Problems from Equivalent fractions: simplification and irreducible fraction

Development:

1. First of all we have to divide $840$ by $7$ and then multiply the result by $3$: $$(840:7)\cdot 3=120\cdot 3=360$$
2. $(-45:9)\cdot (-2)=-5\cdot (-2)=10$

Solution:

1. $360$
2. $10$

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Development:

1. In order to calculate the irreducible fraction we must calculate the factorizations into prime numbers for numerator and denominator: $18=2\cdot 3^2$ and $24=2^3\cdot 3.$ The other fraction: $45=3^2\cdot 5$ and $50=2\cdot 5^2.$

   Now, it's time to calculate the greatest common divisor of numerator and denominator: $gcd(18,24)=2\cdot 3=6$ and $gcd(45,50)=5.$

   And finally, we divide numerator and denominator by gcd: ${\displaystyle \frac{18}{24}}={\displaystyle \frac{18:6}{24:6}}={\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{45}{50}}={\displaystyle \frac{45:5}{50:5}}={\displaystyle \frac{9}{10}}$.

2. In order to change the denominator $7$ to $-98$, it's necessary to multiply $7$ by $-14$, because: $-98=-2\cdot 7^2=7\cdot (-14).$ So, we can do the equivalent fraction: ${\displaystyle \frac{-2}{7}}={\displaystyle \frac{-2\cdot (-14)}{7\cdot (-14)}}={\displaystyle \frac{28}{-98}}$.

Solution:

1. ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{9}{10}}$.
2. ${\displaystyle \frac{28}{-98}}.$

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Development:

We start finding which fractions are equivalent to the fraction ${\displaystyle \frac{3}{4}}$. To do it, we must check it for every fraction:

• ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{4}{5}}$ are not equivalent because $3\cdot 5=15$ and $4\cdot 4=16$
• ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{-3}{4}}$ are not equivalent because $3\cdot 4=12$ and $4\cdot (-3)=-12$
• ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{4}{-3}}$ are not equivalent because $4\cdot 4=16$ and $3\cdot (-3)=-9$
• ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{-3}{-4}}$ are equivalent because $3\cdot (-4)=4\cdot (-3).$
• ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{12}{16}}$ are equivalent because ${\displaystyle \frac{3}{4}}={\displaystyle \frac{3\cdot 4}{4\cdot 4}}={\displaystyle \frac{12}{16}}.$
• ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{3}{4}}$ are equivalent because every fraction is equivalent to itself, (reflexive property).

Now, using the transitive property, we have that ${\displaystyle \frac{3}{4}}$, ${\displaystyle \frac{-3}{-4}}$ and ${\displaystyle \frac{12}{16}}$ are equivalent and the other fractions ${\displaystyle \frac{4}{5}}$, ${\displaystyle \frac{-3}{4}}$ and ${\displaystyle \frac{4}{-3}}$, are not equivalent to the last ones. But we must check if they are equivalent to themselves: ${\displaystyle \frac{4}{5}}$ is neither equivalent to ${\displaystyle \frac{-3}{4}}$ because $4\cdot 4=16$ and $5\cdot (-3)=-15$, nor to ${\displaystyle \frac{4}{-3}}$. And the last pair ${\displaystyle \frac{-3}{4}}$ and ${\displaystyle \frac{4}{-3}}$, which are not equivalent.

Solution:

The fractions ${\displaystyle \frac{3}{4}}$, ${\displaystyle \frac{-3}{-4}}$ and ${\displaystyle \frac{12}{16}}$ are equivalent. The others are not.

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