In this
post we’re going to do some examples of exercises with fractions:
Example 1:
If we have
a cake divided in four parts and one of the parts was eaten we want to know:
What
fraction of the cake we would eat if we eat the half of one piece?
How much of the cake lefts?
First we
must know how much cake we have:
We have 3 pieces of cake left from the entire sweet, that was divided in
four pieces. That means ¾ of the cake.
Now we take one half of one piece. One piece is ¼ , that is the same
that:
So
there are 2 pieces of cake with a size of 1/8 in one piece of cake ( 1/4 ). The answer to the first
question is that we would eat a fraction of a value of 1/8.
Now
the second question, how much of the cake lefts?
We
have eaten 1/8 of a quarter, and we had 3/4 of the cake, so we must
differentiate them:
To
solve this we can use the least common multiple:
So
the least common multiple is 8. Now we multiply by 2 the numerator and
denominator:
Now
we can do the difference:
So
we have 5/8 of the original cake left. This doesn’t mean that all the pieces
of the cake have the same size, this means that the size of the actual cake is
5/8 of the initial cake.
Example 2:
There
are 30 people in a classroom, 10 boys and 20 girls. Furthermore, in the
classroom 10 of the girls play soccer and 6 of the boys too.
If this
proportionality is the same for all the courses and there are 520 students in the
school answer
these questions:
How
many students in the school play soccer?
How
many girls in the school play soccer?
If
we choose 1/30 of the boys and 2/30 of the girls to create random teams, How
many five people teams can we create?
First
we know that in one classroom we have 16 people of 30 that play soccer so the
proportion would be 16/30. This means that for each 30 students 16 play soccer.
With fractional calculus:
It’s
an insane fraction, why don’t simplify it?
That’s
great but we can do better:
It
is hard to explain what I just did. First you divides numerator and denominator
by 10 and you obtain 832/3. Next, you cannot divide 832/3 so I seek the next
number divisible by 3 (that is 831). Then I’ve separated the number 832=831+1
and we obtain now a number divisible between 3.
As
you suppose we cannot cut a person to obtain 1/3 of that person, so in the
school there are 277 people that play soccer.
To
obtain the girls we have two ways: Obtain the girls from the total number of
students or, because we know the proportion of girls that play soccer,
calculate the number from the soccer players.
As we have space we’ll write both
but the two of them are alright.
First
for the total number of students. We have 520 students and the girls are 10/30
so:
The
other way. We have 277+1/3 students that play soccer, from this group 10 for
each 16 soccer players are girls, so:
It’s
not magic, it’s not a trap and of course you can do it with a calculator
easily, this calculus were though for teaching so you can understand what is
the logic behind the calculation.
So there are 173 girls that play soccer.
Finally,
How many teams can we create?
Well,
we have that in total 520 people and we get 1/30 of the boys and 2/30
of the girls, so:
So 5 boys and
23 girls.
So the total is 5+23=28. If we want to know how
many five members teams we can create we only must divide by 5:
We
have 5 teams and 3 students are out of the teams or are extra player.