The learning goal
Ask how many portions of the given size fit into the total. Here, 3/4 L ÷ (1/8 L per bottle) = 6 bottles. Derive the reciprocal rule from portion size × portion count = total, with a nonzero divisor, then verify by multiplication and units.
Represent the total in eighths
This is an original editorial bottling exercise, not a copied textbook problem or an experimental measurement. Assume 3/4 liter of practice liquid, 1/8 liter per full bottle and no losses. Divide one liter into eight equal parts: 3/4 liter is 6/8 liter, containing six portions of 1/8 liter.
The result 6 does not mean the liquid increased. The divisor describes a portion smaller than one, and the answer counts how many portions fit. Its unit changes from liters to bottles.
| Quantity | In eighths of a liter | Role |
|---|---|---|
| Total 3/4 L | Six portions of 1/8 L | All available liquid |
| 1/8 L per bottle | One portion of 1/8 L | Requirement for one bottle |
| Bottle count | 6÷1=6 bottles | No remainder under the no-loss assumption |
The reciprocal follows from an equation
Let q be the portion count, b the portion size and a the total, so b×q=a with b nonzero. Multiply both sides by 1/b to get q=a×(1/b). Thus a÷b=a×(1/b). The OpenStax fraction chapter explains reciprocals and fraction division; the bottling values and worked presentation here are original.
With b=1/8, its reciprocal is 8 and 3/4×8=6. A nonzero fraction m/n has reciprocal n/m, with both m and n nonzero. A reciprocal is not an additive opposite: the opposite of 1/8 is −1/8.
A reciprocal does not mean adding a minus sign
3/4÷1/8=3/4×(−1/8), because the number is “reversed.”
3/4÷1/8=3/4×8=6, checked by 6×1/8=3/4.
The same total can answer two different questions
Sharing 3/4 liter equally among three bottles asks for the amount per bottle: 3/4÷3=1/4 liter per bottle. Filling bottles of 1/8 liter each asks for a count: 3/4÷1/8=6 bottles.
Both use division, but their divisors and answer units have different meanings. Appending liters to every answer makes the second one wrong. Write portion size × portion count = total and identify the unknown first.
Identify the unknown
Equal sharing
Known total and bottle count: 3/4 L÷3 bottles=1/4 L per bottle.
Measuring division
Known total and portion size: 3/4 L÷(1/8 L per bottle)=6 bottles.
Substitution check
Portion size multiplied by count must reconstruct the original total.
Change the total to 5/6 L and each bottle to 1/4 L
Under the same no-loss assumption, 5/6 L÷(1/4 L per bottle)=10/3 bottles, or 3 and 1/3 bottle-capacities. If the task permits only full bottles, do not round up to four: three bottles can be filled completely.
Those bottles use 3×1/4=3/4 liter. The remainder is 5/6−3/4=10/12−9/12=1/12 liter. That equals one third of a bottle’s capacity, not one third of a liter—a distinction the final answer must preserve.
| Quantity | Result | Check |
|---|---|---|
| Mathematical capacity count | 10/3 bottles | (10/3)×1/4=5/6 L |
| Completely filled bottles | 3 bottles | Uses 3/4 L |
| Liquid remainder | 1/12 L | 3/4+1/12=5/6 L |
| Remainder relative to capacity | 1/3 of one bottle | (1/3)×1/4=1/12 L |
Explain the first incorrect step during review
Hide the reference answer and explain why the original yields six bottles while the variation fills only three completely. Verify by units and multiplication. If you can only repeat invert and multiply, return to b×q=a; see explain the reason behind each worked step.
A useful error note says “I confused one third of a bottle with one third of a liter,” followed by the corrected 1/12-liter calculation. Continue with unit cancellation and map-scale checks for dimensional practice. No fixed review interval is prescribed; revisit according to whether you can explain the work independently.
Explain these without the reference
- Identify total, portion size and portion count in the original task.
- Derive multiplication by the reciprocal for a nonzero divisor.
- Distinguish the variation’s mathematical count from full bottles.
- Convert the remaining fraction of a bottle into the correct volume unit.
Common questions
Does dividing by a fraction always increase the number?
No. A positive number divided by a positive divisor between zero and one increases; division by a number above one decreases it. Units and other sign conditions still matter.
Does zero have a reciprocal?
No number multiplied by zero gives one, so zero has no reciprocal and division by zero is not defined by this rule.
Should every bottling answer be rounded down?
Only when asking for completely filled bottles. A capacity count can be 10/3 bottles; a question about containers needed to hold all the liquid is a different task.
