Multiply proper fractions and mixed numbers by whole numbers ... - Free Printable

Let’s solve each problem step by step, using the methods shown.

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Alex’s Method: Counting Sixths

We are given:
3 × 5/6

This means we have 3 groups of 5 sixths.
Each group has 5 blue boxes (each box = 1/6).
So total sixths = 3 × 5 = 15 sixths → 15/6

But we can simplify 15/6:
15 ÷ 3 = 5, 6 ÷ 3 = 2 → so 5/2 or 2 1/2? Wait — let’s check:

Actually, 15/6 = 2 and 3/6 = 2 and 1/2. But maybe they want it as an improper fraction? Let’s see what the model shows.

Looking at the diagram: 3 rows, each with 5 shaded sixths → total 15 shaded sixths → 15/6.

But 15/6 simplifies to 5/2? No — 15÷3=5, 6÷3=2 → yes, 5/2. But 5/2 is 2.5. However, in the context of counting sixths, maybe leave as 15/6? But usually we simplify.

Wait — let’s look at the other problems. They probably expect simplified fractions or mixed numbers if needed.

But for now, let’s just compute numerically:

3 × 5/6 = (3×5)/6 = 15/6 = 5/2 = 2 1/2

But let’s hold on — maybe the worksheet expects unreduced form? Looking at Jemma’s method later, she uses 2/12 etc., which reduces. So likely reduce.

But let’s do all Alex’s problems first.

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Use Alex’s method:

1. 2 × 3/8
→ 2 groups of 3 eighths → 6 eighths → 6/8 = 3/4

2. 3 × 4/9
→ 3 × 4 = 12 ninths → 12/9 = 4/3 = 1 1/3

3. 4 × 3/7
→ 4 × 3 = 12 sevenths → 12/7 = 1 5/7

4. 3 × 2/11
→ 3 × 2 = 6 elevenths → 6/11 (already simplified)

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Jemma’s Method: Using model with twelfths

Given: 3 × 2/12

Model shows 3 groups of 2 twelfths → total 6 twelfths → 6/12 = 1/2

So answer is 1/2

Now use Jemma’s method:

1. 2 × 4/7
→ 2 groups of 4 sevenths → 8 sevenths → 8/7 = 1 1/7

2. 3 × 5/9
→ 3 × 5 = 15 ninths → 15/9 = 5/3 = 1 2/3

3. 4 × 2/6
→ 4 × 2 = 8 sixths → 8/6 = 4/3 = 1 1/3

Note: 2/6 is same as 1/3, but we’ll follow the method as written.

4. 2 × 6/11
→ 2 × 6 = 12 elevenths → 12/11 = 1 1/11

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Malachi’s Method: Number line with tenths

Given: 3 × 2/10

On number line: start at 0, jump 2/10 three times → lands on 6/10 → simplify to 3/5

So answer is 6/10 or 3/5? Probably simplify → 3/5

Now use Malachi’s method:

1. 2 × 3/6
→ 2 × 3 = 6 sixths → 6/6 = 1

2. 3 × 2/6
→ 3 × 2 = 6 sixths → 6/6 = 1

3. 4 × 2/5
→ 4 × 2 = 8 fifths → 8/5 = 1 3/5

4. 3 × 8/10
→ 3 × 8 = 24 tenths → 24/10 = 12/5 = 2 2/5

Wait — let me double-check all calculations.

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Let me list all answers clearly:

Alex’s main example:
3 × 5/6 = 15/6 = 5/2 = 2 1/2 → but since the model counts sixths, maybe write as 15/6? But I think simplifying is expected. Looking at the other problems, they don’t specify, but in math class, we usually simplify.

Actually, looking back at the worksheet, in Jemma’s part, she writes “2/12” under each group, then adds them. So perhaps they want the unsimplified version first? But the blank is for final answer.

I think safest is to simplify unless told otherwise.

But let’s check one: In Malachi’s example, 3 × 2/10 = 6/10 = 3/5 — that’s simplified.

Similarly, in Alex’s case, 15/6 reduces to 5/2 or 2 1/2.

But 5/2 is improper, 2 1/2 is mixed. Which to use?

Looking at the problems, some will be improper, some mixed. I think either is fine, but perhaps mixed number if greater than 1.

But let’s see the format of the blanks — no specification. I’ll go with simplified fraction, and if improper, leave as improper unless it’s a whole number.

Actually, in elementary school, often mixed numbers are preferred for values >1.

To be safe, I’ll convert to mixed numbers where applicable.

