Problem Type
Fractions
Diagram
Showing
Relationship
Sara and Amy went to the mall.
After Sara spent 3/7 of her
money and Amy spent $45, they
each had the same amount left.
If they had a total of $375 when
they started, how much do they
each have left?
Sara
If 4/9 of the money is $164,
what is 2/5 of the money?
Step 1: 4/9 = $164
W R
3 g
G
G
Amy
$45
$375 in total
Description of
the
Relationship
Tim has a bag of
marbles. One-fifth of
the marbles are white,
and there are 3 more
red marbles than white
ones. The remaining
marbles are green. If
there are 35 marbles in
the bag how many are
green?
1. $375– 45=$330
2. 11u = $330
1u = $30
4u = $120
There are 35 marbles
total.
1. 5u = 35
1u = 7
2. 1u-3 = 4
3. 2u + 4 = 18
Step 2:
? ?
1. $164÷4 = $41
1u = $41
2. 9 x $41 = $369
3. $369 ÷ 5 = $73.80
4. 2 x $73.80 = $147.60
They each have $120 remaining.
Number
Sentence
X + 4/7 x + $45 = $375 What
Sara started with.
1/5x + (1/5x+3)+2/5x +
(1/5x-3) = 35
4/9x = $164
2/5x = ?
Fractions
When Erin and Amy went shopping,
they started with a total of $91.
Amy spent $25 and Erin spent 3/5
of her money. At that point, Amy
realized her remaining money was
3 times Erin’s remaining money.
How much money did Amy have
when she started shopping?
Bob and Jane have a
combined age of 82
years. 5 years ago,
Jane’s age was 1/5 of
Bob’s age. How old are
each of them currently?
Diagram
Showing
Relationship
Erin:
5 years ago
Jane:
Stephanie had 20 more
clients than Jeff. After
Jeff gave Stephanie 10
of his clients he noticed
that he had 1/2 the
number of clients as
Stephanie. How many
clients did they both
end up with?
To start with:
Stephanie:
20
Amy:
$25
Bob:
Jeff:
Currently:
Jane:
5
After:
Stephanie:
Bob:
Jeff’s:
Erin and Amy had a total of $91.
10
5
Description
of
Relationship
Amy has $36
Combined 82 years
Step 1:
5 + 5 = 10
80 – 10 = 72
Step 2:
72 ÷ 6 = 12
Jane’s age is 5 + 12=17
Bob’s age 5 *12 + 5 = 65
2/5x + 6/5x +$25 = $91
X + 5 + 5x + 5 = 82
6x + 10 = 82
Step 1:
$91 - $25 = $66
Step 2:
11u = $66 so 1u = $6
Number
Sentence
20
10
10
Step 1:
10 + 20 + 10 = 40
Step 2:
1u = 40
2u= 80
Jeff has 40 clients and
Stephanie has 80
clients.
Stephanie started with
50 and Jeff had 60, 20
less than Stephanie.
Fractions
And Ratios
2/5 of all the boys at school and ¾ of
all of the girls went to a concert. If
there were the same number of boys
and girls from the school at the
concert, what fraction of the school’s
students attended the concert?
Paul baked 144 cookies and
brownies. Initially the ratio of
cookies to brownies was 5:3.
After Steve ate 2/5 of the
cookies and some of the
brownies, the cookies
outnumbered the brownies 6:1.
How many brownies did she
eat?
Diagram
Showing the
Relationship
2/5 of the boys
Before: 144 total 5 : 3 ratio
Cookies
¾ of the girls
Brownies
When Tim and Ken left
for vacation, the ratio of
Tim’s money to Ken’s
money was 7:3. After
the first day of the trip,
Tim gave Ken $30
making the new ratio of
their cash positions 3:2.
At the start of their
vacation, how much
more money did Tim
have than Ken?
Before: 7:3 ratio
Tim’s Money
? ? ? ?
Ken’s Money
After: Cookies to Brownies is in
a 6 : 1 ratio
Cookies
Brownies
Description
of
Relationship
For this situation, 2/5 of the boys is
equivalent to ¾ of the girls.
First, a common multiple of 6 is used.
So the 2 shaded units of boys needs to
be divided into 6 units and the 3
shaded units of girls needs to be
divided into 6 units so they are
equivalent.
a. 6 + 6 = 12
b. 15 + 8 = 23
c. 12/23 of the school attended
the concert
Number
Sentence
a. 8u = 144 so 1u = 18 and
1/2u = 9
b. Barbara ate 9 brownies
After: 3:2 ratio
Tim’s Money
Ken’s Money
This problem works
because the $30
transferred from Tim to
Ken leaves their cash
points in a 3:2 ratio.
a. 1u = $30
b. 4u = $120
Tim started with $120
more than Ken.