Dividing fractions: 3/5 ÷ 1/2 (video) | Khan Academy Dividing fractions Video transcript Divide and write the answer as a mixed number. And we have 3/5 divided by 1/2. Now, whenever you're dividing any fractions, you just have to remember that dividing by a fraction is the same thing as multiplying by its reciprocal. So this thing right here is the same thing as 3/5 times-- so this is our 3/5 right here, and instead of a division sign, you want a multiplication sign, and instead of a 1/2, you want to take the reciprocal of 1/2, which would be 2/1-- so times 2/1. So dividing by 1/2 is the exact same thing as multiplying by 2/1. And we just do this as a straightforward multiplication problem now. 3 times 2 is 6, so our new numerator is 6. 5 times 1 is 5. So 3/5 divided by 1/2 as an improper fraction is 6/5. Now, they want us to write it as at mixed number.

So we divide the 5 into the 6, figure out how many times it goes. That'll be the whole number part of the mixed number. And then whatever's left over will be the remaining numerator over 5. So what we'll do is take 5 into 6. 5 goes into 6 one time. 1 times 5 is 5. Subtract. You have a remainder of 1. So 6/5 is equal to one whole, or 5/5, and 1/5. This 1 comes from whatever is left over. And now we're done! 3/5 divided by 1/2 is 1 and 1/5. Now, the one thing that's not obvious is why did this work? Why is dividing by 1/2 the same thing as multiplying essentially by 2. 2/1 is the same thing as 2. And to do that, I'll do a little side-- fairly simple-- example, but hopefully, it gets the point across. Let me take four objects. So we have four objects: one, two, three, four.

So I have four objects, and if I were to divide into groups of two, so I want to divide it into groups of two. So that is one group of two and then that is another group of two, how many groups do I have? Well, 4 divided by 2, I have two groups of two, so that is equal to 2. Now, what if I took those same four objects: one, two, three, four. So I'm taking those same four objects. Instead of dividing them into groups of two, I want to divide them into groups of 1/2, which means each group will have half of an object in it. So let's say that would be one group right there. That is a second group. That is a third group. I think you see each group has half of a circle in it. That is the fourth. That's the fifth. That's the sixth. That's the seventh, and then that's the eighth. You have eight groups of 1/2, so this is equal to 8. And notice, now each of the objects became two groups. So you could say how many groups do you have?

Well, you have four objects and each of them became two groups. I'm looking for a different color. Each of them became two groups, and so you also have eight. So dividing by 1/2 is the same thing as multiplying by 2. And you could think about it with other numbers, but hopefully, that gives you a little bit of an intuition. Dividing Fractions By Whole Numbers Dividing Fractions By Whole Numbers Multiply the bottom number of the fraction by the whole number. To Divide a Fraction by a Whole Number: Step 1. Multiply the bottom number of the fraction by the whole number Step 2. Simplify the fraction (if needed) Example: Step 1. Multiply the bottom number of the fraction by the whole number: 1 2 × 3 = 1 6 Step 2. Fraction is already as simple as possible, so no need for step 2. Answer: 1 2 ÷ 3 = 1 6 With Pen and Paper And here is how to do it with a pen and paper (press the play button): Does it make sense? Parts of natural numbers -- A complete course in arithmetic To those collections we give a sequence of names and symbols.

1, 2, 3, 4, and so on, are the familiar numerals for the natural numbers. We calculate with those symbols, and so it has become conventional to call the symbols themselves "numbers." Yet a symbol is not what it symbolizes, what it stands for, which in this case is a number of discrete units. By a number in what follows, we will mean a natural number. Cardinal and ordinal The names of the natural numbers have two forms: cardinal and ordinal. The cardinal forms are One, two, three, four, and so on. They answer the question How many? The ordinal forms are First, second, third, fourth, and so on. They answer the question Which one? We will see that those ordinal numbers express division into equal parts. With the exception of "half," an ordinal number will name which part -- the third part, the fourth, the fifth, and so on. To each cardinal number except 1, there will correspond the name of a part. 2 Here are the first few multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40.

5 is the first multiple of 5; 10 is the second multiple; 15, the third; and so on. 3. What does it mean to say that a smaller number is a part of a larger number? It means that the larger number is a multiple of the smaller. Equivalently, the smaller is contained in the larger an exact number of times. 5, then, will be a certain part of each one of its multiples. 5, 10, 15, 20, 25, 30, 35, 40. Since 15 is the third multiple of 5, we say that 5 is the third part of 15. We use that same ordinal number to name the part. Since 20 is the fourth multiple of 5, we call 5 the fourth part of 20. 5 is the fifth part of 25, the sixth part of 30; and so on. But, 5 is half of 10. (We do not say the second part.) And 5 is not a part of itself; there is no such thing as the first part.

