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4.1.1: Size of Divisor and Size of Quotient - Mathematics


Lesson

Let's explore quotients of different sizes.

Exercise (PageIndex{1}): Number Talk: Size of Dividend and Divisor

Find the value of each expression mentally.

(5,000div 5)

(5,000div 2,500)

(5,000div 10,000)

(5,000div 500,000)

Exercise (PageIndex{2}): All Stacked Up

  1. Here are several types of objects. For each type of object, estimate how many are in a stack that is 5 feet high. Be prepared to explain your reasoning.

Cardboard boxes

Notebooks

Bricks

Coins

  1. A stack of books is 72 inches tall. Each book is 2 inches thick. Which expression tells us how many books are in the stack? Be prepared to explain your reasoning.
    • (72cdot 2)
    • (72-2)
    • (2div 72)
    • (72div 2)
  2. Another stack of books is 43 inches tall. Each book is (frac{1}{2})-inch thick. Write an expression that represents the number of books in the stack.

Exercise (PageIndex{3}): All in Order

Your teacher will give you two sets of papers with division expressions.

  1. Without computing, estimate the quotients in each set and order them from greatest to least. Be prepared to explain your reasoning.
    Pause here for a class discussion.
    Record the expressions in each set in order from the greatest value to the least.
    1. Set 1
    2. Set 2
  2. Without computing, estimate the quotients and sort them into the following three groups. Be prepared to explain your reasoning.

(egin{array}{lllllll}{30divfrac{1}{2}}&{qquad}&{9div 10}&{qquad}&{18div 19}&{qquad}&{15,000div 1,500,000}{30div 0.45}&{qquad}&{9div 10,000}&{qquad}&{18div 0.18}&{qquad}&{15,000div 14,500}end{array})

  • Close to 0
  • Close to 1
  • Much larger than 1

Are you ready for more?

Write 10 expressions of the form (12div ?) in a list ordered from least to greatest. Can you list expressions that have value near 1 without equaling 1? How close can you get to the value 1?

Summary

Here is a division expression: (60div 4). In this division, we call 60 the dividend and 4 the divisor. The result of the division is the quotient. In this example, the quotient is 15, because (60div 4=15).

We don’t always have to make calculations to have a sense of what a quotient will be. We can reason about it by looking at the size of the dividend and the divisor. Let’s look at some examples.

  • In (100div 11) and in (18div 2.9) the dividend is larger than the divisor. (100div 11) is very close to (99div 11), which is (9). The quotient (18div 2.9) is close to (18div 3) or (6).
    In general, when a larger number is divided by a smaller number, the quotient is greater than (1).
  • In (99div 101) and in (7.5div 7.4) the dividend and divisor are very close to each other. (99div 101) is very close to (99div 100), which is (frac{99}{100}) or (0.99). The quotient (7.5div 7.4) is close to (7.5div 7,5), which is (1).
    In general, when we divide two numbers that are nearly equal to each other, the quotient is close to (1).
  • In (10div 101) and in (50div 198) the dividend is smaller than the divisor. (10div 101) is very close to (10div 100), which is (frac{10}{100}) or (0.1). The division (50div 198) is close to (50div 200), which is (frac{1}{4}) or (0.25).
    In general, when a smaller number is divided by a larger number, the quotient is less than (1).

Practice

Exercise (PageIndex{4})

Order from smallest to largest:

  • Number of pennies in a stack that is 1 ft high
  • Number of books in a stack that is 1 ft high
  • Number of dollar bills in a stack that is 1 ft high
  • Number of slices of bread in a stack that is 1 ft high

Exercise (PageIndex{5})

Use each of the numbers 4, 40, and 4000 once to complete the sentences.

  1. The value of (underline{qquad}div 40.01) is close to (1).
  2. The value of (underline{qquad}div 40.01) is much less than (1).
  3. The value of (underline{qquad}div 40.01) is much greater than (1).

Exercise (PageIndex{6})

Without computing, decide whether the value of each expression is much smaller than 1, close to 1, or much greater than 1.

  1. (100divfrac{1}{1000})
  2. (50frac{1}{3}div 50frac{1}{4})
  3. (4.7div 5.2)
  4. (2div 7335)
  5. (2,000,001div 9)
  6. (0.002div 2,000)

Exercise (PageIndex{7})

A rocking horse has a weight limit of 60 pounds.

