A red light flashes every 2 minutes and a blue light flashes every 3.5 minutes. Suppose both lights flash together at noon. What is the first time after 1 p.m. that both lights will flash together?
After 1:00, when they last flashed together:
When they first coincide again, let the red light have flashed x time
When they first coincide again, let the blue light have flashed y time
then 2x = 3.5y
4x = 7y
smallest value: x = 7 and y = 4
time = 4x = 28 minutes , so they will coincide at 1:28 pm
check:
red will flash at: 1:00, 1:02, ... 1:26, 1:28, 1:30, ...
blue will flash at 1:00, 1:03.5, 1:07, 1:10.5, 1:14, 1:17.5, 1:21, 1:24.5, 1:28, 1:31.5 ...
To find the first time after 1 p.m. that both lights will flash together, we need to find the least common multiple (LCM) of the two flashing intervals.
The red light flashes every 2 minutes, and the blue light flashes every 3.5 minutes. To find the LCM, we can write the two intervals as fractions with a common denominator:
Red light: 2/1 (flashes every 2 minutes)
Blue light: 7/2 (flashes every 3.5 minutes)
Now, let's find the LCM of these fractions:
The prime factorization of 2 is 2.
The prime factorization of 7 is 7.
The prime factorization of 2 is 2.
The LCM is found by taking the highest power of each prime factor:
LCM = 2^1 * 7^1 = 14.
So, the LCM of 2/1 and 7/2 is 14.
Since both lights flash together at noon, we need to find the first multiple of 14 after 1 p.m.
1 p.m. can be represented as 60 minutes.
To find the next multiple of 14 after 60 minutes, we divide 60 by 14 and take the ceiling value to get the smallest whole number greater than or equal to the result:
ceil(60/14) = 5.
Therefore, the next multiple of 14 after 60 minutes is 14 * 5 = 70 minutes.
Converting 70 minutes back to the 12-hour clock format, we have 1 p.m. + 70 minutes = 2:10 p.m.
So, the first time after 1 p.m. that both lights will flash together is at 2:10 p.m.