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 $$3$$: $$$(840:7)\cdot3=120\cdot3=360$$$
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: $$18=2\cdot3^2$$ 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: $$\dfrac{-2}{7}=\dfrac{-2\cdot(-14)}{7\cdot(-14)}=\dfrac{28}{-98}$$.

Solution:

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

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Development:

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.

Solution:

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

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