  1. What percentage of the weight limit is 33 pounds?
  2. What percentage of the weight limit is 114 pounds?
  3. What weight is 95% of the limit?

(From Unit 3.4.7)

Exercise (PageIndex{8})

Compare using (>), (=), or (<).

  1. (0.7) ______ (0.70)
  2. (0.03+frac{6}{10}) ______ (0.30+frac{6}{100})
  3. (0.9) ______ (0.12)

(From Unit 3.4.5)

Exercise (PageIndex{9})

Diego has 90 songs on his playlist. How many songs are there for each genre?

  1. 40% rock
  2. 10% country
  3. 30% hip-hop
  4. The rest is electronica

(From Unit 3.4.4)

Exercise (PageIndex{10})

A garden hose emits 9 quarts of water in 6 seconds. At this rate:

  1. How long will it take the hose to emit 12 quarts?
  2. How much water does the hose emit in 10 seconds?

(From Unit 3.3.4)


Lesson 1

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Problem 2

Use each of the numbers 4, 40, and 4000 once to complete the sentences.

The value of (underline> div 40.01) is close to 1.

The value of (underline> div 40.01) is much less than 1.

The value of ( underline>div 40.01) is much greater than 1.

Solution

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Problem 3

Without computing, decide whether the value of each expression is much smaller than 1, close to 1, or much greater than 1.

  1. (100 div frac<1><1000>)
  2. (50frac13 div 50frac14)
  3. (4.7 div 5.2)
  4. (2 div 7335)
  5. (2,!000,!001 div 9)
  6. (0.002 div 2,!000)

Solution

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Problem 4

A rocking horse has a weight limit of 60 pounds.

  1. What percentage of the weight limit is 33 pounds?
  2. What percentage of the weight limit is 114 pounds?

What weight is 95% of the limit?

Solution

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Problem 5

Solution

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Problem 6

Diego has 90 songs on his playlist. How many songs are there for each genre?

Solution

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Problem 7

A garden hose emits 9 quarts of water in 6 seconds. At this rate:

  1. How long will it take the hose to emit 12 quarts?
  2. How much water does the hose emit in 10 seconds?

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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1.3 All in Order

Record the expressions in each set in order from largest to smallest.

Without computing, estimate each quotient and arrange them in three groups: close to 0, close to 1, and much larger than 1. Be prepared to explain your reasoning.

Are you ready for more?

Write 10 expressions of the form 12 div ? in a list ordered from least to greatest. Can you list expressions that have value near 1 without equaling 1? How close can you get to the value 1?


Long Division - One-Digit Divisor and a One-Digit Quotient with No Remainder (A)

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THE MEANING OF DIVISION

-- the multiplier and the multiplicand, and we have to name their product.

But in what is called the inverse of multiplication, we are given the product and the multiplicand --

-- and we have to name the multiplier.

"What number times 15 equals 60?"

"How many times do we have to add 15 to get 60?"

We call that division because 60 is being divided by -- cut into equal parts -- by 15.

"60 divided by 15 equals 4."

Equivalently, we could subtract 15 from 60 four times. Multiplication is repeated addition. Therefore we can think of division as repeated subtraction.

then 15, the number on the right of the division sign ÷ , is called the divisor . It is the number we have to multiply to get 60.

60 is called the dividend it is the number being divided by 15.

4 is called the quotient . It is the number of times we must multiply 15 to get 60. In the figure, the quotient is the number of equal parts into which the dividend has been divided.


1. What is the problem of "division"?
Dividend ÷ Divisor = Quotient.
Quotient × Divisor = Dividend.
We are to name the number of times one number, called the Divisor, is contained in another number, called the Dividend.
That number of times is called the Quotient.
Equivalently, we are to say what number times the Divisor will equal the Dividend. That is, how many times do we have to add the Divisor so that it will equal the Dividend?
Or, how many times we could subtract the Divisor from the Dividend?

A divisor may not be 0 -- 6 ÷ 0 -- because any number
times 0 will still be 0. Division by 0 is an excluded operation.

