Math Detours · Grade 7 · Sharpen · 3

Half a mile in a quarter of an hour

A rate is "per one". When the amounts are fractions, find the amount for one whole and the question answers itself.

Name ____________________Date ____________

Tonight's idea

You walk half a mile in a quarter of an hour. How fast is that in miles per hour?

"Per hour" means "in one whole hour". So the real question is: how far do you go in 1 hour? Four quarter-hours make an hour, and each one covers half a mile: \(4 \times \tfrac12 = 2\) miles. Your speed is 2 miles per hour.

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Each quarter-hour step is half a mile. Four steps reach 1 hour, and 2 miles.

A rate compares two amounts, such as miles and hours. The unit rate is the amount for one whole of the second thing: per 1 hour, per 1 litre, per 1 batch. Once you have it, every other question is one multiplication. In \(1\tfrac12\) hours you walk \(2 \times 1\tfrac12 = 3\) miles.

The shortcut is to divide: \(\tfrac12 \div \tfrac14 = 2\). It is the same question in disguise: how many quarters fit into a half? (Dividing by a fraction was in Fractions that fight back.)

Unit rate = first amount ÷ second amount. Miles ÷ hours gives miles per hour.

Watch one: a snail crawls \(\tfrac34\) m in \(\tfrac25\) of an hour

The time is two fifths of an hour, so take two steps.

First find one fifth of an hour. Halve both amounts: \(\tfrac38\) m in \(\tfrac15\) hour.

Then five fifths make a whole hour. Multiply both by 5: \(5 \times \tfrac38 = \tfrac{15}{8}\) m in 1 hour. The snail's speed is \(1\tfrac78\) m per hour.

Dividing agrees: \(\tfrac34 \div \tfrac25 = \tfrac34 \times \tfrac52 = \tfrac{15}{8}\). Halve, then times 5. That is all "flip and multiply" ever does.

Your turn

  1. Find each unit rate. (a) \(\tfrac13\) km in \(\tfrac16\) hour, in km per hour. (b) \(\tfrac34\) of a page in \(\tfrac12\) a minute, in pages per minute. (c) \(2\tfrac12\) litres in \(\tfrac14\) minute, in litres per minute.
  2. \(\tfrac34\) cup of sugar makes \(\tfrac12\) a batch of biscuits. (a) How much sugar is that per batch? (b) How much sugar do \(2\tfrac12\) batches need?
  3. Ana walks \(\tfrac23\) mile in \(\tfrac14\) hour. Ben walks \(\tfrac34\) mile in \(\tfrac{3}{10}\) hour. Ben went further. Who is faster? Find both speeds.
  4. A hose fills \(\tfrac25\) of a pool in \(\tfrac34\) hour. (a) How much of the pool does it fill per hour? (b) How many hours does one whole pool take? (c) Multiply your two answers. Explain what you get.

Stretch: two taps

One tap fills a bath in \(\tfrac14\) hour. A slower tap fills it in \(\tfrac13\) hour. Find each tap's rate in baths per hour. With both taps on, how long does one bath take? Give the answer as a fraction of an hour, then in minutes.

Answers

  1. (a) \(\tfrac13 \div \tfrac16 = 2\) km per hour. (b) \(\tfrac34 \div \tfrac12 = 1\tfrac12\) pages per minute. (c) \(2\tfrac12 \div \tfrac14 = 10\) litres per minute.
  2. (a) Two halves make a batch, so \(2 \times \tfrac34 = 1\tfrac12\) cups per batch. (b) \(1\tfrac12 \times 2\tfrac12 = \tfrac32 \times \tfrac52 = \tfrac{15}{4} = 3\tfrac34\) cups.
  3. Ana: \(\tfrac23 \div \tfrac14 = \tfrac83 = 2\tfrac23\) miles per hour. Ben: \(\tfrac34 \div \tfrac{3}{10} = \tfrac{30}{12} = 2\tfrac12\) miles per hour. Ana is faster. Ben went further only because he walked for longer.
  4. (a) \(\tfrac25 \div \tfrac34 = \tfrac25 \times \tfrac43 = \tfrac{8}{15}\) of the pool per hour. (b) \(\tfrac34 \div \tfrac25 = \tfrac{15}{8} = 1\tfrac78\) hours per pool. (c) \(\tfrac{8}{15} \times \tfrac{15}{8} = 1\). Pools per hour and hours per pool are the same fact read both ways, so one is the other flipped.

Stretch. The first tap fills \(1 \div \tfrac14 = 4\) baths per hour, the second \(1 \div \tfrac13 = 3\). Together they fill 7 baths per hour, so one bath takes \(\tfrac17\) hour. That is \(60 \div 7 = 8\tfrac47\) minutes, a little over eight and a half.

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