math problem, not answer but ways to solve it .?

Question

A student finishes the first half of an exam in 2/3 of the time it takes to finish the second half. If the entire exam takes him an hour, how many minutes does he spend on the first half of the exam?


The answer is 24, what I need is different, quick and varied ways to solve it, "lots of ideas " sometimes I use the longest ways and that is not helpful during any kind of test. I appreciate your help!

Answer 1

Well, the way I do it is with algebra. Time spent on first half = x, say, and time on second half = y.

Then:
2/3*x = y (since the first half takes 2/3 the time the second does)
and
x+y = 60 (since the two halves of the test take 60 minutes)

Solve for x, and you've got your answer. =) I would solve by substitution-- since y = 2/3 * x, just plug that into the second equation to get x + 2/3 * x = 60. Then 5/3*x = 60, so multiply both sides by 3/5 to get x.

All my other ways of doing it are variants of the above, really. You can divide the time taken for the test into fractions-- Since the first half takes 2/3 the time of the second half, I treat the second half as 3 fractions of the total and the first part as 2. Divide the total time by 5, (the total number of fractions) and then multiply by 2, to give the time needed for the first half.

Answer 2

First, what is the total time the student does the test in?

If you take the time he does the second half of the exam in as [x], then the time for the first half is 2/3 [x]. The total time would be:
[x]+ 2/3 [x] = 5/3[x]
This is the key to solving this equation. Find the total time expressed in the variable units.

So, then we find that the total time is 60 minutes, giving us:

5/3 [x] = 60
or
[x] = 60* 3/5 = 36

And thus the time for the first half would be 2/3* 36 = 24 minutes.

Good luck.

Answer 3

Let the first half of the paper be denoted as x

and the latter half as y

Now, in an hour there are 60 min, hence

x + y = 60 ..... (1)

From the data,

x = 2/3y .... (2)

Substitute the value of x in (1), we get,

2/3*y + y = 60

5y/3 = 60

5y = 180

y = 36 minutes

substituting the value of y in (2) we get,

x = 2y/3

x = (2 * 36)/3

x = 24 minutes

Answer 4

Let X be the time it takes him to solve the second half.

So the time it takes him to solve the first half is 2/3 times X.

Now, the total time is 1 hour (60 minutes), so:

X + (2/3)X = 60 minutes

or

(5/3) x = 60 minutes


From there, you get X.

And then (2/3) times X is the time it takes him to solve the first half.

Answer 5

Whatever you want to find (you can apply in future problems) is the variable and we will call it x.

therefore 1 hour = (2/3)x + x

(5/3)x = 60 mins

x = 60*3/5 = 36mins
(2/3)x = 24mins

Answer 6

let he finishes 2nd half in x min.
then he finishes 1st half in 2/3 *x min
total time = 1 hour = 60 min
or, 2/3 x + x = 60
or, 5/3 x = 60
or, x = 3/5*60
or, x =36 min.
he finishes 2nd half in 36 min
he finishes 1st half in (2/3 *36 =) 24 mins

Answer 7

Just divide that hour into 5 equal time periods. The first half took two of them, and the second half took three of them.

Answer 8

let x = time to finish 1st half
y = time to finish 2nd half
x = 2y/3
x + y = 1 hr = 60 min.
Substitute for the variable you DONT want:
y = 3x/2
x + 3x/2 = 60
2x + 3x = 120
5x = 120
x = 24

Answer 9

let x= time for 1st half of exam
then 3/23 x= time for 2nd //2
(3/2)x+x=1hr or 60min
(5/2) x=60
x=60*2/5=24
Fast & direct.

Answer 10

You could look at it as (2+3)time=60minutes
then it's obvious that time=12minutes and you need 2*time?

best of luck - Mike