As for 0 ÷ 0, that is ambiguous because it could be any number. Any number times the divisor 0 will equal the dividend 0.

Example 1. This figure shows how the numbers 1, 2, 3, 4, and 6 go into -- are contained in -- 12.

6 goes into 12 two times. 12 ÷ 6 = 2.

4 goes into 12 three times. 12 ÷ 4 = 3.

3 goes into 12 four times. 12 ÷ 3 = 4.

2 goes into 12 six times. 12 ÷ 2 = 6.

1 goes into 12 twelve times. 12 ÷ 1 = 12.

Example 2. A bottle of juice contains 18 ounces. How many times could you fill a 6 ounce glass?

Answer . Any problem that asks "How many times?" is a division problem. So the question is: How many times is 6 ounces contained in 18?

You could fill the glass 3 times.

Here is the picture of 18 ÷ 6 :

6 goes into 18 three times. That is,

Equivalently, we could subtract 6 from 18 three times.

Here, on the other hand, is the picture of 18 ÷ 3:

18 can be divided into six 3's.

Dividend ÷ Divisor = Quotient
Quotient × Divisor = Dividend

For such a simple problem the student should not write the division box.

It is not necessary to prepare for long division in the way they had to 100 years ago.

Example 3. What number times 10 will equal 72?

Answer . Again, this is a division problem. The number that follows the word "times" is the divisor. We have to divide 72 by 10.

Example 4. If it takes 3 yards of material to make a suit, how many suits could be made from a piece of material that is 15 yards?

Answer . We have to cut 3 yards from 15 yards as many times as we can. That number of times is 15 ÷ 3.

This problem illustrates the following: The dividend and divisor must be units of the same kind . We can only divide yards by yards, dollars by dollars, hours by hours. We cannot divide 8 apples by 2 oranges --

-- because there is no number times 2 oranges that will equal 8 apples

What is more, we see that the quotient is always a pure number.

Dividend ÷ Divisor = Quotient.

It is the number that multiplies the divisor to produce the dividend.

Example 5. A bus is scheduled to arrive every 12 minutes. In the course of 2 hours, how many buses will arrive?

Answer . How many times is 12 minutes contained in 2 hours? But the units must be the same. Since 1 hour = 60 minutes, then 2 hours = 2 × 60 = 120 minutes.

120 minutes ÷ 12 minutes = 10.

10 times 12 minutes = 120 minutes. (Lesson 4.)

In the course of 2 hours, 10 buses will arrive.

(See Problem 6 at the end of the Lesson.)

Division into equal parts

2. If we divide a number into equal parts, how can we know how many there are in each part?
Divide by the number of equal parts.

If we divide into 2 equal parts,

then to know how many there are in each part, divide by 2.

If we divide into 3 equal parts,

That is why to divide the whole of something, which is 100%, into 100 equal parts -- that is, to find 1% of a number -- we divide by 100. (Lesson 4, Question 6.)

Example 6. If we divide 28 people into four equal parts, then how many will be in each part?

In Lesson 15 we will see that we are taking a "quarter" or a "fourth" of 28 people.

Solution . Divide by 4. 28 ÷ 4 = 7 .

There will be 7 people in each part.

But that is the picture of 28 ÷ 7. Why does 28 ÷ 4 give the right answer?

Because of the order property of multiplication. 28 ÷ 4 = 7 means

That means 28 is made up of four 7 's.

Example 7. Christopher bought 3 shirts for a total of $66. Each one cost the same. How much did each one cost?

Solution . If we divide $66 into 3 equal parts, we will know the answer.

What number times 3 will be 66?

Equivalently, 3 times what number will be 66?

In Lesson 15, we will speak simply of taking a third of $66, and the question of division never comes up.

A problem in which we relate units of different kinds -- dollars per shirt, for example -- is called a rate problem, as we are about to see.


3. What is a rate ?
A rate is a relationship between units of different kinds. Miles per hour. Dollars per person. And so on.

A rate is typically indicated by per , which means for each or in each .

In a calculation, per always indicates division.

Example 8. In a certain country, the unit of currency is the corona. With $11 Ana was able to buy 55 coronas. What was the rate of exchange? That is, how many coronas per dollar?

Solution . Follow the sequence: coronas per dollar: 55 ÷ 11 = 5.

