What Is A Ratio Table? How Can It Help Solve Problems? - Free Printable

Let's analyze the problem step by step.

We are given two 2×2 grids. The left grid has some numbers, and the right grid shows the same layout but with one number changed — the bottom-right cell is now 175, and there are arrows indicating a transformation involving multiplication by $ \frac{5}{8} $.

---

Left Grid:

```
| 32 | 20 |
| 280 | ? |
```

Right Grid:

```
| 32 | 20 |
| 280 | 175 |
```

And we see:
- An arrow from the top-right (20) to the bottom-right (175), labeled $ \times \frac{5}{8} $
- But wait:
$ 20 \times \frac{5}{8} = 12.5 $, not 175. So that can't be it.

But look again:
The arrow goes from the top-right to the bottom-right, labeled $ \times \frac{5}{8} $, but the value changes from 20 to 175. That doesn’t make sense because $ 20 \times \frac{5}{8} = 12.5 $, not 175.

Wait — maybe the arrow is not showing how to get from 20 to 175, but rather showing a relationship between rows or columns?

Alternatively, perhaps the entire grid is transformed by multiplying each element by $ \frac{5}{8} $? But then:

- $ 32 \times \frac{5}{8} = 20 $
- $ 20 \times \frac{5}{8} = 12.5 $
- $ 280 \times \frac{5}{8} = 175 $

Ah! That’s interesting.

Let’s check:

$$
280 \times \frac{5}{8} = \frac{280 \times 5}{8} = \frac{1400}{8} = 175
$$

So, the bottom-right cell in the right grid (175) is obtained by multiplying the bottom-left cell (280) by $ \frac{5}{8} $.

Similarly:
- Top-left: $ 32 \times \frac{5}{8} = 20 $
- Top-right: $ 20 \times \frac{5}{8} = 12.5 $

But in the right grid, the top-right is still 20, not 12.5. Contradiction.

Wait — no. The right grid has:
- Top-left: 32 → unchanged
- Top-right: 20 → unchanged
- Bottom-left: 280 → unchanged
- Bottom-right: 175

But the arrow from top-right (20) to bottom-right (175) says $ \times \frac{5}{8} $, which would imply:

$ 20 \times \frac{5}{8} = 12.5 $, but 175 ≠ 12.5

That doesn’t work.

But what if the arrow is pointing the other way? From bottom-right to top-right?

No — the arrow is curved from top-right to bottom-right, with $ \times \frac{5}{8} $, so it suggests:
top-right × 5/8 = bottom-right

But $ 20 × 5/8 = 12.5 $, not 175.

Wait — unless the arrow means something else.

Let’s consider this differently.

Maybe the right grid is the result of applying a transformation to the left grid, and the arrows indicate how values are derived.

But in the left grid, the bottom-right is unknown, and in the right grid, it's 175.

Also, the arrows show:
- A red curved arrow from top-right (20) to bottom-right (175), labeled $ \times \frac{5}{8} $
- Another arrow from bottom-left (280) to bottom-right (175), also labeled $ \times \frac{5}{8} $

Wait — both arrows point to 175, and both are labeled $ \times \frac{5}{8} $

But:
- $ 20 \times \frac{5}{8} = 12.5 $
- $ 280 \times \frac{5}{8} = 175 $

So only the bottom-left times $ \frac{5}{8} $ gives 175.

So why is there an arrow from top-right to bottom-right with $ \times \frac{5}{8} $? That would suggest $ 20 \times \frac{5}{8} = 175 $, which is false.

Unless the label is wrong, or the arrow is misread.

Wait — maybe the arrow is from bottom-left to bottom-right, and the label $ \times \frac{5}{8} $ applies to that?

Yes! Look carefully at the diagram:

There is a curved red arrow from the bottom-left (280) to the bottom-right (175), labeled $ \times \frac{5}{8} $

And another curved red arrow from the top-right (20) to the bottom-right (175), also labeled $ \times \frac{5}{8} $

But only one of them makes sense:
$ 280 \times \frac{5}{8} = 175 $ ✔
$ 20 \times \frac{5}{8} = 12.5 $ ✘

So the second arrow must be incorrect, or perhaps it's meant to represent a different relationship.

