Question

which numbers are the extremes of the proportion shown below?

\\(\frac{4}{7} = \frac{20}{35}\\)

a. 7 and 20
b. 4 and 35
c. 4 and 20
d. 7 and 35

Explanation:

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<reasoning>

Define a proportion and its terms

A proportion is an equation stating that two ratios are equal. It can be written in two common forms:

1. As fraction ratios:

\[\frac{a}{b} = \frac{c}{d}\]

1. As colon ratios:

\[a : b = c : d\]

Identify the means and extremes

In the colon representation \(a : b = c : d\):

• The outer terms, \(a\) and \(d\), are called the extremes.
• The inner terms, \(b\) and \(c\), are called the means.

When translated back to the fractional form \(\frac{a}{b} = \frac{c}{d}\):

• The numerator of the first fraction (\(a\)) and the denominator of the second fraction (\(d\)) are the extremes.
• The denominator of the first fraction (\(b\)) and the numerator of the second fraction (\(c\)) are the means.

Apply to the given proportion

The given proportion is:
\[\frac{4}{7} = \frac{20}{35}\]

Writing this in colon form gives:
\[4 : 7 = 20 : 35\]

From this form, we identify:

• The extremes (outer numbers) are \(4\) and \(35\).
• The means (inner numbers) are \(7\) and \(20\).

Therefore, the extremes of the proportion are \(4\) and \(35\).
</reasoning>

<answer>
<mcq-option>A. 7 and 20</mcq-option>
<mcq-correct>B. 4 and 35</mcq-correct>
<mcq-option>C. 4 and 20</mcq-option>
<mcq-option>D. 7 and 35</mcq-option>
</answer>

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Answer:

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<reasoning>

Define a proportion and its terms

A proportion is an equation stating that two ratios are equal. It can be written in two common forms:

1. As fraction ratios:

\[\frac{a}{b} = \frac{c}{d}\]

1. As colon ratios:

\[a : b = c : d\]

Identify the means and extremes

In the colon representation \(a : b = c : d\):

• The outer terms, \(a\) and \(d\), are called the extremes.
• The inner terms, \(b\) and \(c\), are called the means.

When translated back to the fractional form \(\frac{a}{b} = \frac{c}{d}\):

• The numerator of the first fraction (\(a\)) and the denominator of the second fraction (\(d\)) are the extremes.
• The denominator of the first fraction (\(b\)) and the numerator of the second fraction (\(c\)) are the means.

Apply to the given proportion

The given proportion is:
\[\frac{4}{7} = \frac{20}{35}\]

Writing this in colon form gives:
\[4 : 7 = 20 : 35\]

From this form, we identify:

• The extremes (outer numbers) are \(4\) and \(35\).
• The means (inner numbers) are \(7\) and \(20\).

Therefore, the extremes of the proportion are \(4\) and \(35\).
</reasoning>

<answer>
<mcq-option>A. 7 and 20</mcq-option>
<mcq-correct>B. 4 and 35</mcq-correct>
<mcq-option>C. 4 and 20</mcq-option>
<mcq-option>D. 7 and 35</mcq-option>
</answer>

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