Employing the Definition of a Fraction to Solve Problems

1
Employing the Definition of a Fraction to Solve
Problems
2

• Given the fraction a/b,
• the denominator, b, tells how many equal parts to
  separate the whole into and
• the numerator, a, tells how many of those parts
  this fraction represents.

We will use this understanding of a fraction to
solve three problems.
3
Problem 1
Three-fifths of an estate goes to the deceaseds
three children. The remainder is divided among
six charities. What fractional part of the total
estate will go to one of the six charities?
Solve using a rectangular model.
Let the rectangle below represent the whole
estate.
Break the estate into five equal parts.
4
Three-fifths of an estate goes to the deceaseds
three children. The remainder is divided among
six charities. What fractional part of the total
estate will go to one of the six charities?
Solve using a rectangular model.
Three of the fifths represent the childrens
share.
The remaining two fifths represents the portion
shared
by the six charities.
Childrens share
Charities share
Consider how the charities share can be broken
into six equal parts.
5
Three-fifths of an estate goes to the deceaseds
three children. The remainder is divided among
six charities. What fractional part of the total
estate will go to one of the six charities?
Solve using a rectangular model.
Draw three horizontal lines to divide the
charities share
into six equal parts.
One of the six parts represents the share of one
charity.
One charity
Childrens share
Charities share
Consider what fractional part of the total estate
this green rectangle represents.
6
Three-fifths of an estate goes to the deceaseds
three children. The remainder is divided among
six charities. What fractional part of the total
estate will go to one of the six charities?
Solve using a rectangular model.
Extend the lines.
Now the whole estate is broken into equal parts
the
size of one charity's share.
One charity
Childrens share
Charities share
So, what fractional part of the total estate does
green rectangle represents?
7
Three-fifths of an estate goes to the deceaseds
three children. The remainder is divided among
six charities. What fractional part of the total
estate will go to one of the six charities?
Solve using a rectangular model.
Simply add up the little rectangles.
You get 15. The single green rectangle
represents 1/15 of
the large rectangle.
One charity
Childrens share
Charities share
One charity will receive 1/15 of the estate.
8
Notice that this was solve by simply applying the
definitions of the numerator and the denominator.
The denominator tells us how many equal parts to
divide the whole into.
The numerator tells us how many of those equal
parts this particular fraction represents.
9
Problem 2
Three-fifths of the students in Ms. Estevezs
class play on the soccer team. If there are four
more students in class who play soccer than dont
play soccer, how many students are there in Ms.
Estevezs class? Solve using a rectangular
model.
Let the rectangle below represent the all the
students
in Ms. Estevezs class.
Break the class into five equal groups of
students.
10
Three-fifths of the students in Ms. Estevezs
class play on the soccer team. If there are four
more students in class who play soccer than dont
play soccer, how many students are there in Ms.
Estevezs class? Solve using a rectangular
model.
Three of the fifths represent the students who
play soccer.
The remaining two fifths represents students who
do Not play soccer.
Students who play soccer.
Students who do NOT play soccer.
What in our model can represent the four more
students?
11
Three-fifths of the students in Ms. Estevezs
class play on the soccer team. If there are four
more students in class who play soccer than dont
play soccer, how many students are there in Ms.
Estevezs class? Solve using a rectangular
model.
Note that the students who play soccer are
represented by.
one more rectangular part than those who do not
play.
Here are the four more students who play soccer!
four students
Students who play soccer.
Students who do NOT play soccer.
Now use the definition of the fraction to
determine how many students are represented by
each other rectangle.
12
Three-fifths of the students in Ms. Estevezs
class play on the soccer team. If there are four
more students in class who play soccer than dont
play soccer, how many students are there in Ms.
Estevezs class? Solve using a rectangular
model.
If one of the parts has four students, all the
other parts
must have four students.
This is because each of the parts must be equal.
four students
4
4
4
4
4
Students who play soccer.
Students who do NOT play soccer.
Now answer the question posed by the problem.
13
Three-fifths of the students in Ms. Estevezs
class play on the soccer team. If there are four
more students in class who play soccer than dont
play soccer, how many students are there in Ms.
Estevezs class? Solve using a rectangular
model.
Simply add up the students.
four students
4
4
4
4
4
Students who play soccer.
Students who do NOT play soccer.
There are 20 students in Ms. Estevezs class.
14
Problem 3
Mr. OLeary started the school year with a large
box of pencils. By November he had given out 3/8
of his pencils. Of those left, 7/10 were given
out by March. He now had 30 pencils left in his
box. How many pencils did Mr. OLeary start
with?
Let the rectangle below represent the number of
pencils
that Mr. OLeary starts the school year with.
Break the pencils into eight equal parts.
15
Problem 3
Mr. OLeary started the school year with a large
box of pencils. By November he had given out 3/8
of his pencils. Of those left, 7/10 were given
out by March. He now had 30 pencils left in his
box. How many pencils did Mr. OLeary start
with?
Three eighths of the pencils are gone by
November.
Five eighths of the pencils are still left as of
November.
Pencils gone by November
Pencils left as of November
How are we to represent the 7/10 gone by March?
16
Problem 3
Mr. OLeary started the school year with a large
box of pencils. By November he had given out 3/8
of his pencils. Of those left, 7/10 were given
out by March. He now had 30 pencils left in his
box. How many pencils did Mr. OLeary start
with?
Note that five rectangles represent the pencils
remaining in
November. We need to represent 7/10 of these.
Begin by drawing a horizontal line through the
remaining
pencils to break them into ten parts.
Pencils gone by November
Pencils left as of November
17
Problem 3
Mr. OLeary started the school year with a large
box of pencils. By November he had given out 3/8
of his pencils. Of those left, 7/10 were given
out by March. He now had 30 pencils left in his
box. How many pencils did Mr. OLeary start
with?
Shade the seven of these tenths that represent
the pencils
given out between November and March.
Pencils given out between November and March.
Pencils gone by November
Pencils left as of November
What do the three remaining white rectangles
represent?
18
Problem 3
Mr. OLeary started the school year with a large
box of pencils. By November he had given out 3/8
of his pencils. Of those left, 7/10 were given
out by March. He now had 30 pencils left in his
box. How many pencils did Mr. OLeary start
with?
Ten pencils belong in each white rectangle,
and in each other rectangle of the same size.
30 pencils left in March.
Pencils given out between November and March.
10
10
10
10
10
10
10
10
10
10
Pencils gone by November
Pencils left as of November
19
Problem 3
Mr. OLeary started the school year with a large
box of pencils. By November he had given out 3/8
of his pencils. Of those left, 7/10 were given
out by March. He now had 30 pencils left in his
box. How many pencils did Mr. OLeary start
with?
We can see that the remaining rectangles are
twice the
size and so they each represent 20 pencils.
30 pencils left in March.
Pencils given out between November and March.
10
10
10
10
10
20
20
20
10
10
10
10
10
Pencils gone by November
Pencils left as of November
20
Problem 3
Mr. OLeary started the school year with a large
box of pencils. By November he had given out 3/8
of his pencils. Of those left, 7/10 were given
out by March. He now had 30 pencils left in his
box. How many pencils did Mr. OLeary start
with?
Or, we can extend the horizontal line.
Now each rectangle is the same size and
represents
10 pencils.
30 pencils left in March.
Pencils given out between November and March.
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Pencils gone by November
Pencils left as of November
Adding up, we find that Mr. OLeary began the
school year with 160 pencils.
21

• On the exam you will need to use the
  rectangular model method demonstrated here to
  solve similar problems.

Right click and select End Show. Then Close to
return.