Lesson Advanced problems on ratios

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Advanced problems on ratios

Problem 1

Alan and Bob share a sum of money. If Alan gives 1/4 of his money to Bob, Bob will have twice as much as Alan.
What was the ratio of Alan's money to Bob's money at first.

Solution

Let A = amount of Alan's money,
    B = amount of Bob's  money.


The condition says that  = 2.    (1)


Simplify the ratio in the left side, step by step. You will have

 =  =  +  =  +  =  +  =  + .


So we can write (1) in an equivalent form

 +  = 2,


which implies  

 =  - ,  ====>   =   ====>   =  = .


Answer.  The ratio of Alan's money to Bob's money at first was   = .

Problem 2

At an after school event, 5/9 of the students were 8th graders and the rest were 7th graders.
Of the students who were in 7th grade, the ratio of boys to girls is 3:5.
What fraction of ALL students at the after school event were boys who are in the 7th grade?

Solution

Let x = the number of the boys who were in 7th grade.


Let Z = the number of ALL students.


They want you determine the ratio x/Z.



    Let Y be the number of all 7th grade students.  


    Notice that  = , as it follows from the condition.


    Also from the condition  =  = .



Now we have EVERYTHING to calculate 


     =  =  =  = .


Thus we just found the ratio   = .


Answer.  The ratio under the question is .

Problem 3

In a condo complex, 2%2F3

of the men were married to 3%2F4

of the women. What is the ratio of married people
to the total adult population of the condo complex?

Solution

1.  From the condition,

         = .     (1)


    Multiply both sides by 12. You will get

        8M = 9W.           (2)

    or

        M =           (3)



2.  They ask you to evaluate the ratio of   +   (the amount of married people) to

    (M+W)  (the total adult population of the condo complex).  


    The numerator is   +  =    (4)


    In (4), replace 8M by 9W,  based on (2).  Then you can continue for the numerator

         +  =  =  =  = .



     The denominator is  M + W =  = .       ( <<<---=== I replaced  here M  by   based on (3). )



3.  Now the ratio under the question

        =  =  =  =  = .


Answer.  The ratio under the question = .

Problem 4

One night, a guest at an inn couldn't sleep and wanted a snack. He went down to the kitchen where he found a plate of cupcakes.
He was hungry and ate 1/6 of the cupcakes.
Later, his wife was hungry, couldn't sleep, and so she went to the kitchen and ate 1/5 of the remaining cupcakes.
At 10, the oldest son ate 1/4 of what his mother had left.
At 11, the second son ate 1/3 of the leftover cupcakes, and just after him,
the daughter ate 1/2 of what her brother had left, leaving only three cupcakes for the Inn Keeper's birthday party the next day.
How many cupcakes were originally on the plate?

Solution

1.  The amount which was left after the 1-st guest was .


2.  The amount which was left after his wife was  = .


3.  The amount which was left after the oldest son was  = .


3.  The amount which was left after the second son was  = .


4.  The amount which was left after the daughter was  = .


5.  We are given that   = 3.


    Hence,  c = 6*3 = 18.


Answer.  How many cupcakes were originally on the plate? - 18.

My other lessons on fractions in this site are
    - Using fractions to solve word problems
    - Using fractions to solve Travel problems
    - Calculations with fractions
    - Entertainment problems on fractions
    - OVERVIEW of my lessons on fractions

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