Printer A can print a certain number of pages in 7 hours and printer B can print the same number in 3 hours. Both printed for 5 hours, and printer A produced 1800 pages in that time. How many pages can printer B print in 5 hours?

Step-by-step solution

1. 1

   Convert times into rates. Let $N$ be the size of the shared job in pages. Then

   $r_A=\frac{N}{7}\ \text{pages/h},\qquad r_B=\frac{N}{3}\ \text{pages/h}.$

   Rates, not times, are what add and scale linearly — this is the move that makes every work problem routine.

2. 2

   Use A's actual output to find its rate. Printer A produced $1800$ pages in $5$ h, so

   $r_A=\frac{1800}{5}=360\ \text{pages/h}.$

3. 3

   Recover the job size N. Since $r_A=\tfrac{N}{7}$,

   $N=7r_A=7(360)=2520\ \text{pages}.$

4. 4

   Find B's rate.

   $r_B=\frac{N}{3}=\frac{2520}{3}=840\ \text{pages/h}.$

   Notice $r_B/r_A=\tfrac{7}{3}$: B is faster in exactly the inverse ratio of the times, which is the shortcut route to the same place.

5. 5

   Multiply by the 5 hours.

   $1800\times\frac{7}{3}=840\times 5=4200\ \text{pages}.$

6. 6

   Sanity-check the result. In $5$ h, B does $\tfrac53$ of the whole job ($5$ h out of the $3$ h it needs), and $\tfrac53\times 2520=4200$ ✓. Likewise A does $\tfrac57$ of the job: $\tfrac57\times 2520=1800$ ✓, matching the given figure.