So with the exception of "half," an ordinal number names into which parts a number has been divided. 15 has been divided into Thirds; that is, into three equal parts. The third part of 15 is 5. This use of ordinal numbers does not imply a sequence: the first part, the second, the third, and so on. See below . If we divide a number into four equal parts, then we have divided it into fourths (or quarters); if into five equal parts, into fifths. But if we divide into two equal parts, then we have divided it in half. It is important to understand that we are not speaking here of proper fractions -- numbers that are less than 1 and that we need for measuring. We are explaining how the ordinal numbers --- third, fourth, fifth, etc. -- name the equal parts into which a number has been divided. When answering the questions of this Lesson, the student should not write fractions. We will come to those symbols in Lesson 20. It should be clear that the ordinal names of the parts belong to language itself, and are prior to the names of the proper fractions, which are the parts of 1.

Why is the number we write as 1 over 3 -- - called "one-third"? Because the numerator is one third -- the third part -- of the denominator. p That must be understood first. We can then explain that the number we call is the third part of 1. p Example 1. 3 is which part of 18? Answer. The sixth part. 3 is contained in 18 six times. Note that 1 is a part of every number (except itself) because every number is a multiple of 1. Which part is it? The part that says the number's name. 1 is the third part of 3, the fourth part of 4, the fifth part of 5, the hundredth part of 100. 1 is half of 2. Example 2. What number is the fourth part, or a quarter, of 28? Answer. 7. Because 28 is made up of four sevens. Four times 7 is 28. Example 3. 2 is the fifth part of what number? Answer. 10. Because five 2's are 10.

Every number is the fifth part of five times itself 4 is the fifth part of 5 × 4, which is 20. 9 is the fifth part of 5 × 9, which is 45. 20 is the fifth part of 5 × 20, which is 100. 4. See Lesson 11, Question 2 , and especially Example 6. Example 4. How much is an eighth of $72? Answer. 72 ÷ 8 = 9. An eighth of $72 is $9. Example 5. Tenths, hundredths. How much is a tenth of $275? How much is a hundredth? Answer. To find a tenth, divide by 10. 275 ÷ 10 = 27.5 Since this is money, we report the answer as $27.50.

( Lesson 3 .) As for a hundredth, we will separate two decimal digits : $275 ÷ 100 = $2.75 In Lesson 4, we saw that when we divide by 10, we have taken 10% of the number. And when we divide by 100, we have taken 1% . Therefore, 10% of $275 is $27.50. 1% is $2.75. Note: Whenever we divide by any power of 10 -- the digits do not change. 275 ÷ 100 = 2.75 Conversely, then, if two numbers have the same digits, they differ by a power of 10. Example 6. $85 is which part of $850? Answer. Apart from the 0 at the end of $850, those numbers have the same digits. Therefore, they differ by a power of 10. 850 is in fact 10 times 85. ( Lesson 4, Question 1 .) Therefore, $85 is the tenth part of $850.

To say the same thing, $85 is 10% of $850. Example 7. $.98 is which part of $98? Answer. They have the same digits. They differ by a power of 10. $.98 is the hundredth part of $98. It is 1% of it. Divisors We say that a smaller number is a divisor of a larger number if the larger is a multiple of the smaller. 3 is a divisor of 12 because 12 is a multiple of 3. 5 is not a divisor of 12. We say however that a number is a divisor of itself. With the exception of the number itself, the divisors of a number are the only parts that a number has. 3 is the fourth part of 12. 6 is half of 12. 5 is not any part of 12. You cannot divide 12 people into groups of 5. Example 8. Find all the divisors of 30 in pairs. Each divisor (except 30) is which part of 30? Answer.

Here are all the divisors of 30 in pairs: 1 and 30. (Because 1 × 30 = 30.) 2 and 15. (Because 2 × 15 = 30.) 3 and 10. (Because 3 × 10 = 30.) 5 and 6. (Because 5 × 6 = 30.) On naming which part of 30, each divisor will say the ordinal name of its partner: 1 is the thirtieth part of 30. 2 is the fifteenth part of 30. 15 is half of 30. 3 is the tenth part of 30. 10 is the third part of 30. 5 is the sixth part of 30. 6 is the fifth part of 30. Divisors always come in pairs. And that implies the following: Theorem. For every divisor (except 1) that a number has, it will have a part with the ordinal name of that divisor. (Euclid, VII.

