Our software, Algebra Buster solves any algebra problem you enter (including all the problems found in tutorials below and much more! ). It gives you all the solution steps and clear explanations. Click here for demo or  to find out more about this incredible program!

 
        
 
Dividing Whole Numbers by Fractions

Dividing Whole Numbers by Fractions

Question: What is the value of ?

Misconception

The value of ¸ is equivalent to 3 ÷ 4 ¸ and hence has value or 0.75 Similarly ¸ is equivalent to 5 ÷ 2 ¸ and hence has value 2.5 or .

Correct

The division , means how many are there in the number 3. Clearly there are 4 quarters in 1 and hence 3×4 (=12) in 3. So .

 

Further Explanation

We must learn NEVER to be influenced by what things look like: the meaning of dividing by 2, dividing by 5, etc. is clear: the concept of dividing by a quarter is, however, less straightforward and requires more thought.

Think of ¸ as 'the number of 's that fit into 3'. There are 4 quarters in 1, so in 3 there are 3 × 4 quarters in 3 as can be seen in the diagram below.

So

Generally

Hence, for example,

There is another way to approach this task logically which we will demonstrate with . Use the problem solving method – 'if you are having difficulties, find something similar which you know you CAN do and work out the difference between this and the problem given'.

The difficult part here is dividing by a fraction .

Start with something similar which is straightforward: just divide the 6 by 3 . Now continue by examining the effect of the difference between what we did and what was given (using clearer terminology to refer to division, i.e. divide between).

We divided the 6 by 3 instead of by the given (which is, of course, less than 3). When a cake is divided between a certain number of people, each gets a certain portion. Dividing it between fewer people results in each one receiving a larger portion. How much larger? If it is divided between, say, 5 times fewer people, each portions would become 5 times larger. We arrived at 2 by dividing the 6 by 3. We should have divided by something that is 5 times smaller than the 3, (by ), so, the result should be 5 times larger than the . Thus we deduce that our ¸ must mean . Generalising,

Yet another way of determining is to forget about the unclear meaning of dividing by a fraction and to do whatever yields a result which doesn't contradict other things that are already established.

Whatever we mean by , we already know that its result, r, must be such that r × k will be equal to p. i.e. in , r must be such that r × k = p, (e.g. is 17 because 17 × 11 = 187 ). Following this for , we simply seek a result which gives 3 when multiplied by a . The question then becomes: "what times a quarter is 3?" , or using a familiar rephrasing "a quarter of what is 3?" (The answer is of course 12.) In summary to determine the value of r in , find which value of r satisfies .


TUTORIAL HOME
difference squares
fractions
dividing rational expressions
adding substracting like fractions
arithmetics
factoring polynomials
multiplying fractions
equations lines slope intercept
arithmetic operations
adding substracting rational numbers
adding substracting rational expressions
sum roots quadratics
multiplying numbers
adding substracting rational expressions unlike denominators
radicals
solving quadratic inequalities
expansion product binomials
laws exponents
simplifying fractions
adding substracting polynomials
multiplying mixed numbers
mathematical terms
calculations negative numbers
comparing decimals
multipliying increases decreases number
solving inequalities fractions parentheses
multiplying dividing monomials
inequalities
decimals fractions
distributive law brackets parentheses
improper fractions mixed numbers
evaluating simple formulas
algebraic operations simplification
adding substracting fractions
adding fractions
equations
multiplying polynomials
algebraic expresions containing radicals
scientific notation
solving systems equations elimination
adding algebraic fractions
operations fractions
dividing mixed numbers
subtracting mixed numbers remaining
solving quadratic equations completing square
percents
factoring expressions
decimals
estimating sums differences mixed numbers
square roots real numbers
adding substracting square roots
fractions percents decimals
collecting like terms algebraic expressions
graphing inequalities
solving compound inequalities
graphing systems equations
multiplying multiples numbers
solving rational equations
dividing whole numbers fractions
multiplying monomials
simplifying complex fractions
quadratic inequalities
algebraic fractions
equations lines point slope
coordinate system
multiplying decimals
adding substracting mixed numbers
graphing systems inequalities
graphing parabolas
fractional exponents
mixed numbers complex fractions
simplifying rational expressions
estimating products quotients mixed numbers
multiplying dividing rational numbers
monomial factors
positive integral divisors
multiplying rational expressions
dividing monomials
literal numbers
adding substracting unlike fractions
parallel perpendicular lines
sum squares
solving systems equations substitution
solving systems equations elimination multiplication
relatively prime numbers
powers ten
prime composite numbers
prime factors
equivalent fractions reducing cancelalation
evaluating expressions fractions
multiplying dividing square roots
pythagoras theorem
rational expressions
powers
adding substracting rational expressions like denominators
arithmetic operations numerical fractions
calculations hundreds thousands
equivalent fractions
arithmetic aproximate numbers
dividing fractions
rational numbers
operations fractions mixed numbers
simplifying square roots
exponents
solving linear equations graphically
roots radicals
solving inequalities
graph lines
brackets
prime numbers
multiplying dividing fractions
slope lines
negative exponents
special products
decimals equivalent fractions
rationalizing denominators
straight lines
subtracting fractions
simple partial fractions
sum difference cubes
powers roots
factoring binomials trinomials
variables expressions

 










 
 

 

 

 

 
Home    Why Algebra Buster?    Guarantee    Testimonials    Ordering    FAQ    About Us
Tutoring    Forum    Bibliography of Textbooks
 

Click here for a comprehensive guide to algebra textbooks, including descriptions and student reviews!

2008-11-20 04:25:35