The rate of exchange was 5 coronas per dollar.

Again, a rate problem involves dividing a number into equal parts. In this example, we divided 55 coronas into 5 equal parts of 11 coronas each.

5 × 11 coronas = 55 coronas.

But that implies 11 × 5 coronas = 55 coronas:

Each group of 5 was worth $1. That was the rate of exchange. 5 coronas per dollar.

To preserve the meanings of multiplication and division, we must relate units of the same kind, even though that is not how it might appear.

Exact versus inexact division

The numbers exactly divisible by 3 are the multiples of 3:

And since they are divisible by 3, so are

Since those are the multiples of 3, we say that 3 is their divisor . In other words, we say that one number is a divisor of a second if the second is its multiple.

A number will go evenly into every one of its multiples.

The numbers exactly divisible by 8 are the multiples of 8:

Example 9. A bottle holds 35 ounces. A glass holds 8 ounces. How many glasses can you fill from that bottle?

Solution . We must calculate 35 ÷ 8. Now, 8 goes into 32 exactly, but 8 does not go into 35 exactly:

There is a remainder of 3.

Therefore you could fill 4 glasses, and 3 ounces will remain in the bottle.

The remainder 3 is what we have to add to 4 × 8 to get 35.

Say there are a large number of people, and we want to divide them into groups of 5.

But say we discover that there is not an exact number of 5's. Then how many people might we not be able to group? How many people might remain?

Answer: Either 1, or 2, or 3, or 4. Because if more than 4 remained, we could make another group of 5

The remainder is always less than the divisor.

If we divide by 5, then the possible remainders are 1, 2, 3, or 4.

a) If 7 is the divisor, what are the possible remainders?

b) How many 7's are there in 61?

Answer. 8. 8 × 7 is 56 -- plus 5 is 61.

The remainder 5 is what we must add to 56 to get 61.

Example 11. Prove: 47 ÷ 9 = 5 R 2.

Proof . 5 × 9 + 2 = 45 + 2 = 47.

Say the whole number quotient and the remainder. Do not write the division box.

"8 goes into 53 six times -- 48 -- with 5 left over."

The remainder is the number you have to add to 48 to get 53.

How would you know that you have to add 5? 4 8 plus what number ends in 3? 8 plus 5 ends in 3. (13.) 5 is the remainder.

48 plus 2 is 50, plus 3 is 53.

"4 goes into 31 seven times -- 28 -- with 3 left over."

In what follows, we will signify division in this way:

Dividend
Divisor
= Quotient

Quotient × Divisor = Dividend

The horizontal line separating 16 and 8 is called the division bar . The division bar is also used to signify a fraction, because a fraction sometimes requires division of the numerator by the denominator. (Lessons 20 and 24.) We also use the division bar to indicate the ratio of two numbers. (Lesson 20.)

"280 divided by 7 is what number?"

Answer . Ignore the 0. 7 goes into 28 four ( 4 ) times. Therefore 7 goes into 280 forty ( 40 ) times.

In other words, since 28 is divisible by 7, then so is '28' followed by any number of 0's.


Handbook of Algebra

Proof

According to the Martindale theorem, RC has a primitive idempotent, but the ring Q(R ) with no zero divisors can have only one nonzero idempotent it is 1. Therefore, RC = 1 · RC · 1 = T is a finite dimensional sfield over C.

Let us consider a linear over C projection l: T → o n C and assume that l(T) ∩ R = 0. Then, by Lemma 2.2.6 there are elements ai, bi ∈ T, such that


Examples

To extend round to new numeric types, it is typically sufficient to define Base.round(x::NewType, r::RoundingMode) .

A type used for controlling the rounding mode of floating point operations (via rounding / setrounding functions), or as optional arguments for rounding to the nearest integer (via the round function).

Currently supported rounding modes are:

The default rounding mode. Rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer.

Rounds to nearest integer, with ties rounded away from zero (C/C++ round behaviour).

Rounds to nearest integer, with ties rounded toward positive infinity (Java/JavaScript round behaviour).

round using this rounding mode is an alias for trunc .

Rounds away from zero. This rounding mode may only be used with T == BigFloat inputs to round .

round using this rounding mode is an alias for ceil .

round using this rounding mode is an alias for floor .

