Fractions of an hour and fractions of a minute

We are very familiar with certain fractions of an hour — half an hour is 30 minutes, or three quarters of an hour is 45 minutes. As 60 seconds make a minute, fractions of a minute work the same way. Three quarters of a minute is 45 seconds.

Big Ben displays 25 minutes past 12 but what fraction of an hour is this?
Is this 12:25 or five twelfths past twelve?

Getting used to other fractions of an hour or minute helps solve many practical arithmetic questions. For instance, to find three fifths of an hour, we calculate one fifth by 60 ÷ 5 = 12 minutes, then multiply by three to give 3 × 12 = 36 minutes. Similarly to find 7/10 of one minute, we find one tenth as 60 ÷ 10 = 6 seconds, then seven tenths is 7 × 6 = 42 seconds. The important fractions of sixty follow easily learned patterns, which you can memorise from the sheet below. Like learning your common percentage, fraction and decimal conversions, this saves calculations — and vital time — in mental arithmetic tests.

If a minibus drives at an average speed of 50 miles per hour, how many minutes does it take to reach its destination 30 miles away? Using the formula time = distance ÷ speed and writing the division as a fraction gives 30/50 which simplifies to 3/5 of an hour, which we know is 36 minutes.

A teacher can write 18 reports in one hour. How long will it take her to write 42 reports? We know 18 reports take 1 hour, 36 reports take 2 hours and so on. We need to know how many eighteens fit into 42. Written as a fraction, \frac{42}{18} = 2 \frac{6}{18} = 2 \frac{1}{3}, and one third of an hour is 20 minutes, so the answer is 2 hours 20 minutes.

An alternative to using fractions of sixty in speed and rates questions is to use proportional reasoning. If a minibus drives at an average speed of 50 miles per hour, how many minutes does it take to reach its destination 30 miles away? We know the minibus goes 50 miles in 60 minutes; we can simplify this ratio down (divide each each side by five) to 5 miles in 6 minutes, then build the ratio up (multiply both sides by six) to get the desired 30 miles in 36 minutes.

A teacher can write 18 reports in one hour. How long will it take her to write 42 reports? We know 18 reports take 60 minutes but can’t easily build this up to 42 as that isn’t divisible by 18. However, we can simplify the ratio by dividing both sides by 3, to see 6 reports take 20 minutes. Then we multiply both sides by 7 to build up to the required 42 reports in 140 minutes, which is 2 hours 20 minutes. (It’s even easier to spot that 36 of the reports take 2 hours and we need to know how long the remaining 6 reports take. Since 18 reports take 60 minutes, dividing by 3 shows the last 6 reports take 20 minutes.)