Problems involving time and rates can vary drastically in difficulty. Let’s practice with some.

• A train travels at 60 miles per hour. How long will it take to go 180 miles?

A rate of 60 miles per hour means that the train will travel 60 miles in 1 hour. 180 / 60 equals 3, so the train will have to travel 60 miles 3 times to travel 180 miles. So, the train will travel for 1 hour 3 times, or 3 hours in total.

• A faucet fills 2 liters of water every 5 minutes. How much water will it fill in 15 minutes?

15 divided by 5 equals 3, so the faucet will fill water for three 5-minute durations. In 5 minutes, the faucet will fill 2 liters of water. So, in 15 minutes, the faucet will fill 2 x 3 = 6 liters of water.

Now, let’s try a problem that is a little more complicated.

• Two people start biking from the same point. Adam bikes at 5 mph and Bella at 7 mph. How far apart are they after 4 hours?

In one hour, Adam will bike 5 miles and Bella will bike 7 miles. The distance between them is 2 miles. After another hour, Adam will have biked 10 miles and Bella will have biked 14. Their distance increased by 2 again.

Every hour, the distance between Adam and Bella increases by 2. This means that after 4 hours, the distance between Adam and Bella will be 2 x 4, which is 8 miles.

Let’s try an even more complicated problem!

• A swimming pool fills in 12 hours with Pipe A and in 18 hours with Pipe B. If both pipes are open, how long will it take to fill the pool?

This problem seems impossible at first, but it is very doable if we think about it intuitively. Pipe A fills one entire pool in 12 hours. This means that in one hour, it fills 1/12 of a pool. Similarly, Pipe B fills 1/18 of a pool in one hour.

Now, since both rates have been converted to identical units, we can add them. Both pipes together fill 1/12 + 1/18 of a pool in one hour, which equals 5/36 of a pool in one hour.

To completely fill up a pool, it will take 36/5 hours, since 36/5 x 5/36 = 1. If we want to simplify 36/5, we can convert it into a mixed number, which is 7 and 1/5 hours. 1/5 hours = 12 minutes, so the total time to fill the pool will be 7 hours and 12 minutes.

As you can see, time and rate problems can be difficult, but many similar tactics are widely applicable.