Return the nearest integral value of the same type as the complex-valued z to z , breaking ties using the specified RoundingMode s. The first RoundingMode is used for rounding the real components while the second is used for rounding the imaginary components.

ceil(x) returns the nearest integral value of the same type as x that is greater than or equal to x .

ceil(T, x) converts the result to type T , throwing an InexactError if the value is not representable.

digits , sigdigits and base work as for round .

floor(x) returns the nearest integral value of the same type as x that is less than or equal to x .

floor(T, x) converts the result to type T , throwing an InexactError if the value is not representable.

digits , sigdigits and base work as for round .

trunc(x) returns the nearest integral value of the same type as x whose absolute value is less than or equal to x .

trunc(T, x) converts the result to type T , throwing an InexactError if the value is not representable.

digits , sigdigits and base work as for round .

Return the nearest integral value of type T whose absolute value is less than or equal to x . If the value is not representable by T , an arbitrary value will be returned.

Return the minimum of the arguments. See also the minimum function to take the minimum element from a collection.

Return the maximum of the arguments. See also the maximum function to take the maximum element from a collection.

Return (min(x,y), max(x,y)) . See also: extrema that returns (minimum(x), maximum(x)) .

Return x if lo <= x <= hi . If x > hi , return hi . If x < lo , return lo . Arguments are promoted to a common type.

Clamp x between typemin(T) and typemax(T) and convert the result to type T .

Clamp x to lie within range r .

This method requires at least Julia 1.6.

Restrict values in array to the specified range, in-place. See also clamp .

When abs is applied to signed integers, overflow may occur, resulting in the return of a negative value. This overflow occurs only when abs is applied to the minimum representable value of a signed integer. That is, when x == typemin(typeof(x)) , abs(x) == x < 0 , not -x as might be expected.

Calculates abs(x) , checking for overflow errors where applicable. For example, standard two's complement signed integers (e.g. Int ) cannot represent abs(typemin(Int)) , thus leading to an overflow.

The overflow protection may impose a perceptible performance penalty.

Calculates -x , checking for overflow errors where applicable. For example, standard two's complement signed integers (e.g. Int ) cannot represent -typemin(Int) , thus leading to an overflow.

The overflow protection may impose a perceptible performance penalty.

Calculates x+y , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates x-y , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates x*y , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates div(x,y) , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates x%y , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates fld(x,y) , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates mod(x,y) , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates cld(x,y) , checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Calculates r = x+y , with the flag f indicating whether overflow has occurred.

Calculates r = x-y , with the flag f indicating whether overflow has occurred.

Calculates r = x*y , with the flag f indicating whether overflow has occurred.

Squared absolute value of x .

Return z which has the magnitude of x and the same sign as y .

Return zero if x==0 and $x/|x|$ otherwise (i.e., ±1 for real x ).

Returns true if the value of the sign of x is negative, otherwise false .

Return x with its sign flipped if y is negative. For example abs(x) = flipsign(x,x) .

Return $sqrt$ . Throws DomainError for negative Real arguments. Use complex negative arguments instead. The prefix operator √ is equivalent to sqrt .

Integer square root: the largest integer m such that m*m <= n .

Return the cube root of x , i.e. $x^<1/3>$ . Negative values are accepted (returning the negative real root when $x < 0$ ).

The prefix operator ∛ is equivalent to cbrt .

Return the real part of the complex number z .

Return the imaginary part of the complex number z .

Return both the real and imaginary parts of the complex number z .

Compute the complex conjugate of a complex number z .

Compute the phase angle in radians of a complex number z .

Compute $exp(ipi x)$ more accurately than cis(pi*x) , especially for large x .

This function requires Julia 1.6 or later.

The binomial coefficient $inom$ , being the coefficient of the $k$ th term in the polynomial expansion of $(1+x)^n$ .

If $n$ is non-negative, then it is the number of ways to choose k out of n items:

If $n$ is negative, then it is defined in terms of the identity

External links

Factorial of n . If n is an Integer , the factorial is computed as an integer (promoted to at least 64 bits). Note that this may overflow if n is not small, but you can use factorial(big(n)) to compute the result exactly in arbitrary precision.

