adding mixed numbers © math as a second language all rights reserved next #7 taking the fear out of...

Adding Mixed

Numbers

Adding Mixed

Numbers© Math As A Second Language All Rights Reserved

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#7

Taking the Fearout of Math

131122+

Students have probably added mixed numbers many times and not even

realized that they were doing it.

© Math As A Second Language All Rights Reserved

A Non-Technical Introductionnextnext

For example, suppose you rent a machineat a cost of $12 per hour and you have to pay for it to the nearest minute. Suppose further that you used the machine for 6 hours and

40 minutes one day and for 3 hours and 45 minutes the next day, and you want toocompute how much it cost you to rent the

machine during the two days.

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One way to proceed would be to start by computing the total time you used

the machine, namely…

6 hours + 40 minutes + 3 hours + 45 minutes

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Without thinking about it consciously you would probably use the properties of

arithmetic to conclude that…

6 hours + 40 minutes + 3 hours + 45 minutes =

6 hours + 3 hours + 40 minutes + 45 minutes =

9 hours + 85 minutes

Recognizing that 85 minutes is more than 1 hour, the sum could be

rewritten as…


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9 hours + 60 minutes + 25 minutes =

9 hours + 1 hour + 25 minutes =

10 hours + 25 minutes

Since 10 hours and 25 minutes is more than 10 hours but less than 11 hours, we

see that at a cost of $12 per hour, the total cost is more than 10 × $12 or $120 but less

than 11 × $12 or $132.


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To find the exact answer, we might next observe that…

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25 minutes =

2560

of an hour =5

12of an hour

…and therefore that the additional cost is 5/12 of $12 or $5.

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In summary, the cost for 10 hours is $120 and the cost for the additional

25 minutes is $5. Therefore the total cost is $120 + $5 or $125.

If we suffered from “fraction aversion”, we might have decided to work solely with

whole numbers. In that case, we would have converted the times into minutes

rather than hours.

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Since there are 60 minutes in an hour, 10 hours would equal 600 minutes, and…

Using the fact that 100 cents equals one dollar, the total cost is...

625 × 20 cents = 12,500 cents = $125.00.This checks with the answer,

we obtained previously.

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10 hours + 25 minutes = 600 minutes + 25 minutes

= 625 minutes

Although the arithmetic appears to be more abstract, adding mixed numbers is no more

complicated than what we did above.


Adding Mixed Numbersnextnext

As we saw in our previous presentation, if 38 corn breads are divided equally

among 7 people the number of corn breads each person gets is 5 + 3/7.

The fact that the plus sign separates the whole number from the fraction makes

it easy for one to assume that only the fraction is modifying “corn breads”1.


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To avoid the possibility of misinterpretation, we should use

parentheses and write (5 + 3/7) corn breads.

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1 This type of misinterpretation happens frequently to beginning students in algebra. For example, when they are trying to simplify an expression such as

(a + b)x they rewrite the expression as a + bx rather than as ax + bx.

However, because it’s cumbersome to write the answer in this form, we agree to omit the plus sign and write the fractional part immediately to the right of the whole

number; that is, 53/7 means (5 + 3/7).


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In any event, the associative and commutative properties of addition now

show us that to add two mixed numbers…

We simply have to add the two whole numbers to get the whole number part of the sum and the two fractions to get the

fraction part of the sum.

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For example, let’s find the sum of 52/7 and 64/7 as a mixed number.

First, we may rewrite the problem in the form…

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(5 + 2/7) + (6 + 4/7)

Then using the associative and commutative properties of addition, we may rewrite the

above expression in the form…

(5 + 6) + (2/7 + 4/7)

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…from which we see, the sum is 11 + 6/7 or in more standard form, 116/7.

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If we are uncomfortable with mixed numbers but are more comfortable with

common fractions, we can translate every mixed number problem into an equivalent

(improper) fraction problem.

Conversely, if we prefer, we can convert improper fractions into mixed numbers.

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Hence, 52/7 + 64/7 = 37/7 + 46/7 = 83/7 = 116/7.

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52/7 = 5 + 2/7

= 5/1 + 2/7

= 35/7 + 2/7

= 37/7

64/7 = 6 + 4/7

= 6/1 + 4/7

= 42/7 + 4/7

= 46/7

Mixed Numbers to Improper Fractions

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However, in this case it is much simpler just to add the whole numbers and to add the

fractional parts.

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For example, if one person had 52/7 corn breads and another person has 64/7 corn breads, the total corn breads is

116/7 corn breads.


Using the Corn Breadnextnext

123456

corn breadcorn bread1/71/7

1/71/7

1/71/7

1/7

52/7 64/7

1/71/7

1/71/7

1/71/7

1/7

1/71/7

116/7+ =

12345

6789

1011

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As another example, let’s find the sum of 62/3 and 33/4 as a mixed number.

First, we may rewrite the problem in the form…

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(6 + 2/3) + (3 + 3/4)

Then using the associative and commutative properties of addition, we may rewrite the

above expression in the form…

(6 + 3) + (2/3 + 3/4)

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A common denominator for 2/3 and 3/4 is 12.

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62/3 + 33/4

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Since 2/3 equals 8/12, and 3/4 equals 9/12, we

can combine the fractions using the common denominator 12…

= (6 + 3) + (2/3 + 3/4)

= (6 + 3) + (8/12 + 9/12)

= 9 + (17/12)

= 9 + (12/12 + 5/12)

= 9 + (1 + 5/12)

= 10 + 5/12

= 105/12

We would not write the answer as 917/12. Remember that to be a mixed number, the fractional part has to be less than 1.


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Notice that because the denominators were not the same, we had to find a

common denominator before we could add the fractions.

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Notice that because the denominator was 12, we traded 12 twelfths for 1 whole

rather ten twelfths.


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Very often students make the mistake of not looking at the denominator of a mixed number and continue to “trade in” by tens

no matter what the denominator was.

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Once again we could have solved this problem by converting the two mixed

numbers to improper fractions and then using our algorithm for adding fractions.


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However, this process is both more cumbersome and more “mechanical”.

This problem brings us back full circle to our opening discussion.


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If we assume that 62/3 and 3 3/4 modify “hours”, 40 minutes is 2/3 of an hour and

45 minutes is 3/4 of an hour. So if you rented a machine for 6 hours and 40

minutes one day and for 3 hours and 45 minutes the next day, the total amount of

time that you rented the machine was 10 hours and 25 minutes.

If we convert our minutes to hours we would obtain…


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6 hours + 40 minutes + 3 hours + 45 minutes =

6 hours + 3 hours + 40 minutes + 45 minutes =

9 hours + 85 minutes =

9 hours + 60 minutes + 25 minutes =

9 hours + 1 hour + 25 minutes =

10 hours + 25 minutes =

10 hours + 25/60 hours =

10 hours + 5/12 hours =

105/12 hours

105/12 hours is exactly what we obtained earlier in our presentation.


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Sometimes using such nouns as hours and minutes, as we just did above, allows

students to better visualize what is happening when we add mixed numbers.

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In our next section we will discuss the process of subtracting one mixed

number from another.


62/3 – 33/4 = ?

adding mixed numbers © math as a second language all rights reserved next #7 taking the fear out of...

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