Sure, let's find the equivalent fractions for [tex]\(\frac{5}{7}\)[/tex] under the given conditions:
### Part (a): Finding the Equivalent Fraction with Numerator 25
We start with the fraction [tex]\(\frac{5}{7}\)[/tex] and need to find an equivalent fraction that has a numerator of 25.
Let the required fraction be [tex]\(\frac{25}{x}\)[/tex], where [tex]\(x\)[/tex] is the new denominator we need to find.
To find this equivalent fraction, we use the principle of equivalent fractions where:
[tex]\[
\frac{5}{7} = \frac{25}{x}
\][/tex]
We can find [tex]\(x\)[/tex] by setting up the equation:
[tex]\[
5 \cdot x = 25 \cdot 7
\][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[
5x = 175
\][/tex]
[tex]\[
x = \frac{175}{5} = 35
\][/tex]
So, the fraction equivalent to [tex]\(\frac{5}{7}\)[/tex] with a numerator of 25 is:
[tex]\[
\frac{25}{35}
\][/tex]
### Part (b): Finding the Equivalent Fraction with Denominator 42
Next, we need to find the equivalent fraction that has a denominator of 42.
Let the required fraction be [tex]\(\frac{y}{42}\)[/tex], where [tex]\(y\)[/tex] is the new numerator we need to find.
To find this equivalent fraction, we use the principle of equivalent fractions where:
[tex]\[
\frac{5}{7} = \frac{y}{42}
\][/tex]
We can find [tex]\(y\)[/tex] by setting up the equation:
[tex]\[
5 \cdot 42 = 7 \cdot y
\][/tex]
Solving for [tex]\(y\)[/tex], we get:
[tex]\[
210 = 7y
\][/tex]
[tex]\[
y = \frac{210}{7} = 30
\][/tex]
So, the fraction equivalent to [tex]\(\frac{5}{7}\)[/tex] with a denominator of 42 is:
[tex]\[
\frac{30}{42}
\][/tex]
In summary:
(a) The fraction equivalent to [tex]\(\frac{5}{7}\)[/tex] with a numerator of 25 is [tex]\(\frac{25}{35}\)[/tex].
(b) The fraction equivalent to [tex]\(\frac{5}{7}\)[/tex] with a denominator of 42 is [tex]\(\frac{30}{42}\)[/tex].