External links

Greatest common (positive) divisor (or zero if all arguments are zero). The arguments may be integer and rational numbers.

Rational arguments require Julia 1.4 or later.

Least common (positive) multiple (or zero if any argument is zero). The arguments may be integer and rational numbers.

Rational arguments require Julia 1.4 or later.

Computes the greatest common (positive) divisor of a and b and their Bézout coefficients, i.e. the integer coefficients u and v that satisfy $ua+vb = d = gcd(a, b)$ . $gcdx(a, b)$ returns $(d, u, v)$ .

The arguments may be integer and rational numbers.

Rational arguments require Julia 1.4 or later.

Bézout coefficients are not uniquely defined. gcdx returns the minimal Bézout coefficients that are computed by the extended Euclidean algorithm. (Ref: D. Knuth, TAoCP, 2/e, p. 325, Algorithm X.) For signed integers, these coefficients u and v are minimal in the sense that $|u| < |y/d|$ and $|v| < |x/d|$ . Furthermore, the signs of u and v are chosen so that d is positive. For unsigned integers, the coefficients u and v might be near their typemax , and the identity then holds only via the unsigned integers' modulo arithmetic.

Test whether n is an integer power of two.

Support for non- Integer arguments was added in Julia 1.6.

The smallest a^n not less than x , where n is a non-negative integer. a must be greater than 1, and x must be greater than 0.

The largest a^n not greater than x , where n is a non-negative integer. a must be greater than 1, and x must not be less than 1.

Next integer greater than or equal to n that can be written as $prod k_i^$ for integers $p_1$ , $p_2$ , etcetera, for factors $k_i$ in factors .

The method that accepts a tuple requires Julia 1.6 or later.

Take the inverse of n modulo m : y such that $n y = 1 pmod m$ , and $div(y,m) = 0$ . This will throw an error if $m = 0$ , or if $gcd(n,m) eq 1$ .

Compute the number of digits in integer n written in base base ( base must not be in [-1, 0, 1] ), optionally padded with zeros to a specified size (the result will never be less than pad ).

Multiply x and y , giving the result as a larger type.

Evaluate the polynomial $sum_k x^ p[k]$ for the coefficients p[1] , p[2] , . that is, the coefficients are given in ascending order by power of x . Loops are unrolled at compile time if the number of coefficients is statically known, i.e. when p is a Tuple . This function generates efficient code using Horner's method if x is real, or using a Goertzel-like [DK62] algorithm if x is complex.

This function requires Julia 1.4 or later.

Evaluate the polynomial $sum_k z^ c[k]$ for the coefficients c[1] , c[2] , . that is, the coefficients are given in ascending order by power of z . This macro expands to efficient inline code that uses either Horner's method or, for complex z , a more efficient Goertzel-like algorithm.

Execute a transformed version of the expression, which calls functions that may violate strict IEEE semantics. This allows the fastest possible operation, but results are undefined – be careful when doing this, as it may change numerical results.

This sets the LLVM Fast-Math flags, and corresponds to the -ffast-math option in clang. See the notes on performance annotations for more details.


More Division Stories

Videos and solutions to help Grade 6 students demonstrate further understanding of division of fractions by creating their own word problems.

New York State Common Core Math Grade 6, Module 1, Lesson 6

Lesson 6 Student Outcomes

&bull Students demonstrate further understanding of division of fractions when they create their own word problems.
&bull Students choose a partitive division problem, draw a model, find the answer, choose a unit, and then set up a situation. Further, they practice trying several situations and units before finding which are realistic with given numbers.

Lesson 6 Summary

The method of creating division stories has five steps, to be followed in order:

Step 1: Decide on an interpretation (measurement or partitive). Today we used only measurement division.
Step 2: Draw a model.
Step 3: Find the answer.
Step 4: Choose a unit.
Step 5: Set up a situation. This means writing a story problem that is interesting, realistic, short, and clear and that has all the information necessary to solve it. It may take you several attempts before you find a story that works well with the given dividend and divisor.