37.) Since 18, for example, has a divisor 3, then 18 has a third part. Since 18 has a divisor 6, then 18 has a sixth part. But 18 does not have a fifth part, because 5 is not a divisor of 18. Here is an illustration that 18 has a divisor 3: 18 = 6 × 3. But according to the order property of multiplication : 18 = 3 × 6. This shows that 6 -- the partner of 3 -- is the third part of 18. In other words, since 18 has a divisor 3, then 18 has a third part. Example 9. Into which parts could 12 people be divided? Answer. The divisors of 12 are 1, 2, 3, 4, 6, and 12. Corresponding to each divisor (except 1), there will be a part with the ordinal name of the divisor. 12 people, therefore, could be divided into Halves, thirds, fourths, sixths, and twelfths.

You cannot take a fifth of 12 people. 12 does not have a divisor 5. Percent: Parts of 100% A percent is another way of naming a part . Because whatever part the percent is of 100%, that is the part we are naming. Since 50% is half of 100%, then 50% means half. 50% of 12 -- half of 12 -- is 6. Since 25% is a quarter, or a fourth, of 100% -- -- then 25% is another way of saying a quarter. 25% of 40 -- the fourth part of 40 -- is 10. In the next Lesson, Question 10 , we will see how to take 25% by taking half of half. Since 20% is the fifth part of 100% -- (100 is made up of five 20's) -- then 20% is another way of saying a fifth. 20% of 15 -- the fifth part of 15 -- is 3.

Repeated division in half Every time we take half of something, we get twice as many parts. Half of a whole -- -- -- results in two equal parts. Each part is Half. If we divide each Half in half -- -- the whole will be in four equal parts, or Quarters. If we divide each Quarter in half -- -- the whole will then be in twice as many, that is, eight equal parts, or Eighths. Half of an Eighth is a Sixteenth. Half of a Sixteenth is a Thirty-second. And so on. Now, here are the number of equal parts that result when we repeatedly take half: 2, 4, 8, 16, 32, 64, and so on. Those numbers are called the powers of 2. Repeated division in half is very common. The student should know the names of the sequence of those parts: Halves, Quarters, Eighths, Sixteenths, Thirty-seconds, and so on. * When we say "5 is the third part of 15," we do not imply a sequence: the first part, the second part, the third, and so on.

When we speak of the third part, that is a different meaning for the word "third." It means each one of three equal parts that together make up the whole. We say that we have divided 15 into thirds. Yet "third" still retains its ordinal character. Because to the question, "Which part of 15 is 5?", we answer, "The third part." We use that ordinal number because 15 is the third multiple of 5. At this point, please "turn" the page and do some Problems . or Fractions Fractions Unit 14 > Lesson 1 of 11 A circle is a geometric shape that we have seen in other lessons. The circle to the left can be used to represent one whole. We can divide this circle into equal parts as shown below. This circle has been divided into 2 equal parts. This circle has been divided into 3 equal parts. This circle has been divided into 4 equal parts. We can shade a portion of a circle to name a specific part of the whole as shown below. Definition: A fraction names part of a region or part of a group. The top number of a fraction is called its numerator and the bottom part is its denominator.

So a fraction is the number of shaded parts divided by the number of equal parts as shown below: number of shaded parts denominator Looking at the numbers above, we have: There are two equal parts, giving a denominator of 2. One of the parts is shaded, giving a numerator of 1. There are three equal parts, giving a denominator of 3. Two of the parts are shaded, giving a numerator of 2. There are four equal parts, giving a denominator of 4. One of the parts is shaded, giving a numerator of 1. Note that the fraction bar means to divide the numerator by the denominator. Let's look at some more examples of fractions. In examples 1 through 4 below, we have identified the numerator and the denominator for each shaded circle. We have also written each fraction as a number and using words. Example 1 four-fifths Why is the number written as three-fourths? We use a hyphen to distinguish a fraction from a ratio. For example, "The ratio of girls to boys in a class is 3 to 4." This ratio is written a 3 to 4, or 3:4. We do not know how many students are in the whole class.

