Question

Compare the ratios 7:4
and 12:5
using the following tables. Finish the tables by filling in the missing values A and B.

These tables compare two sets of ratios, with missing values A and B to be determined.
7 4
14 8
21 12
28 16
A B
12 5
24 10
36 15
48 20
60 25


(1 point)
Responses

A = 32
and B = 20
.
upper A equals 32 and upper B equals 20 .

A = 35
and B = 18
.
upper A equals 35 and upper B equals 18 .

A = 35
and B = 20
.
upper A equals 35 and upper B equals 20 .

A = 32
and B = 18
.

Answer 1

To find the missing values A and B, we need to observe the relationships between the values in the two given ratios: \(7:4\) and \(12:5\).

The first step is to determine the relationship and pattern of the ratios given in each table. We can express the values in the first table in terms of the ratio \(7:4\):

\[
\frac{7}{4} = \frac{A}{B}
\]

Next, we can see how the first table progresses:

- For \(7:4\):
- \(7 \rightarrow 14\) (doubled)
- \(4 \rightarrow 8\) (doubled)
- \(21 \rightarrow 28\) (increased by \(7\))
- \(4 \rightarrow 16\) (increased by \(8\))

Using the ratio \(7:4\), we find the proportional relationship of each entry to confirm:

For example:
- From the first row \(7:4\), the ratio is \(1.75\).
- From the second row \(14:8\), the ratio remains \(1.75\).
- From the third row \(21:12\), the ratio is also \(1.75\).
- Likewise for \(28:16\).

Continuing this pattern, we find that for the next entry:
- \(A\) corresponds to \(7\) multiplied by \(4\) (i.e., \(7 * 4 = 28\)) gives us \(A = 32\) from the fourth row calculations.

Next, we can calculate \(B\) using the same ratio.

Now looking at the second table corresponding to the ratio \(12:5\):

Confirming the ratios there:
- For all the provided entries \(12:5\), the ratios are consistently confirming.

Now let's solve for \(A\) and \(B\):
The corresponding ratio formula for \(12:5\) is:
\[
\frac{12}{5} = \frac{A}{B}
\]

We already established that:
- In this proportion, if \(12\) corresponds to \(A\) in the same way as \(7\) does to the other ratios, we can determine the missing values \(A\) and \(B\) using simple multiples.

Thus the proportional calculations confirm:
- If \(A = 32\):
\[
\text{Then: } \frac{12}{5} = \frac{32}{B}
\]

So, using cross-multiplication:
\[
5 \cdot 32 = 12 \cdot B \implies 160 = 12B \implies B = \frac{160}{12} \approx 13.33
\]

This approach however can be tested quicker using our pattern, returning to verify against decisions:
Otherwise multiple-choice confirms:
- **A = 32 and B = 20.**

Thus the final answers are:
A = 32 and B = 20.