Let me redo with that in mind.

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Final Answers:

Alex’s Example:
3 × 5/6 = 15/6 = 2 3/6 = 2 1/2 → so 2 1/2

But wait — 15/6 divide numerator and denominator by 3: 5/2 = 2 1/2. Yes.

Alex’s Problems:

1. 2 × 3/8 = 6/8 = 3/4 → 3/4

2. 3 × 4/9 = 12/9 = 4/3 = 1 1/3 → 1 1/3

3. 4 × 3/7 = 12/7 = 1 5/7 → 1 5/7

4. 3 × 2/11 = 6/11 → 6/11 (no simplification)

Jemma’s Example:
3 × 2/12 = 6/12 = 1/2 → 1/2

Jemma’s Problems:

1. 2 × 4/7 = 8/7 = 1 1/7 → 1 1/7

2. 3 × 5/9 = 15/9 = 5/3 = 1 2/3 → 1 2/3

3. 4 × 2/6 = 8/6 = 4/3 = 1 1/3 → 1 1/3

4. 2 × 6/11 = 12/11 = 1 1/11 → 1 1/11

Malachi’s Example:
3 × 2/10 = 6/10 = 3/5 → 3/5

Malachi’s Problems:

1. 2 × 3/6 = 6/6 = 1 → 1

2. 3 × 2/6 = 6/6 = 1 → 1

3. 4 × 2/5 = 8/5 = 1 3/5 → 1 3/5

4. 3 × 8/10 = 24/10 = 12/5 = 2 2/5 → 2 2/5

All seem correct.

Let me verify one more: 4 × 2/5 = 8/5 = 1.6, which is 1 3/5 — yes.

3 × 8/10 = 24/10 = 2.4 = 2 4/10 = 2 2/5 — yes.

Okay, confident.

Final Answer:
For Alex's example: 2 1/2
Alex's problems: 3/4, 1 1/3, 1 5/7, 6/11
Jemma's example: 1/2
Jemma's problems: 1 1/7, 1 2/3, 1 1/3, 1 1/11
Malachi's example: 3/5
Malachi's problems: 1, 1, 1 3/5, 2 2/5

But the question is to solve the problems in the image. Since the user didn't specify which one, but the image has multiple, I need to provide all answers as per the worksheet.

Looking back at the user's request: "Solve the problem accurately." and the image contains several problems.

In the response format, it says "Final Answer" — probably meaning all answers.

But how to present? Perhaps list them in order.

Since the worksheet has sections, I'll list the answers in the order they appear.

First section: Alex's method example: 3 × 5/6 = ?

Then four problems under "Use Alex’s method"

Then Jemma's example: 3 × 2/12 = ?

Then four problems under "Use Jemma’s method"

Then Malachi's example: 3 × 2/10 = ?

Then four problems under "Use Malachi’s method"

So total 1 + 4 + 1 + 4 + 1 + 4 = 15 answers.

I'll list them in order.

Final Answer:
2 1/2, 3/4, 1 1/3, 1 5/7, 6/11, 1/2, 1 1/7, 1 2/3, 1 1/3, 1 1/11, 3/5, 1, 1, 1 3/5, 2 2/5

But to make it clear, perhaps label them.

Since the instruction is to put final answer after reasoning, and not to explain, I'll just list the numerical answers in sequence.

But let me confirm the very first one: 3 × 5/6.

Is it 15/6 or 5/2 or 2 1/2? I think 2 1/2 is best.

Similarly, others.

Another way: perhaps keep as improper fractions? But in the context, mixed numbers are used in the models sometimes.

I think mixed numbers are appropriate here.

So I'll go with that.

Final Answer:
2\frac{1}{2}, \frac{3}{4}, 1\frac{1}{3}, 1\frac{5}{7}, \frac{6}{11}, \frac{1}{2}, 1\frac{1}{7}, 1\frac{2}{3}, 1\frac{1}{3}, 1\frac{1}{11}, \frac{3}{5}, 1, 1, 1\frac{3}{5}, 2\frac{2}{5}

But in text, without LaTeX.

Since this is plain text, I'll write as:

2 1/2, 3/4, 1 1/3, 1 5/7, 6/11, 1/2, 1 1/7, 1 2/3, 1 1/3, 1 1/11, 3/5, 1, 1, 1 3/5, 2 2/5

Yes.

Now, to box it.

But the instruction says "Final Answer:" and then the answer.

So I'll write it as a list.

Perhaps comma-separated.

I think that's fine.