However, the fraction is written as three-fourths (with a hyphen) because 3 is 3/4 of one whole. Thus a ratio names a relationship, whereas, a fraction names a number that represents the part of a whole. When writing a fraction, a hyphen is always used. It is important to note that other shapes besides a circle can be divided in equal parts. For example, we can let a rectangle represent one whole, and then divide it into equal parts as shown below. two equal parts four equal parts five equal parts Remember that a fraction is the number of shaded parts divided by the number of equal parts. In the example below, rectangles have been shaded to represent different fractions. Example 5 one-fourth one-fifth The fractions above all have the same numerator. Each of these fractions is called a unit fraction. Definition: A unit fraction is a fraction whose numerator is one. Each unit fraction is part of one whole (the number 1). The denominator names that part. Every fraction is a multiple of a unit fraction. In examples 6 through 8, we will identify the fraction represented by the shaded portion of each shape. Example 6 a In example 6, there are four equal parts in each rectangle.

Three sections have been shaded in each rectangle, but not the same three. This was done intentionally to demonstrate that any 3 of the 4 equal parts can be shaded to represent the fraction three-fourths. b a b In example 7, each circle is shaded in different sections. However, both circles represent the fraction two-thirds. The value of a fraction is not changed by which sections are shaded. Example 8 a In example 8, each rectangle is shaded in different sections. However, both rectangles represent the fraction two-fifths. Once again, the value of a fraction is not changed by which sections are shaded. b In the examples above, we demonstrated that the value of a fraction is not changed by which sections are shaded. This is because a fraction is the number of shaded parts divided by the number of equal parts. Let's look at some more examples. Example 9 answer 4 Summary: A fraction names part of a region or part of a group. A fraction is the number of shaded parts divided by the number of equal parts. The numerator is the number above the fraction bar, and the denominator is the number below the fraction bar.

Exercises In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction two-thirds, enter 2/3 into the form. 1.

��ࡱ�>�� FH����E��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������%` ��,7bjbjN�N� .F,�,�,/

���������������HHHH$l�k �������

�� � � � � � � $s h�� ������ ����% � � � 

������ � �� � � ��� �� 

`9ت���H��� � ; 0k � � (� �� ��� ��

���  � ���k ��������H��

�H������������� 



Level G Lesson 9 Divide Fractions The objective for lesson 9 is the student will work with division of fractions.

We will have three essential questions that will be guiding our lesson. Question 1, why is it important to know how to build a model for division of fractions? Question 2, what does a division sentence mean? Question 3, what are the steps for dividing a fraction by a fraction? The SOLVE problem for this lesson is, Ms. Sherrill has three fourths of a yard of red material. She needs to cut the material into pieces that are one eighth of a yard in length. How many pieces of material will she have? We�re going to S the problem and we�re going to start by underlining the question. How many pieces of material will she have? The second part of S is to complete the sentence this problem is asking me to find, the number of pieces of material. The first problem we are going to look at is 2 divided by one half. The meaning of this problem is, how many groups of one half are in 2 whole units? We�re going to model this problem using our fraction strips a manipulative model and I�m going to start by representing the first fraction. I have two whole units. In a division problem the first number is called a dividend.

The second number is called the divisor. The divisor tells us what section we�re going to divide the dividend into. So this time our divisor is one half. So we�re going to divide our whole units into groups of one-half. There�s my first whole unit divided into one half and my second whole unit divided into groups of one half. When I add those 1, 2, 3, 4, I have 4 groups of one half in 2 whole units. I�m going to go up here and now I�m going to model it pictorially. I have 1 group, 2 whole groups and I�m going to divide it into groups of One half unit. I have 1, 2, 3, 4, 2 divided by one half is equal to 4. The second problem we�re going to model is 2 divided by one third. The meaning of this problem is, how many groups of one third are in 2 whole units? We�re going to model this problem by following our fraction strip by starting with our dividend which is two whole units. We have one unit and then we have a second unit.

We�re going to divide our whole units into groups of one third. By one third, I take my green fraction strip, I have 3 units in 1 whole unit and I have 3 one third units in the second unit. So for 2 whole units, I have 1, 2, 3, 4, 5, 6, one third units. I can also model this pictorially by representing my first fraction, my dividend, and dividing it into groups of one third. I have 1 group, 2 groups, 3 groups, 4 groups, 5, groups and a total of 6 groups, 2 divided by one third is 6. We�re now going to model dividing fractions by another fraction. Our problem is one third divided by one sixth. The meaning of this is, how many groups of one sixth are in one third? In division we represent our first number our dividend with our fraction strip one third unit. We then need to find out how many groups of our one sixth we can fit into that one third unit. We take our fraction strips and we see that there are 2 groups of one sixth in one third.

