Example of ratios and proportions with a math problem on a blackboard.

Let's solve the problem step by step.

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Problem:

We are given a ratio proportion:

$$
8 : 7 \rightarrow 40 : x
$$

This means that the ratio $ 8:7 $ is equivalent to the ratio $ 40:x $. We need to find the value of $ x $ that makes this proportion true.

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Step 1: Write as a Proportion

We can write the proportion as:

$$
\frac{8}{7} = \frac{40}{x}
$$

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Step 2: Solve for $ x $

To solve for $ x $, we use cross-multiplication:

$$
8 \cdot x = 7 \cdot 40
$$

$$
8x = 280
$$

Now divide both sides by 8:

$$
x = \frac{280}{8} = 35
$$

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✔ Answer:

$$
x = 35
$$

So, the proportion is:

$$
8 : 7 = 40 : 35
$$

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Check:

Let’s verify:
- $ \frac{8}{7} \approx 1.1429 $
- $ \frac{40}{35} = \frac{8}{7} \approx 1.1429 $

✔ They are equal — so the solution is correct.

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Additional Note (Bottom Row):

The bottom part shows some fractions:

$$
\frac{3}{n} = \frac{4}{d} = \frac{7}{9} = \frac{14}{112}
$$

But notice: $ \frac{7}{9} \neq \frac{14}{112} $, because:

- $ \frac{14}{112} = \frac{1}{8} $
- $ \frac{7}{9} \approx 0.777... $

So these are not all equal. There might be a typo or error in that row. But since your main question was about $ 8:7 \to 40:x $, we focused on that.

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✔ Final Answer:

$$
\boxed{x = 35}
$$