Math / Algebra

Question46 On a circular track of unknown length L meters, three runners-A, B, and $C$ -start running simultaneously from the same point in the same direction. Their speeds are distinct positive integers $\mathrm{v}_{\mathrm{A}}>\mathrm{V}_{\mathrm{B}}>\mathrm{v}_{\mathrm{C}}$ meters per second respectively. They all meet at the starting point exactly after 1 hour. During this time A overtakes B exactly 15 times, B overtakes C exactly 10 times. Then find the minimum possible value of L in meters

Studdy Solution

STEP 1

1. The track is circular with an unknown length $L$ .
2. Runners A, B, and C start simultaneously and meet at the starting point after 1 hour.
3. The speeds $v_A > v_B > v_C$ are distinct positive integers.
4. A overtakes B 15 times, and B overtakes C 10 times in 1 hour.

STEP 2

1. Establish the relationship between the speeds and the number of overtakes.
2. Use the conditions to set up equations.
3. Solve the equations to find the minimum possible value of $L$ .

STEP 3

The runners meet at the starting point after 1 hour, implying that the least common multiple (LCM) of their lap times equals 1 hour (3600 seconds).

STEP 4

The number of overtakes is determined by the relative speeds: - A overtakes B 15 times, so $v_A - v_B = \frac{15L}{3600}$ . - B overtakes C 10 times, so $v_B - v_C = \frac{10L}{3600}$ .

STEP 5

Simplify the equations: - $v_A - v_B = \frac{L}{240}$ - $v_B - v_C = \frac{L}{360}$

STEP 6

Express $L$ in terms of $v_A, v_B, v_C$ : - From $v_A - v_B = \frac{L}{240}$ , $L = 240(v_A - v_B)$ . - From $v_B - v_C = \frac{L}{360}$ , $L = 360(v_B - v_C)$ .
Set the two expressions for $L$ equal: $240(v_A - v_B) = 360(v_B - v_C)$

STEP 7

Solve for the relationship between $v_A, v_B, v_C$ : $2(v_A - v_B) = 3(v_B - v_C)$ $2v_A - 2v_B = 3v_B - 3v_C$ $2v_A = 5v_B - 3v_C$

STEP 8

Find the smallest integer values for $v_A, v_B, v_C$ that satisfy the equation and the conditions $v_A > v_B > v_C$ : - Assume $v_C = 1$ , then $v_B = 2$ , and solve for $v_A$ : $2v_A = 5(2) - 3(1)$ $2v_A = 10 - 3$ $2v_A = 7$ $v_A = 3.5$ (not an integer)
Try $v_C = 2$ , $v_B = 3$ : - $2v_A = 5(3) - 3(2)$ - $2v_A = 15 - 6$ - $2v_A = 9$ - $v_A = 4.5$ (not an integer)
Try $v_C = 3$ , $v_B = 4$ : - $2v_A = 5(4) - 3(3)$ - $2v_A = 20 - 9$ - $2v_A = 11$ - $v_A = 5.5$ (not an integer)
Try $v_C = 4$ , $v_B = 5$ : - $2v_A = 5(5) - 3(4)$ - $2v_A = 25 - 12$ - $2v_A = 13$ - $v_A = 6.5$ (not an integer)
Try $v_C = 5$ , $v_B = 6$ : - $2v_A = 5(6) - 3(5)$ - $2v_A = 30 - 15$ - $2v_A = 15$ - $v_A = 7$ (integer)

STEP 9

Calculate $L$ using $v_A = 7$ , $v_B = 6$ , $v_C = 5$ : - $L = 240(v_A - v_B) = 240(7 - 6) = 240$ - $L = 360(v_B - v_C) = 360(6 - 5) = 360$
The minimum possible value of $L$ is the least common multiple of 240 and 360, which is 720.
The minimum possible value of $L$ is:
$\boxed{720}$

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Studdy Solution

STEP 1

1. The track is circular with an unknown length $L$ .
2. Runners A, B, and C start simultaneously and meet at the starting point after 1 hour.
3. The speeds $v_A > v_B > v_C$ are distinct positive integers.
4. A overtakes B 15 times, and B overtakes C 10 times in 1 hour.

STEP 2

1. Establish the relationship between the speeds and the number of overtakes.
2. Use the conditions to set up equations.
3. Solve the equations to find the minimum possible value of $L$ .

