Question

Compare the ratios 4:5
and 8:10
using the following tables. Finish the tables by filling in the missing values A and B.

These two tables compare two sets of ratios, with missing values A and B to be determined.
4 5
A B
8 10
16 20

Answer 1

To compare the ratios \(4:5\) and \(8:10\), let's first simplify the ratios and find the values of \(A\) and \(B\).

The ratio \(8:10\) simplifies to \(4:5\) because both numbers can be divided by 2:

\[
\frac{8}{10} = \frac{4}{5}
\]

Now let's populate the tables with the missing values based on these ratios.

1. For the first table:
- The ratio is \(4:5\). If we keep the first value as \(4\), then the second value in the same proportion would be calculated as follows:

Using the relationship:

\(\frac{4}{5} = \frac{A}{B}\)

We know \(B\) corresponds with \(5\) when corresponding to \(A\).

2. For the second table:
- The second table shows the ratio \(8:10\).

- Here, if \(A\) corresponds to the proportion \(8\) (the same step as before, using \(4:5\)), we can find \(A\) and \(B\) as follows:

Putting it in the ratio:

\(A = 8\) and \(B = 10\) correspond directly since both align with the \(4:5\) and \(8:10\) ratios.

Now, we fill in the tables as follows:

**Table 1:**
\[
\begin{array}{c|c}
4 & 5 \\
A & B \\
\end{array} =
\begin{array}{c|c}
4 & 5 \\
8 & 10 \\
\end{array}
\]

From this setup, we see:

- \(A = 8\)
- \(B = 10\)

So the final tables would look like this:

**Filled Tables:**
- Table 1:
- \(A = 8\)
- \(B = 10\)

- Table 2:
- Compare:
- \(8:10\) which confirms it simplifies to \(4:5\).

Hence:

\[
\begin{array}{c|c}
4 & 5 \\
8 & 10 \\
16 & 20 \\
\end{array}
\]

This allows us to conclude the ratio comparisons are consistent through both tables. Thus both ratios \(4:5\) and \(8:10\) ultimately are the exact same ratio expressed differently.