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Fill in the unknown quantities.

\[
\begin{array}{c}
\frac{80}{100} = \frac{16}{20} = \frac{32}{40} \\
\frac{60}{100} = \frac{6}{10} = \frac{24}{40} \\
\frac{40}{100} = \frac{8}{20} = \frac{16}{40}
\end{array}
\]

Answer :

To solve the question and fill in the unknown quantities, let's analyze each row of fractions separately:

First Row:
[tex]\[
\frac{80}{100} = \frac{16}{20} = \frac{32}{40}
\][/tex]

Let's break down each fraction:

- [tex]\(\frac{80}{100}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex].
- [tex]\(\frac{16}{20}\)[/tex] also simplifies to [tex]\(\frac{4}{5}\)[/tex].
- [tex]\(\frac{32}{40}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex].

Since all these fractions simplify to the same value, they are equivalent.

Second Row:
[tex]\[
\frac{60}{100} = \frac{6}{\text{unknown}} = \frac{\text{unknown}}{1}
\][/tex]

For the fraction [tex]\(\frac{60}{100}\)[/tex], let's simplify it:

- [tex]\(\frac{60}{100}\)[/tex] simplifies to [tex]\(\frac{3}{5}\)[/tex].

Now, let's fill in the unknowns:

1. The first unknown [tex]\(\frac{6}{\text{unknown}}\)[/tex] should also simplify to [tex]\(\frac{3}{5}\)[/tex].

To find the unknown denominator, set up the equation:
[tex]\[
\frac{6}{x} = \frac{3}{5}
\][/tex]
Cross multiplying gives us:
[tex]\[
6 \times 5 = 3 \times x \implies 30 = 3x \implies x = 10
\][/tex]

2. The second unknown [tex]\(\frac{\text{unknown}}{1}\)[/tex] should be equal to [tex]\(\frac{3}{5}\)[/tex] as well. Therefore, the numerator must be:
[tex]\[
\text{unknown} = \frac{3}{5} \times 1 = 0.6
\][/tex]

Third Row:
[tex]\[
\frac{40}{100} = \frac{24}{20} = \frac{24}{6}
\][/tex]

Let's simplify each fraction:

- [tex]\(\frac{40}{100}\)[/tex] simplifies to [tex]\(\frac{2}{5}\)[/tex].
- [tex]\(\frac{24}{20}\)[/tex] simplifies to [tex]\(\frac{6}{5}\)[/tex].
- [tex]\(\frac{24}{6}\)[/tex] simplifies to [tex]\(4\)[/tex].

These fractions are not equivalent because they simplify to different values.

In summary, we have:

1. All fractions in the first row are equivalent.
2. The unknown denominator in the second row is 10, and the unknown numerator is 0.6.
3. The fractions in the third row are not equivalent.

Therefore, the results are:
- First row equivalence: True
- Second row unknowns: 10 (denominator), 0.6 (numerator)
- Third row equivalence: False

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