Equivalent fractions: simplification and irreducible fraction

1. First of all we have to divide $840$ by $7$ and then multiply the result by $3$: $$(840:7)\cdot3=120\cdot3=360$$
2. $(-45:9)\cdot(-2)=-5\cdot(-2)=10$

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

• $\dfrac{3}{4}$ and $\dfrac{4}{5}$ are not equivalent because $3\cdot5=15$ and $4\cdot4=16$
• $\dfrac{3}{4}$ and $\dfrac{-3}{4}$ are not equivalent because $3\cdot4=12$ and $4\cdot(-3)=-12$
• $\dfrac{3}{4}$ and $\dfrac{4}{-3}$ are not equivalent because $4\cdot4=16$ and $3\cdot(-3)=-9$
• $\dfrac{3}{4}$ and $\dfrac{-3}{-4}$ are equivalent because $3\cdot(-4)=4\cdot(-3).$
• $\dfrac{3}{4}$ and $\dfrac{12}{16}$ are equivalent because $\dfrac{3}{4}=\dfrac{3\cdot4}{4\cdot4}=\dfrac{12}{16}.$
• $\dfrac{3}{4}$ and $\dfrac{3}{4}$ are equivalent because every fraction is equivalent to itself, (reflexive property).

Now, using the transitive property, we have that $\dfrac{3}{4}$, $\dfrac{-3}{-4}$ and $\dfrac{12}{16}$ are equivalent and the other fractions $\dfrac{4}{5}$, $\dfrac{-3}{4}$ and $\dfrac{4}{-3}$, are not equivalent to the last ones. But we must check if they are equivalent to themselves: $\dfrac{4}{5}$ is neither equivalent to $\dfrac{-3}{4}$ because $4\cdot4=16$ and $5\cdot(-3)=-15$, nor to $\dfrac{4}{-3}$. And the last pair $\dfrac{-3}{4}$ and $\dfrac{4}{-3}$, which are not equivalent.

1. In order to calculate the irreducible fraction we must calculate the factorizations into prime numbers for numerator and denominator: $18=2\cdot3^2$ and $24=2^3\cdot3.$ The other fraction: $45=3^2\cdot5$ and $50=2\cdot 5^2.$

Now, it's time to calculate the greatest common divisor of numerator and denominator: $gcd(18,24)=2\cdot3=6$ and $gcd(45,50)=5.$

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

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: $\dfrac{-2}{7}=\dfrac{-2\cdot(-14)}{7\cdot(-14)}=\dfrac{28}{-98}$.