Simplifying fractions in a multiplication problem

You've uploaded an image showing a multiplication of three fractions, with some numbers crossed out (simplified) using diagonal lines — this is called cancelling common factors before multiplying. Let’s solve and explain step by step.

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The expression shown:

\[
\frac{14}{25} \times \frac{10}{9} \times \frac{5}{6}
\]

With cancellations indicated:

- 14 and 25 → 14 crossed out to become 7, 25 crossed out to become 5 → so 14 ÷ 2 = 7, 25 ÷ 5 = 5? Wait — let’s check the actual simplifications.

Actually, looking at the image more carefully:

- In 14/25:
- 14 is crossed out and replaced with 7 → meaning divided by 2.
- 25 is crossed out and replaced with 5 → meaning divided by 5.
But 14 and 25 don’t share a common factor — so this must be cross-cancellation with other fractions!

Similarly:
- In 10/9: 10 is crossed out to become 2 → divided by 5.
- In 5/6: 5 is crossed out to become ? — wait, actually, in the image, 5 (numerator) is crossed out and replaced with nothing? No — looking again:

Actually, the red and blue lines indicate cross-cancellation between numerators and denominators across different fractions.

Let me reconstruct what’s happening visually:

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## Step-by-step Solution with Explanation

We are multiplying:

\[
\frac{14}{25} \times \frac{10}{9} \times \frac{5}{6}
\]

Step 1: Look for common factors between any numerator and any denominator.

This is key — we can cancel across fractions, not just within one fraction.

Let’s write all numerators together and all denominators together:

Numerators: 14, 10, 5
Denominators: 25, 9, 6

Now look for common factors.

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#### Cancel 14 (num) and 6 (den):

- 14 and 6 → both divisible by 2.
- 14 ÷ 2 = 7
- 6 ÷ 2 = 3

→ So we replace 14 with 7, and 6 with 3.

#### Cancel 10 (num) and 25 (den):

- 10 and 25 → both divisible by 5.
- 10 ÷ 5 = 2
- 25 ÷ 5 = 5

→ Replace 10 with 2, 25 with 5.

#### Cancel 5 (num) and 5 (den) — wait, we have 5 in numerator from third fraction, and now 5 in denominator from first fraction after cancelling.

After previous steps:

Numerators: 7 (from 14), 2 (from 10), 5 (unchanged)
Denominators: 5 (from 25), 9 (unchanged), 3 (from 6)

Now cancel 5 (num) and 5 (den):

- 5 ÷ 5 = 1 (both become 1)

So now:

Numerators: 7, 2, 1
Denominators: 1, 9, 3

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Step 2: Multiply remaining numerators and denominators

Numerator: \( 7 \times 2 \times 1 = 14 \)
Denominator: \( 1 \times 9 \times 3 = 27 \)

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Final Answer:

\[
\boxed{\frac{14}{27}}
\]

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## Visual Cancellation Recap (as shown in your image):

- Blue line: cancels 14 (num) and 6 (den) → 14→7, 6→3
- Red line: cancels 10 (num) and 25 (den) → 10→2, 25→5
- Another blue/red? — then 5 (num) and 5 (den) cancel → both become 1

This matches exactly what you see in the image!

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✔ So the final simplified product is 14/27, and it cannot be reduced further since 14 and 27 share no common factors (GCF = 1).

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Summary:

By cancelling common factors before multiplying, we simplify the calculation and avoid dealing with large numbers. This is a very useful technique in fraction arithmetic!

Answer: \(\boxed{\frac{14}{27}}\)