Wait — could the bottom-right cell be derived from the top-right cell via some operation, and the arrow is just indicating that both cells are related through multiplication by 5/8?

But 20 → 175 is not ×5/8.

Alternative idea: Maybe the entire grid follows a pattern such that:

- The product of diagonals is equal?
- Or the ratio between rows/columns is consistent?

Let’s try to find a pattern.

Assume that the missing number in the left grid is x, and in the right grid, it's 175, and the transformation is that each cell is multiplied by 5/8 to get the corresponding cell in the right grid?

But let’s test:

If we apply $ \times \frac{5}{8} $ to the left grid, we get:

- 32 → $ 32 \times \frac{5}{8} = 20 $
- 20 → $ 20 \times \frac{5}{8} = 12.5 $
- 280 → $ 280 \times \frac{5}{8} = 175 $
- x → $ x \times \frac{5}{8} $

But in the right grid, the values are:
- 32, 20, 280, 175

So:
- Top-left: 32 → 32 ⇒ unchanged? But $ 32 \times \frac{5}{8} = 20 $, not 32
- So not all cells are multiplied by 5/8

Wait — unless the right grid is not a scaled version of the left grid.

But notice: In the right grid, the bottom-right is 175, and the bottom-left is 280, and $ 280 \times \frac{5}{8} = 175 $

Also, top-left is 32, top-right is 20

Now, $ 32 \times \frac{5}{8} = 20 $, which matches the top-right

So:
- $ 32 \times \frac{5}{8} = 20 $
- $ 280 \times \frac{5}{8} = 175 $

So the top-right is $ \text{top-left} \times \frac{5}{8} $
and the bottom-right is $ \text{bottom-left} \times \frac{5}{8} $

So the right column is obtained by multiplying the left column by $ \frac{5}{8} $

So in general:

- Top-right = top-left × $ \frac{5}{8} $
- Bottom-right = bottom-left × $ \frac{5}{8} $

Therefore, in the left grid, if we know:
- Top-left = 32
- Bottom-left = 280
- Top-right = 20
- Bottom-right = ?

Then:
- Bottom-right = bottom-left × $ \frac{5}{8} = 280 \times \frac{5}{8} = 175 $

But wait — that's already given in the right grid.

But in the left grid, the bottom-right is ?, and in the right grid, it's 175

So the left grid has:
- Top-left: 32
- Top-right: 20
- Bottom-left: 280
- Bottom-right: ?

And we want to find ?

But from the pattern above, ? = 280 × 5/8 = 175

So the answer is 175

But wait — then why is the right grid showing the same number? Is the right grid just showing the result of the calculation?

But in the left grid, the bottom-right is missing, and the right grid shows it as 175, with arrows explaining how it was calculated.

So the question is: What should go in the bottom-right of the left grid?

Answer: $ 280 \times \frac{5}{8} = 175 $

But let’s verify consistency.

We also have:
- Top-right = 20
- Top-left = 32
- $ 32 \times \frac{5}{8} = 20 $ → matches

So the rule is: Each entry in the right column is the corresponding left column entry multiplied by $ \frac{5}{8} $

So:
- Top-right = top-left × $ \frac{5}{8} $
- Bottom-right = bottom-left × $ \frac{5}{8} $

Thus:
- Bottom-right = $ 280 \times \frac{5}{8} = \frac{1400}{8} = 175 $

✔

So the missing number is 175

But wait — in the left grid, the bottom-right is ?, and in the right grid, it's shown as 175, so likely the left grid is incomplete, and the right grid is the completed version, with the arrows showing how the value was computed.

Therefore, the answer is 175

But let’s double-check: Is there any contradiction?

In the left grid:
- 32 and 20: $ 32 \times \frac{5}{8} = 20 $ → consistent
- 280 and ?: $ 280 \times \frac{5}{8} = 175 $ → so ? = 175

So yes.

Final Answer:

$$
\boxed{175}
$$

Explanation:

The pattern is that each number in the right column is obtained by multiplying the corresponding number in the left column by $ \frac{5}{8} $.
- Top-right: $ 32 \times \frac{5}{8} = 20 $
- Bottom-right: $ 280 \times \frac{5}{8} = 175 $

Thus, the missing number is $ \boxed{175} $.