This problem can be modeled pictorially by drawing the one third fraction bar, and the 2 groups of one sixth underneath. I have a total of 2 groups, 2 groups of one sixth. The next problem we�re going to model is two thirds divided by one twelfth. The meaning of the problem is, how many groups of one twelfth are in two thirds? When we are modeling division multiplication we represent our first fraction. I have my two thirds or 2, one third units. I need to divide that into groups of one twelfth. So I�m going to place my pink fraction strips from my manipulative kit underneath my one third units to see how many groups of one twelfth are in two thirds. I have 1, 2, 3, 4, 5, 6, 7, 8. I can model this pictorially by drawing my 2 one third fraction strips and then drawing the one twelfth units underneath to show that there are 8. I�m then going to add those up for a total of 8. How many groups of one twelfth are in two thirds unit, 8 groups of one twelfth.

We are now gong to model the problem three fourths divided by one half. In division we always start with our first fraction or our dividend. The meaning of or problem is, how many groups of one half are in three fourths? I�m going to take my one half units and place them underneath my one fourth units. You can see that we have a second brown strip that goes beyond our 3, one fourth units. What we�re going to do is we�re going to take the one half unit and underneath that I�m going to place the legal trade of two fourths. Now I have completely covered my three fourths unit, 1 full one half unit covers my two fourths and one half of my brown unit covers my third fourth in this picture. Because we�ve divided the one half into 2 sections, 1 out of the 2 makes up the third fourth. When we go up to the pictorial model we have our 3 fourths and we are dividing into groups of one half. Our one half extends over the last fourth so we divide the one half into 2 pieces, one out of those two pieces or one half of the one half completes our model.

So we have 1 whole group of one half, and then one half of a group which is our one fourth. How many groups of one half are in three fourths? 1 and one half groups of one half. We are going to talk about a method of dividing fractions without using models. We are going to be learning about reciprocals. One fourth and 4 are reciprocals because they have a product of 1. Any whole number can b e written as a fraction by placing it over one, so the 4 whole number can be written as an improper faction of 4 over 1. A number is considered to be a reciprocal if the product of 2 fractions is equal to 1. We found the reciprocal of one fourth by inverting the fraction, that is flipping the fraction so that the denominator is now the numerator and the numerator is now the denominator. In example 2, we�re going to look at one fifth and 5. These 2 values are reciprocals. Remember we are going to first change our whole number to an improper fraction by writing it over 1.

Any whole number can be written as a fraction by placing it over 1 as the denominator because 5 divided by 1 is still 5. When we multiply one fifth times 5 over 1 we see that our product is 1. These 2 numbers are reciprocals. The last example we�re going to look at is example 3. The question is what is the reciprocal of two thirds. The reciprocal of two thirds is 3 halves. We have inverted or flipped our fraction so that the denominator is now the numerator and the numerator is now the denominator. We know that these two values are reciprocals because when we multiply them because our product is 1. We�re going to model a division of fraction problems using reciprocals. Remember a reciprocal is any two pairs of fractions that are multiplied with a product of 1. We find the reciprocal of the fractions by inverting it. We are going to start our problem three fourths divided by five sixth by rewriting it vertically.

Remember we have two fractions here but that the fraction line is also an indicator of the operation of division, 3 / 4 is actually 3 divided by 4, five sixths is actually 5 divided by 6, and we can change this division sign to the vertical model. When we are dividing two values we need to get a denominator of 1 here. We can do that by multiplying five sixths by it�s reciprocal which is six fifths. Remember you find the reciprocal by inverting the fraction. If we�re going to multiply by six fifths we also to have to multiply our numerator by six fifths, and we can do that because if we look at these two values six fifths divided by six fifths is the value of 1. If we divide something by 1 we do not change the final value. We�re going to go ahead and multiply our numerator�s, 3 times 6 is 18, 4 times 5 is 20, 5 times 6 is 30, 6 times 5 is 30, 30 over 30 is equivalent to 1.

