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# How long will it take for Henry to catch up?

Sarah and Henry are reading the same book. Sarah reads 1/3 page per minute and Henry reads 3/4 page per minute. Sarah has already read 70 pages while Henry has read 30. If they both resume reading eventually Henry will catch up to Sarah. On what page will that occur. How many minutes have they read when Henry catches up?

### 3 Answers by Expert Tutors

Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
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Mental Math:
In each minute, Henry catches up (3/4) - (1/3) = 5/12 pages. In (70-30)/(5/12) = 96 minutes, Henry will catch up to Sarah.

At the time Henry catches up to Sarah, they both read 30+(3/4)(96) = 102 pages. So, Henry read 102/(3/4) = 136 minutes, and Sarah read 102/(1/3) = 306 minutes.

I love more efficient ways to solve a problem. You've wisely calculated the number of pages that the gap is reduced each minute (5/12 pages per minute or 5 pages per 12 minutes). So how many of these 12 minute intervals (call this X) will it take to bring the gap to 0?

40 pages - X*5 pages/12 mins = 0
40 pages * 12 minutes / 5 pages = X. All our units cancel nicely, and:
X = 96 minutes.

Brilliant!
4.9 4.9 (233 lesson ratings) (233)
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Hey Tania -- working in hours might be nice ... S reads 20p/hr, H reads 45p/hr ... STAIR-STEP 'em:

S 70, 90, 110 ... back up in 0.2-hr steps ... 106, 102, 98, 94
H 30, 75, 120 ... back up in 0.2-hr steps ... 111, 102 ====> both land at p102 in 96 mins

S already read for 3.5hr or 210 mins ==> 306 min total
H already read for 2/3 hr or 40 mins ==> 136 min total ... Regards, ma'am :)
Alex S. | Patient and Knowledgeable Math and Economics TutorPatient and Knowledgeable Math and Econo...
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EDIT: SEE ROBERT J'S BETTER, MORE, EFFICIENT AND MORE INTUITIVE ANSWER ABOVE.

To read 70 pages has taken Sarah:
70 pages * (3 minutes / page ) = 210 minutes.

To read 30 pages has taken Henry:
30 pages* (4 minutes / 3 pages) = 40 minutes

This is THE STARTING POINT.

Sarah reads X more pages (total = 70+X) and Henry reads Y more pages (total = 30+Y). Once Henry catches up, they'll have each read a total of:

70 + X = 30 + Y. Simplifying (always do this when possible),
Y = X + 40.

We want to solve for either X or Y to tell us when they've read the same total number of pages. This is THE ENDING POINT.

Now, from THE STARTING POINT it will take the same amount of time for each person to reach THE ENDING POINT. (Please ask if you don't understand why this must be true).

Sarah will have spent: X pages*(3 minutes / page) = 3X minutes
Henry will have spent: Y pages * (4 minutes / 3 pages) = (4/3)Y minutes

So, 3X = (4/3)Y. Simplifying,
Y = 9X/4

Now we have two equations, so we can solve for X or Y.
Y = X+40
Y = 9X/4

Here's how I did it:
X + 40 = 9X/4
4X + 160 = 9X
160 = 5X
X = 32

So Sarah (and Henry) will have read a total of 70 + (32) = 102 pages!
And, from THE STARTING POINT, it will have taken them an additional 3*(32) = 96 minutes.