STEP 3

The runners meet at the starting point after 1 hour, implying that the least common multiple (LCM) of their lap times equals 1 hour (3600 seconds).

STEP 4

The number of overtakes is determined by the relative speeds: - A overtakes B 15 times, so $v_A - v_B = \frac{15L}{3600}$ . - B overtakes C 10 times, so $v_B - v_C = \frac{10L}{3600}$ .

STEP 5

Simplify the equations: - $v_A - v_B = \frac{L}{240}$ - $v_B - v_C = \frac{L}{360}$

STEP 6

Express $L$ in terms of $v_A, v_B, v_C$ : - From $v_A - v_B = \frac{L}{240}$ , $L = 240(v_A - v_B)$ . - From $v_B - v_C = \frac{L}{360}$ , $L = 360(v_B - v_C)$ .
Set the two expressions for $L$ equal: $240(v_A - v_B) = 360(v_B - v_C)$

STEP 7

Solve for the relationship between $v_A, v_B, v_C$ : $2(v_A - v_B) = 3(v_B - v_C)$ $2v_A - 2v_B = 3v_B - 3v_C$ $2v_A = 5v_B - 3v_C$

STEP 8

STEP 9

Calculate $L$ using $v_A = 7$ , $v_B = 6$ , $v_C = 5$ : - $L = 240(v_A - v_B) = 240(7 - 6) = 240$ - $L = 360(v_B - v_C) = 360(6 - 5) = 360$
The minimum possible value of $L$ is the least common multiple of 240 and 360, which is 720.
The minimum possible value of $L$ is:
$\boxed{720}$

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Studdy Solution

STEP 1

1. The track is circular with an unknown length L L L.2. Runners A, B, and C start simultaneously and meet at the starting point after 1 hour.3. The speeds vA>vB>vC v_A > v_B > v_C vA​>vB​>vC​ are distinct positive integers.4. A overtakes B 15 times, and B overtakes C 10 times in 1 hour.

STEP 2

1. Establish the relationship between the speeds and the number of overtakes.2. Use the conditions to set up equations.3. Solve the equations to find the minimum possible value of L L L.

STEP 3

The runners meet at the starting point after 1 hour, implying that the least common multiple (LCM) of their lap times equals 1 hour (3600 seconds).

STEP 4

The number of overtakes is determined by the relative speeds: - A overtakes B 15 times, so vA−vB=15L3600 v_A - v_B = \frac{15L}{3600} vA​−vB​=360015L​. - B overtakes C 10 times, so vB−vC=10L3600 v_B - v_C = \frac{10L}{3600} vB​−vC​=360010L​.

STEP 5

Simplify the equations: - vA−vB=L240 v_A - v_B = \frac{L}{240} vA​−vB​=240L​ - vB−vC=L360 v_B - v_C = \frac{L}{360} vB​−vC​=360L​

STEP 6

STEP 7

Solve for the relationship between vA,vB,vC v_A, v_B, v_C vA​,vB​,vC​: 2(vA−vB)=3(vB−vC) 2(v_A - v_B) = 3(v_B - v_C) 2(vA​−vB​)=3(vB​−vC​) 2vA−2vB=3vB−3vC 2v_A - 2v_B = 3v_B - 3v_C 2vA​−2vB​=3vB​−3vC​ 2vA=5vB−3vC 2v_A = 5v_B - 3v_C 2vA​=5vB​−3vC​

STEP 8

STEP 9

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1. The track is circular with an unknown length $L$ .
2. Runners A, B, and C start simultaneously and meet at the starting point after 1 hour.
3. The speeds $v_A > v_B > v_C$ are distinct positive integers.
4. A overtakes B 15 times, and B overtakes C 10 times in 1 hour.

1. Establish the relationship between the speeds and the number of overtakes.
2. Use the conditions to set up equations.
3. Solve the equations to find the minimum possible value of $L$ .

The number of overtakes is determined by the relative speeds: - A overtakes B 15 times, so $v_A - v_B = \frac{15L}{3600}$ . - B overtakes C 10 times, so $v_B - v_C = \frac{10L}{3600}$ .

Simplify the equations: - $v_A - v_B = \frac{L}{240}$ - $v_B - v_C = \frac{L}{360}$

Solve for the relationship between $v_A, v_B, v_C$ : $2(v_A - v_B) = 3(v_B - v_C)$ $2v_A - 2v_B = 3v_B - 3v_C$ $2v_A = 5v_B - 3v_C$