Our value of 18 over 20 divided by 1 is that same value 18 over 20. We�re going to simplify our fraction by dividing 2 over 2 and again this is the value of 1 so we�re not changing the value of the fraction we�re simply simplifying it, 18 divided by 2 is 9, 20 divided by 2 is 10. And our simplified answer is nine tenths. This is the information that you should include on the division section of your fraction book. You want to start by writing what the division means, �how many groups of blank are in blank?� In division we represent the first fraction and divide it by the value of the second fraction, � how many groups of one fourth are in one half?� Then what we�re going to do is we�re going to show one half and we place one fourth units underneath it and we count the number of groups of one-fourth in one half. There are 2 groups of one fourth in one half. We are going back to our SOLVE problem from the beginning of the lesson. Ms. Sherrill has three fourths of a yard of red material.

She needs to cut the material into pieces that are one eighth of a yard in length. How many pieces of material will she have? We S the problem, we study the problem, we underline the question and we completed the sentence this problem is asking me to find the number of pieces of material. We�re now moving to O, we are going to organize our facts. First we are going to identify the facts. Ms. Sherrill has three fourths of a yard of red material. She needs to cut the material into pieces that are one eighth of a yard in length. The second step is to eliminate the unnecessary facts. Ms. Sherrill has three fourths of a yard of material. We need to know how much material she has, so that fact will be necessary. She needs to cut them into pieces that are one eighth of a yard. We also need to know that so we have no unnecessary facts in this problem. Then we�re going to list the necessary facts. We have three fourths of a yard of material, and we need one eighth yard pieces. We�re going to line up our plan by choosing an operation or operations.

Since we�re cutting material, we�re actually dividing it into pieces we�re going to choose division. Our plan of action will be to divide the total quantity of material by the length of each piece. We are now moving to our V step and we are going to begin to verify our plan with action by estimating our answer. We know we have three fourths of a yard, and we know each piece needs to be one eighth in length, so our estimate is going to be about 5. We�re now going to carry out our plan. We take our three fourths yard and divide it into pieces that are one eighth in length. We�re going to use the concept of reciprocals, remember we divided by one eighth we were multiplying both the fractions by 8 over 1, 3 times 8 is 24, 4 times 1 is 4, when we divide and simplify we have 6 pieces of material. To complete our word problem we are going to go back to the E step and examine our results. Does our answer make sense? We had 6 pieces of material and we were looking for the number of pieces of material. So, yes our answer makes sense. Is your answer reasonable?

Our estimate was 5 and our answer is 6. So, yes our answer is reasonable because our answer is close to our answer of about 5. Is your answer accurate? We�re going to go back and recheck our work and the answer is yes. Then we are going to complete the E step by writing our answer in a complete sentence. Ms. Sherrill will have 6 pieces of material. Now we are going to go back and answer the essential questions from the beginning of our lesson. Why is it important to know how to build a model for division of fractions? So we will know what the problems mean. What does a division sentence mean? How many groups of blank are in blank? What are the steps for dividing a fraction by a fraction? We create a reciprocal of the second fraction and then multiply.

!">F� �  / 1 �

 �  - .

< = h j � � � � < = @ L � � � ] � � � � � 1dl��#JPS����� Pi��� 7:;Hq����]�����(U

�������������������������������������������������������������������������������h` �h %3h�Z�h�*#h� �h�v�h�<hd �hg�h�&hx&h�hI6hTasO"#� � i j ��mn��n o w!x!�"�"$))/+0+�,�,�/�/�0�

��������������������

�������� ,7�



Uh���X����'Bflm|��  f�����;X�

����:�����Trw�������S]������ >Ps������% 9 k � � � G K l m n � �������������������������������������������������������������������������������������h�2�hK�

h�8h^X�hg�hPlFh %�h�v�h�&h` �U� � � Z v � �  ^ � �  d � � !%!4!;!@!D!v!w!x!�!�!"!"�"�"�"�"�"$&$k$p$�$�$�$%+%�%�%�%�%)&>&F&R&v&

y&�&�&�& '+'P'm'�'�' (!("(#(�())) )>)�)9*V*d*x*�*�*�*�*�*��������������������������������������������������������������������������������h V�hV� \UT_�e�Nf�g) g�pg�kB mLm2ym�mqJBsTasays�3tX u�w-K{;� %��>��v��@�^j��!��)��L��:�t�>

^�� ��!�d��[��v� ?@ABCD��������G������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������Root Entry�������� �F0j����I�Data

 ������������$1Table����,WordDocument

����.FSummaryInformation(������������5DocumentSummaryInformation

8��������=CompObj������������q����

�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� ���� �F Microsoft Office Word Document MSWordDocWord.Document.8�9�q

Question: A half divided by a sixth is?
Answer: Three