Question 1100944: The repair department of the bicycle shop repairs three things: flat tires, bent handle bars and
ripped seats. Today in the repair department, 25% of the bikes had flat tires only, 5% had bent
handlebars only, and 10% had ripped seats only. Just 1/12th of the bikes had all three repairs to
do: flat tires, bent handlebars and ripped seats. No bikes were completely fixed and there are a
total of 101 repairs to be made. How many bikes are in the repair department? How many bikes
need two repairs? If less than half of all the bikes have a ripped seat, what is the range of bikes
that need both the tires and handlebars repaired without needing to fix the seat?
Found 2 solutions by KMST, richwmiller:
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! ONE APPROACH:
 = number of bikes in the shop waiting to be repaired.
Of those,
25% have flat tires only,
5% have bent handlebars only, and
10% have ripped seats only.
That adds up to 25%+5%+10%=40% .
So, the number of bikes needing only 1 repair is
 .
The number of bikes needing 3 repairs is  .
The rest of the bikes is the fraction of all bikes that need exactly 2 repairs.
As a fraction of all the bikes, that is
 .
As a number of bikes, it is
 .
Counting the number of repairs needed, we have
 repairs for the bikes that need only 1 repair,
 repairs for the  bikes that need exactly 2 repairs,
and
 repairs for the bikes that need 3 repairs.
The number of repairs adds up to
There are  bikes in the repair department.
Of those  bikes,
 need exactly two repairs,
 , or  bikes, need all 3 repairs including their ripped seats,
25% , or  bikes, only need flat tires fixed,
5% , or  bikes, only need bent handlebars fixed, and
10%, or  bikes only need their ripped seats fixed.
In pictures, we could represent is as
 .
The last question asks for the range of values for value of  ,
The number of "bikes
that need both the tires and handlebars repaired without needing to fix the seat".
From the picture,
if  ,  is the greatest possible ;
If  ,  is the least possible .
Without a picture:
If  is the total number of ripped seat repairs,
we do not know exactly, but
it includes at least the that need only the seat repaired,
and the that need all 3 repairs, so
 or  .
If less than half the bikes have a ripped seat
(meaning  or less bikes),
we also know that  , so
 .
If we exclude
those bikes with ripped seats,
the bikes that only need flat tires fixed,
and the bikes that only need bent handlebars fixed,
we are excluding  ,
what is left is the group of bikes that needs
"both the tires and handlebars repaired without needing to fix the seat."
That number is  .
As ,
 .
There are at least  and at most  bikes that need
"both the tires and handlebars repaired without needing to fix the seat."
The range can be expressed as [13,31], or as 
Answer by richwmiller(17219)  (Show Source):
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