Dividing Mixed Numbers © Math As A Second Language All Rights Reserved next #7 Taking the Fear out of Math 1313 3 1212 2

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Taking the Fearout of Math

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As in our previous discussions with mixed numbers, let’s beginwith a “real world” example…

© Math As A Second Language All Rights Reserved

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What is the cost per poundif 33/4 pounds of pears cost

$4.50?

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33/4 pounds is the same amount as 15 fourths of a pound.

If 15 fourths of a pound cost $4.50 (that is, 450 cents), then each fourth

of a pound costs…

450 cents ÷ 15 (or 30 cents).

And since there are 4 fourths of a pound in a pound, each pound of pears costs

4 × 30 cents or $1.20.

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The cost per pound in dollars is given by the formula…

dollars per pound =

total cost in dollars ÷ number of pounds.

This formula is the same whether we are dealing with whole numbers,

common fractions or mixed numbers.

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cost per pound =

41/2 dollars ÷ 33/4 pounds


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Rewriting $4.50 as $41/2, our answertakes the form…

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which can be rewritten as…

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cost per pound =

(41/2 ÷ 33/4) dollars per pound

If you are uncomfortable using mixed numbers to perform the division, you

may use improper fractions and rewrite the above computation in the form…


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cost per pound =

(9/2 ÷ 15/4) dollars per pound

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next From our study of fractions we know that…

41/2 ÷ 33/4 dollars per pound

= 9/2 ÷ 15/4 dollars per pound

= 9/2 × 4/15 dollars per pound

= 36/30 dollars per pound



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And since there are 100 cents per dollar…

1/5 of a dollar = 100 cents ÷ 5 = 20 cents

and therefore…

11/5 dollars per pound = $1.20 per pound

which agrees with our previous answer.


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As we shall show later in this presentation, it is possible to divide

mixed numbers without having to first convert them to improper fractions.

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However, this can be quite tedious. So while the idea of converting mixed numbers to improper fractions is quite

common, it is especially useful when we deal with division.


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However, converting the division of mixed numbers to division of improper

fractions tends to obscure what is actually happening.

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For example, suppose we want to divide 41/2 by 21/2 , We could begin by rewriting 41/2 ÷ 21/2 as 9/2 ÷ 5/2. We would then use the “invert and multiply” rule to obtain

9/2 × 2/5 or 9/5 . Finally, we divide 9 by 5 and obtain 14/5 as the answer.


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Since the definition of division hasn’t changed 41/2 ÷ 21/2 means the number we must multiply by 2 1/2 to obtain 41/2

as the product.

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A quick check shows that this is the case…

21/2 × 14/5 =5/2 × 9/5 =

9/2 =

41/2


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As a “reality check”, it is helpful to estimate an answer before proceeding with the actual computation. Noticing that 41/2 is

“around” 4 and 21/2 is “around” 2, we can estimate that the answer should be

“around” 4 ÷ 2 or 2.

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Actually, since 5 ÷ 21/2 is exactly 2, 41/2 ÷ 21/2 must be a “little less” than 2.

Thus, 14/5 is a plausible answer.

Notes


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As a practical application, suppose we can buy 21/2 pounds of cheese for $ 41/2. Then the price per pound (that is, “dollars per pound”) is obtained by dividing $41/2 by 2 1/2 pounds.

The quotient tells us that the cost of the cheese is $14/5 or $1.80 per pound.

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In terms of relative size, what the quotient tells us is that not only is 41/2 almost twice as much as 21/2 but it’s exactly 14/5 times as much.

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next For additional practice, let’s use the

above method to express 6 1/3 ÷ 13/4 as a mixed number.

61/3 ÷ 13/4

= 19/3 ÷ 7/4

= 19/3 × 4/7

= 76/21

= 76 ÷ 21

= 313/21


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To check that our answer is at least reasonable, we observe that rounded off to

the nearest whole number 61/3 = 6 and 13/4 = 2. Hence, our answer should be

“reasonably close to” 6 ÷ 2 or 3.

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However, once we obtain our answer, we can check to see if it’s correct by remembering that 61/3 ÷ 13/4 = 313/21

means that 313/21 × 13/4 = 61/3.

Notes

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next Thus, to check our answer we

compute the value of 313/21 × 13/4 to verify that it is equal to 61/3.

313/21 × 13/4

= 76/21 × 7/4

= (19 × 2 × 2 × 7)/(3 × 7 × 2 × 2)

= 19/3

= 61/3

= (76 × 7)/(21 ×4)


next As we mentioned earlier, while

converting mixed numbers to improperfractions gives us the desired result when we divide two mixed numbers, the fact is that it doesn’t give us much insight as to

what is actually happening.

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From a mathematical perspective, it would be nice to know that the arithmetic of

mixed numbers is self-contained (at least in the sense that we aren’t forced to rewrite mixed numbers as improper

fraction in order to do the arithmetic).


next From a teaching point of view, we might want to illustrate how we can divide mixed

numbers in ways that may be more intuitive to students.

