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Dividing by a fraction seems anomalous in the set of arithmetic operations inasmuch at it appears to have no analog in the physical world.

Is this the case? Or is there some physical analog to the arithmetic operation of division by a fraction?

Examples

Addition

Analog: Place one pebble on the ground. Place a second pebble on the ground beside the first. Count the pebbles. That is an analog for the addition operation.

Subtraction

Analog: Place three pebbles on the ground. Remove one pebble. Count the remaining pebbles. That is an analog for subtraction of 3 minus 1.

Multiplication

Analog: Take a group of x pebbles. Place n of those groups on the ground. Count the pebbles. That is an analog for multiplication of x times n.

Division

Analog: Take an apple. Slice it into six pieces. Give one piece to each of six friends. That is an analog for division by six.

One can construct examples of other arithmetic operations. Even powers. But in the world of simple arithmetic, for division by fractions, there seems to be no analog.

Or have I overlooked something?

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    $\begingroup$ Take an apple, how many $6$ths can you divide it into? $\endgroup$ – Michael Burr Mar 6 '16 at 21:37
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    $\begingroup$ Division by fraction is the same as multiplication and then division. $\endgroup$ – Ant Mar 6 '16 at 21:37
  • $\begingroup$ A fifth of whiskey is a bottle containing $\frac{1}{5}$ of a gallon. How many fifths are in a gallon? $5$. If you have $12$ fifths and $6$ people to give them to, how many fifths does each person get? These are examples of the use of division of fractions in common everyday usage. $\endgroup$ – user4894 Mar 6 '16 at 21:51
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    $\begingroup$ I teach half hour guitar lessons. How many lessons can I teach in six hours? $\endgroup$ – zahbaz Mar 6 '16 at 22:37
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You have $24$ apples to distribute on Halloween, and you want to give each child $2$ apples; how many children can you accommodate?

$$\frac{24}2=12\;.$$

Hm. That’s not very many. Suppose that you give each child $\frac12$ an apple; how many children can you then accommodate?

$$\frac{24}{1/2}=48\;.$$

It’s the same process.

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Recently I had to plan moving the stairway in my house. The net rise is 93 inches. According to code, $7\frac14$ inch risers would be good. How many steps will be in my new staircase? $$\frac{93}{7\frac14}=\frac{93}{\frac{29}{4}}$$ asks me to divide by a fraction. However you represent the result (decimal or fraction) it implies I need 13 steps. Now that I know that, I'll make the risers as close as I can to $\frac{93}{13}$ inches tall. I can also use the count of 13 together with what building code is for tread to map out how large the staircases's footprint will be.

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