Question

In the 2006 Winter Olympics, Sweden won 3 medals more than China. Germany secured one more than twice the number of medals bagged by sweden. If 54 medals were won in all, how many medals did Germany secure?

Answer 1

Let the number of medals won by Sweden be \( S \), the number of medals won by China be \( C \), and the number of medals won by Germany be \( G \).

From the problem, we can set up the following equations based on the information given:

1. Sweden won 3 medals more than China:
\[
S = C + 3
\]

2. Germany secured one more than twice the number of medals won by Sweden:
\[
G = 2S + 1
\]

3. The total number of medals won is 54:
\[
S + C + G = 54
\]

Now we can substitute the expression for \( C \) from the first equation into the third equation. From \( S = C + 3 \), we can rearrange it to find \( C \):
\[
C = S - 3
\]

Now substitute \( C \) in the total:
\[
S + (S - 3) + G = 54
\]
This simplifies to:
\[
2S - 3 + G = 54
\]
\[
2S + G = 57 \quad \text{(Equation 1)}
\]

Next, substitute the expression for \( G \) from the second equation into Equation 1:
\[
2S + (2S + 1) = 57
\]
\[
4S + 1 = 57
\]
Subtract 1 from both sides:
\[
4S = 56
\]
Now divide by 4:
\[
S = 14
\]

Now we can find \( C \) using \( C = S - 3 \):
\[
C = 14 - 3 = 11
\]

Now, substituting \( S \) into the equation for \( G \):
\[
G = 2S + 1 = 2(14) + 1 = 28 + 1 = 29
\]

Thus, the number of medals won by Germany is \( \boxed{29} \).

Lastly, let's verify the total:
- Sweden: \( S = 14 \)
- China: \( C = 11 \)
- Germany: \( G = 29 \)

Calculating the total:
\[
S + C + G = 14 + 11 + 29 = 54
\]

The total is correct, confirming \( G = 29 \) is indeed the right answer.