Eels received 19 points, so we can set up a similar equation:
10a + 5b + c = 19 (equation 2)
Now, let's analyze the information given in the question:
1. Every team scored at least 1 goal in each game.
This means that both z and c should be greater than or equal to 1.
2. No shutouts occurred.
This means that both x and a should be greater than or equal to 1.
Next, we can try to eliminate variables and simplify the equations. Since both equations involve three unknowns, we need additional information to solve the system of equations. Let's assume one of the teams won all their games.
If we assume x = 3 (all wins for the Crabs), the first equation becomes:
10(3) + 5y + z = 20
30 + 5y + z = 20
5y + z = -10 (equation 3)
Since y and z should be positive values, equation 3 implies that the assumption x = 3 is incorrect.
Let's try assuming x = 2 (two wins for the Crabs). The first equation becomes:
10(2) + 5y + z = 20
20 + 5y + z = 20
5y + z = 0 (equation 4)
Since both y and z should be positive values, equation 4 implies that the assumption x = 2 is incorrect as well.
Now, let's assume x = 1 (one win for the Crabs). The first equation becomes:
10(1) + 5y + z = 20
10 + 5y + z = 20
5y + z = 10 (equation 5)
Note that when x = 1, the Crabs score a maximum of 10 points from wins alone, which matches the equation on the right side.
Now we can substitute equation 5 into equation 2:
10a + 5b + c = 19
10a + 5b + 5y + z = 19
10a + 5b + 5(10 - z) + z = 19 (since equation 5 states 5y + z = 10)
10a + 5b + 50 - 5z + z = 19
10a + 5b - 4z + 50 = 19
10a + 5b - 4z = -31 (equation 6)
We can now solve equations 5 and 6 simultaneously:
Equation 5: 5y + z = 10 (equation 5)
Equation 6: 10a + 5b - 4z = -31 (equation 6)
By trial and error, we can find the solution to be y = 1, a = 1, b = 1, and z = 3.
Therefore, the score of the game between the Crabs and the Eels is 3-1 in favor of the Crabs.
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Now, let's move on to the second question.
According to the information given, X is defined as one more than the number of digits in the integer x. We are required to find the value of x when it has 900 digits.
To solve this, we can set up the equation:
X = 1 + number of digits in x
Since x has 900 digits, we can express this as:
X = 1 + 900
Simplifying the equation, we find:
X = 901
So, the value of x when it has 900 digits is 901.