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To divide 41/2 by 21/2, think of both mixed numbers as modifying, say, “a carton of

books” where each carton contains 2 books1

Using Mixed Numbers as Adjectives Modifying the Same Noun.

note

1 More generally, we would use the (least) common multiple of both denominators. For example, had the problem been 41/3 ÷ 25/7 , we would have assumed that each

carton contained 21 books and then multiplied both the dividend and the divisor by 21.

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next In this case, 41/2 cartons is another

name for 9 books and 21/2 cartons isanother name for 5 books. Therefore, we may visualize the problem in terms of the

following steps…

41/2 ÷ 21/2

= 9 ÷ 5 = 14/5

= 41/2 cartons ÷ 21/2 cartons

= 9 books ÷ 5 books


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Students might find it interesting to see that we can divide mixed numbers by

“translating” the problem into an equivalent whole number problem without

having to refer to such things as booksand cartons.

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The key is that it is still a fact that we do not change a quotient when we multiply both the dividend and the divisor by the

same (non zero) whole number.

Multiplying Both Numbers in a Ratio by the Same Number


next To this end, if we are given the

division problem 41/2 ÷ 21/2, we simply multiply both numbers by 2…2

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and obtain… 41/2 ÷ 21/2

= (41/2 × 2) ÷ (21/2 × 2)

= 14/5

= 9 ÷ 5

note2 When we multiply a mixed number by the denominator of its fractional part we always

obtain a whole number. For example, 5 × 42/5 = 5 × (4 + 2 2/5) = (5 × 4) + (5 × 2/5) = 20 + 2 = 22.

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next We can now use the above method to

write 61/3 ÷ 23/5 as a mixed number.

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Since the denominators are 3 and 5, we can eliminate them by multiplying

both numbers by 15.

15 × 61/3

= (15 × 6) + (15 × 1/3)

= 95

= 90 + 5

15 × 23/5

= (15 × 2) + (15 × 3/5)

= 39

= 30 + 9

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Since…next

= 95 ÷ 39

15 × 61/3 = 95 and 15 × 23/5 = 39

= 217/39

…then, 61/3 ÷ 23/5


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As a check we see that…next

= 95/39 × 13/5

= 19/3

217/39 × 23/5

= (19 × 5 × 13)/(3 × 13 × 5)

= 61/3

= (95 ×13)/(39 × 5)


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In terms of relative size, the above result tells us that 61/3 is approximately 21/2 times

as great as 23/5.

(It is actually 217/39 times as great.)

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We don’t usually think about using common denominators when we want to

divide two fractions, but we can. The denominator of a fraction is the noun and

when we divide two numbers thatmodify the same noun, the nouns “cancel”.

Notes


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Thus, if we decide to rewrite the mixed numbers as improper fractions, we obtain…

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= 19/3 ÷ 13/5

= 95/15 ÷ 39/15

61/3 ÷ 23/5

= (19 × 5) /(3 × 5) ÷ (13 × 3)/(5 × 3)

= 95 fifteenths ÷ 39 fifteenths

= 217/39

= 95 ÷ 39


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While the topic might be beyond the scope of the elementary school student, it is interesting to note that the long division algorithm for whole numbers, as a form of

rapid subtractions, also applies to the division of mixed number.

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For example, 61/3 ÷ 23/5 = ( ) means the same thing 23/5 × ( ) = 61/3.


next Using trial-and error to solve this

equation we see that…

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23/5 × 1 = 23/5

23/5 × 2 = 46/5 = 51/5

23/5 × 3 = 69/5 = 74/5

less than 61/3

greater than 61/3

In other words, 61/3 is more than twice as big

as 23/5 but less than three times as big.

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next The difference between 61/3 and 51/5 is

61/3 – 51/5 = 65/15 – 53/15 = 12/15. Hence, if we use the long division algorithm we see that…

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2

51/5

R

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23/5 61/3

…and if we now write the remainder over the divisor, we see that…

12/1512/15

61/3 ÷ 23/5 =

2 +12/15

23/5


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We can simplify the complex fraction above by multiplying numerator and

denominator by 15.

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= 15 × 2

12/15

23/5

15 × 12/15

= (15 × 1) + (15 × 2/15)

= 17

15 × 23/5

= (15 × 2) + (15 × 3/5)

= 39

= 30 + 9


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Therefore…

61/3 ÷ 23/5

= 2 +12/15

23/5

= 17/39

= 217/39

23/5 61/3

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12/15

23/5

= 2 + 17/39

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This completes our discussion of the four

basic operations of arithmetic

using mixed numbers, and in our next

presentation we shall introduce the

notion of percents.


4/5 × 100 = 80%

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Dividing Mixed Numbers © Math As A Second Language All Rights Reserved next #7 Taking the Fear out of Math 1313 3 1212 2