Here’s a problem from Khan Academy’s SAT practice section. Please explain this one for me.

A marine aquarium has a small tank and a large tank, each containing only red and blue fish. In each tank, the ratio of red fish to blue fish is 3 to 4. The ratio of fish in the large tank to fish in the small tank is 46 to 5. What is the ratio of blue fish in the small tank to red fish in the large tank?

A ratio of 3 red to 4 blue means the total must be a multiple of 7, so it might help you to first multiply the totals in each tank by 7. The 46:5 ratio for large tank to small tank becomes 322:35. Let’s just pretend those are the actual numbers of fish in the tank.

In the small tank, 4/7 of the 35 fish in the tank are blue, so 20 fish are blue.

In the large tank, 3/7 of the 322 fish are red, so 138 fish are red.

Therefore, the ratio of blue fish in the small tank to red fish in the big tank is 20:138, which simplifies to 10:69.

Note that while multiplying everything by 7 was helpful (in my opinion) in making the math more intuitive, you don’t need to. You can simply enter the following in your calculator and ask it to convert to a fraction:

\left(\dfrac{4}{7}\times 5\right)\div\left(\dfrac{3}{7}\times 46\right)