What time was it when she looked at her watch for the second time?

Question

At a certain time, Janice notices that her digital watch reads a minutes after 2:00. Fifteen minutes later, it reads b minutes after 3:00. She is amused to note that a is six times b. (Source: Mandelbrot)

Answer 1

The problem states that when Janice looked at her digital watch for the second time, it read b minutes after 3:00. 

Therefore we can say that b = a + 15 - 60.

 

We have to subtract 60 from a + 15 because sometime during the 15 minutes it went from 2 something to 3 something.  Once 60 minutes are reached, the hour value increases by 1 hour, and the minutes start at 0 again (or, you can say 60 minutes are subtracted to begin at 0 again).

 

(For example, say Janice first looked at her watch at 2:57.  This would make a equal to 57.  15 minutes later, when Janice looked at her watch a second time, we know that it would show 3:12.  If her watch did not subtract the 60 minutes it would show the time as 2:72, which we know is not the correct time.)

 

Janice first looked at her watch:

a minutes after 2:00

 

Janice looked at her watch for a second time:

b minutes after 3:00

 

What we need to find out:  The time when Janice looked at her watch a second time.  This means we need to find out what b is.

 

Given to us:

a = 6 X b

 

Therefore we have one formula that was given to us and one formula that we came up with:

a = 6 X b  (let's simply this formula to a = 6b)

b = a + 15 - 60 (let's simplify this formula to b = a - 45

 

Reviewing our notes/the question, we need to solve for b.

 

Using the two formulas just mentioned, we can use substitution to solve for b.

 

Since we need to know what b is, let's use the formula b = a - 45:

b = a - 45

 

The other formula we have says a = 6b; so, let's substitute 6b for the a in our formula b = a - 45:

b = a - 45

b = 6b - 45

 

We can now solve for b:

b = 6b - 45

 

(First step: Subtract 6b from each side)

b - 6b = -45

 

(Second step:  Simplify)

-5b = -45

 

(Third step:  Divide by -5) 

b = 9

 

Let's review our notes:

   Janice looked at her watch for a second time:
   "b" minutes after 3:00

 

We can conclude that it was 3:09 when Janice looked at her watch for the second time.

Answer 2

2:00+a+15=3:00+b

a=6b

you have to let 2:00=120 minutes and let 3:00=180 minutes otherwise you would be adding minutes to hours

120+a+15=180+b

substitute

120+6b+15=180+b

135+6b=180+b

6b=180-135+b

6b=45+b

6b-b=45

5b=45

b=9 and a=6*9=54

first time looking at her watch it was 2:54

second time, 15 minutes later, it was 3:09

check: from 2:54 to 3:09 is 15 minutes

Answer 3

Let's define one more variable to better visualize this:

 

c : the time between 2:a and 3:00

 

We know that c + b = 15 (1)

We know that a = 6b (2) 

 

We also can infer that a + c = 60 (3) (because it rounds out the hour)

 

With these three equations, we can solve for three variables

 

Rewrite equation (1)

c = 15 - b

Substitute into equation (3)

a + 15 - b = 60

a - 45 = b

Substitute into equation (2)

a = 6a - 270

5a = 270

 

a = 54

b = 9