Dividing Fractions Worksheets with Answer Key - Free Printable

Let's solve each of these word problems involving division of fractions, step by step, and reduce the answers to their simplest form.

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Problem 1:

Jack needs to make sandwiches for a class picnic with $\frac{2}{9}$ of a kilogram of sugar. If each sandwich needs $\frac{3}{18}$ of a kilogram of sugar, then how many total sandwiches can be made?

#### Step 1: Understand what’s being asked
We are dividing the total amount of sugar by the amount used per sandwich:
$$
\text{Number of sandwiches} = \frac{\frac{2}{9}}{\frac{3}{18}}
$$

#### Step 2: Simplify the divisor
Simplify $\frac{3}{18}$:
$$
\frac{3}{18} = \frac{1}{6}
$$

So now:
$$
\frac{2}{9} \div \frac{1}{6}
$$

#### Step 3: Divide fractions (multiply by reciprocal)
$$
\frac{2}{9} \div \frac{1}{6} = \frac{2}{9} \times \frac{6}{1} = \frac{12}{9}
$$

#### Step 4: Simplify
$$
\frac{12}{9} = \frac{4}{3} = 1\frac{1}{3}
$$

But we can’t make a third of a sandwich — this is a whole number context. So Jack can make 1 full sandwich, but not enough for a second.

Wait — let’s double-check: Is $\frac{4}{3}$ acceptable as an answer?
Since the question asks *how many total sandwiches can be made*, and we're dealing with real-world items, we expect a whole number, but mathematically, we should report the exact value unless told otherwise.

But actually, in fraction word problems like this, if it asks “how many” and the result is a fraction, we interpret it as how many complete units fit.

But here, since it says "how many total sandwiches can be made?" and doesn't specify "complete", but in reality you can’t make a partial sandwich with partial sugar, so likely they want the number of full sandwiches, but let’s see:

Wait — $\frac{2}{9} \div \frac{1}{6} = \frac{4}{3} = 1\frac{1}{3}$ — meaning only 1 full sandwich can be made, with leftover sugar.

But wait — let’s check the calculation again.

Wait: Is $\frac{3}{18} = \frac{1}{6}$ correct? Yes.

Now: $\frac{2}{9} \div \frac{1}{6} = \frac{2}{9} \times 6 = \frac{12}{9} = \frac{4}{3}$ → $1\frac{1}{3}$

So only 1 full sandwich can be made, but the mathematical answer is $\frac{4}{3}$.

But the problem says "how many total sandwiches can be made?" — this implies how many complete ones, so we take the whole number part: 1.

But let’s look at the exact math — perhaps they just want the fractional result.

But wait — let’s re-express:

Total sugar: $\frac{2}{9}$ kg
Per sandwich: $\frac{3}{18} = \frac{1}{6}$ kg

So:
$$
\frac{2}{9} \div \frac{1}{6} = \frac{2}{9} \times \frac{6}{1} = \frac{12}{9} = \frac{4}{3}
$$

So $\frac{4}{3}$ sandwiches can be made.

But that’s 1 and 1/3 sandwiches.

But you can't make a third of a sandwich — so only 1 full sandwich.

However, in math problems, unless specified otherwise, we often accept the fractional answer as the number of times the smaller quantity fits into the larger one.

So, answer: $\boxed{\frac{4}{3}}$ or $1\frac{1}{3}$ — but let's keep it as improper fraction unless asked otherwise.

But let’s confirm: maybe I made a mistake?

Wait — let’s double-check: Is $\frac{2}{9}$ really enough for even one sandwich?

Each sandwich needs $\frac{3}{18} = \frac{1}{6} \approx 0.1667$ kg
Total sugar: $\frac{2}{9} \approx 0.2222$ kg

So yes, $\frac{2}{9} > \frac{1}{6}$, so one sandwich is possible.

Can we do a second?
Two sandwiches need $2 \times \frac{1}{6} = \frac{1}{3} \approx 0.333$ kg — but we only have ~0.222 kg → no.

So only 1 full sandwich can be made.

But mathematically, the number of times $\frac{1}{6}$ fits into $\frac{2}{9}$ is $\frac{2}{9} \div \frac{1}{6} = \frac{4}{3}$

So depending on interpretation:

- Mathematical answer: $\boxed{\frac{4}{3}}$
- Real-world answer: 1 sandwich

But since the problem says "how many total sandwiches can be made?", and doesn't say "how many portions", we go with the maximum number of complete sandwiches.

But in most school-level fraction division problems, they expect the fractional quotient, even if it's not a whole number.

Let’s look at other problems — they may clarify.

But let’s proceed with the mathematical division and reduce to simplest form.

