# Cycling 4482

Cycling races had 4 stages. The length of the first was 1/3 of the length of the entire track. The length of the second stage was 2/9 of the length of the entire track, and the length of the third stage was 1/4 of the track. The fourth stage was 15 km shorter than the second stage. How long was the entire race track?

x =  540 km

## Step-by-step explanation:

x=a+b+c+d
a = x/3
b = 2/9 x
c = 1/4x
d = b-15

x=a+b+c+d
a = x/3
b = 2/9·x
c = 1/4·x
d = b-15

a+b+c+d-x = 0
3a-x = 0
9b-2x = 0
4c-x = 0
b-d = 15

Pivot: Row 1 ↔ Row 2
3a-x = 0
a+b+c+d-x = 0
9b-2x = 0
4c-x = 0
b-d = 15

Row 2 - 1/3 · Row 1 → Row 2
3a-x = 0
b+c+d-0.66667x = 0
9b-2x = 0
4c-x = 0
b-d = 15

Pivot: Row 2 ↔ Row 3
3a-x = 0
9b-2x = 0
b+c+d-0.66667x = 0
4c-x = 0
b-d = 15

Row 3 - 1/9 · Row 2 → Row 3
3a-x = 0
9b-2x = 0
c+d-0.44444x = 0
4c-x = 0
b-d = 15

Row 5 - 1/9 · Row 2 → Row 5
3a-x = 0
9b-2x = 0
c+d-0.44444x = 0
4c-x = 0
-d+0.22222x = 15

Pivot: Row 3 ↔ Row 4
3a-x = 0
9b-2x = 0
4c-x = 0
c+d-0.44444x = 0
-d+0.22222x = 15

Row 4 - 1/4 · Row 3 → Row 4
3a-x = 0
9b-2x = 0
4c-x = 0
d-0.19444x = 0
-d+0.22222x = 15

Row 5 + Row 4 → Row 5
3a-x = 0
9b-2x = 0
4c-x = 0
d-0.19444x = 0
0.02778x = 15

x = 15/0.02777778 = 540
d = 0+0.19444444444444x = 0+0.19444444 · 540 = 105
c = 0+x/4 = 0+540/4 = 135
b = 0+2x/9 = 0+2 · 540/9 = 120
a = 0+x/3 = 0+540/3 = 180

a = 180
b = 120
c = 135
d = 105
x = 540

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