The Sum Of The Digits On A Digital Clock Is 15. The Number Of Minutes Is 5 Times The Number Of Hours What Time Is It?

Case1: the number of hours is single digit

hours --- h
unit digit of minutes --x
tens digit of minutes --y
x+10y = 5h **

x+y+h = 15
h = 15-x-y
5h = 75 - 5x-5y ***

equating ** and ***

x+10y = 75-5x-5y
6x + 15y = 75
2x + 5y = 25
x = (25-5y)/2
--> 25 - 5y must be even so it divides evenly by 2,
values of 25-5y , 0 ≤ y≤ 5 :
25
20 <--- y = 1 , x = 10 , not possible
15
10 <--- y = 3 , x = 5 , h = 15-3-5 = 7
5
0

the time is 7:35

check:
6+3+5 = 15 , check
35 = 5(7), check

To determine the time on a digital clock given the sum of the digits and the relationship between the number of minutes and hours, we can follow the steps below:

Step 1: Understand the problem.
The problem states that the sum of the digits on a digital clock is 15, and that the number of minutes is 5 times the number of hours.

Step 2: Represent the numbers as variables.
Let's assume the hour digits as x and the minute digits as y. Therefore, the sum of the digits becomes x + y = 15.

Step 3: Express the relationship between minutes and hours.
The problem states that the number of minutes is 5 times the number of hours. As there are two hour digits and two minute digits, we can express this relationship as: (10x + y) minutes = 5 hours. However, as there are 60 minutes in an hour, we can rewrite it as: 60(10x + y) minutes = 5 hours.

Step 4: Convert the hours to minutes.
In order to compare the minutes and hours, we need to convert the time from hours to minutes. As there are 60 minutes in an hour, we multiply the number of hours by 60. This gives us 60 * 5 hours = 300 minutes.

Step 5: Set up the equation.
Now we can set up an equation combining the sum of the digits and the relationship between minutes and hours:
x + y = 15
60(10x + y) = 300

Step 6: Solve the equations simultaneously.
By solving these two equations simultaneously, we can find the values of x and y, which represent the hour and minute digits, respectively.

Step 7: Find the time.
Once you have found the values of x and y, you can determine the time on the digital clock. The hour digits are represented by x, while the minute digits are represented by y. Therefore, the time is given as x:y (in a 24-hour format).

By following these steps, you should be able to determine the time on the digital clock.