So:
$$
\frac{2}{9} \div \frac{3}{18} = \frac{2}{9} \div \frac{1}{6} = \frac{2}{9} \times \frac{6}{1} = \frac{12}{9} = \frac{4}{3}
$$

✔ Final Answer: $\boxed{\frac{4}{3}}$ or $1\frac{1}{3}$

But let’s write it as improper fraction reduced: $\boxed{\frac{4}{3}}$

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Problem 2:

John has a piece of metal rod that is $\frac{3}{4}$ of a meter long. He needs to cut pieces from the rod that are $\frac{5}{16}$ of a meter long. How many pieces can John cut?

This is a classic division: total length ÷ length per piece

$$
\frac{3}{4} \div \frac{5}{16}
$$

Step 1: Multiply by reciprocal:
$$
\frac{3}{4} \times \frac{16}{5} = \frac{48}{20}
$$

Step 2: Simplify:
$$
\frac{48}{20} = \frac{12}{5} = 2\frac{2}{5}
$$

So 2 full pieces can be cut, with some leftover.

But again, the question asks how many pieces can John cut? — probably expecting how many complete pieces.

But in math problems, sometimes they want the exact division.

But since he can’t cut a fraction of a piece (unless it's allowed), we take floor of $\frac{12}{5} = 2.4$, so 2 pieces.

But again, let’s see what the mathematical answer is.

The number of times $\frac{5}{16}$ fits into $\frac{3}{4}$ is $\frac{12}{5}$

So mathematically: $\boxed{\frac{12}{5}}$

But in real life: 2 pieces

But since this is a math worksheet, likely wants the fractional answer.

Let’s keep it as $\boxed{\frac{12}{5}}$ or $2\frac{2}{5}$

But reduce: $\frac{12}{5}$ is already simplified.

✔ Final Answer: $\boxed{\frac{12}{5}}$

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Problem 3:

$\frac{3}{7}$ of a 1 liter container is filled with water. If a mug can contain $\frac{9}{84}$ of a liter, then how many mugs of water are needed to fill up the bucket?

Wait — the container is already $\frac{3}{7}$ full. But the question says: *"how many mugs of water are needed to fill up the bucket?"*

So the bucket is not full — it has $\frac{3}{7}$ liters, and we need to add more to make it full (i.e., 1 liter).

So amount needed to fill it:
$$
1 - \frac{3}{7} = \frac{4}{7} \text{ liters}
$$

Each mug holds $\frac{9}{84}$ liters.

Simplify $\frac{9}{84}$:
$$
\frac{9}{84} = \frac{3}{28} \quad (\text{divide numerator and denominator by 3})
$$

Now divide:
$$
\frac{4}{7} \div \frac{3}{28} = \frac{4}{7} \times \frac{28}{3} = \frac{112}{21}
$$

Simplify:
$$
\frac{112}{21} = \frac{16}{3} \quad (\text{divide numerator and denominator by 7})
$$

Because: $112 \div 7 = 16$, $21 \div 7 = 3$

So $\frac{16}{3} = 5\frac{1}{3}$

So 5 full mugs would fill it, and a little more.

But again, the question is: how many mugs of water are needed to fill up the bucket?

So if you can pour partial mugs, then $\frac{16}{3}$ mugs are needed.

But if only full mugs, then 6 mugs.

But since it says “mugs of water”, and doesn’t specify full mugs, likely allows partial.

But the math is clear.

✔ Final Answer: $\boxed{\frac{16}{3}}$

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Problem 4:

A box of table tennis balls weighs $\frac{5}{9}$ of a kg. If each ball weighs $\frac{15}{81}$ of a kg, then how many balls are there in the box?

This is straightforward: total weight ÷ weight per ball

$$
\frac{5}{9} \div \frac{15}{81}
$$

First, simplify $\frac{15}{81}$:
$$
\frac{15}{81} = \frac{5}{27} \quad (\text{divide by 3})
$$

Now:
$$
\frac{5}{9} \div \frac{5}{27} = \frac{5}{9} \times \frac{27}{5}
$$

Cancel the 5s:
$$
\frac{1}{9} \times \frac{27}{1} = \frac{27}{9} = 3
$$

✔ Final Answer: $\boxed{3}$

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✔ Final Answers:

1. $\boxed{\frac{4}{3}}$
2. $\boxed{\frac{12}{5}}$
3. $\boxed{\frac{16}{3}}$
4. $\boxed{3}$

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📝 Summary of Steps:

- For all problems: Divide the total by the unit amount.
- Use reciprocal multiplication.
- Simplify fractions before and after.
- Reduce final answers to simplest form.
- When division results in a fraction, unless specified otherwise, accept the fractional answer as the number of times the smaller unit fits.

Let me know if you'd like explanations in simpler terms or visual models!