Discussion
Partitive division is another interpretation of division problems. What do you recall about partitive division?
&bull We know that when we divide a whole number by a fraction, the quotient will be greater than the whole number we began with (the dividend). This is true regardless of whether we use a partitive approach or a measurement approach.
&bull In other cases, we know what the whole is and how many groups we are making and must figure out what size the groups are.

Example 1
Partitive Division.
Divide 50 ÷ 2/3

Exercise 1
Using the same dividend and divisor, work with a partner to create your own story problem. You may use the same unit, dollars, but your situation must be unique. You could try another unit, such as miles, if you prefer.
Possible story problems:
1. Ronaldo has ridden 50 miles during his bicycle race and is 2/3 of the way to the finish line. How long is the race?
2. Samantha used 50 tickets (2/3 of her total) to trade for a kewpie doll at the fair. How many tickets did she start with?

Example 1
Partitive Division.
Divide 50 ÷ 2/3

1. Write a partitive division story problem for 45 ÷ 3/5

2. Write a partitive division story problem for 100 ÷ 2/5

Lesson 7 Student Outcomes

Students formally connect models of fractions to multiplication through the use of multiplicative inverses as they are represented in models.

The reciprocal , or inverse, of a fraction is the fraction made by interchanging the numerator and denominator.

Two numbers whose product is 1 are multiplicative inverses .

Lesson 8 Student Outcomes

Students divide fractions by mixed numbers by first converting the mixed numbers into a fraction with a value larger than one.
Students use equations to find quotients.

Example 1: Introduction to Calculating the Quotient of a Mixed Number and a Fraction

Carli has 4 1/2 walls left to paint in order for all the bedrooms in her house to have the same color paint. However, she has used almost all of her paint and only has 5/6 of a gallon left. How much paint can she use on each wall in order to have enough to paint the remaining walls?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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The order of a conjugacy class must always divide the order of the group. This follows from a theorem sometimes called the "orbit-stabilizer" theorem: The size of a group = the size or an orbit $ imes$ the size of a corresponding stabilizer.

Consider $G$ acting on itself by conjugation: $g circ x = gxg^<-1>$

Pick some $x in G$. Then $mathrm(x) = < g circ x | g in G>= < gxg^<-1>| g in G>$ (this is the conjugacy class of $x$) and $mathrm(x) = < g in G | g circ x=x >= < g in G | gxg^<-1>=x > = < g in G | gx=xg >$ (this is the centralizer of $x$).

So the size of a conjugacy class times the size of a corresponding centralizer is equal to the size of the group.

So for a group of order 30. The size of the conjugacy classes are limited to divisors of 30: 1, 2, 3, 5, 6, 10, 15, and 30. However, these classes also partition $G$, so their sizes must sum to the order of $G$. The identity is in a conjugacy class by itself, so you have to have a bunch of divisors which add up to 30 which include at least one 1 [this means we can't have a conjugacy class of which exhausts the whole group -- i.e. 30 is not an option]. Another limitation [on any class equation] is that the conjugacy classes of order 1 correspond to elements which commute with everything (i.e. elements of the center). So the 1's in the class equation need to add to a divisor of 30 as well (since the order of the center must divide the order of the group). So for example, $1+1+1+1+2+2+2+10+10=30$ is no good. Since the order of the center (a subgroup) would be $1+1+1+1=4$ which does not divide 30. That's about all the low hanging fruit.

Of course, more can be said. For example, the class equation cannot be $1+1+2+2+2+2+5+15=30$. If it was, the center of the group would have order 2 so that $G$ mod its center would have order 15. However, every group of order 15 is isomorphic to the cyclic group of order 15. This is a problem since if $G$ quotient its center is cyclic, then $G$ must be abelian so that the center is the whole group. Thus this particular equation is ruled out. Other than piecemeal tricks like this, the Sylow theorems tend to help you find lots of restrictions on the class equation.

[A note about orbit-stablizer theorem: If you let $G$ act on the left cosets of some subgroup $H$, then the orbit of $H$ is the set of all left cosets and the stabilizer is $H$ itself. In this case the orbit-stablizer theorem says the size of a subgropu times the number of cosets is the size of the group. That's Lagrange's thoerem. As you might imagine, since Lagrange's thorem is so important, a generalization of his theorem should show up quite often.]


Absolute Value

  1. The absolute value of a number is its distance from zero.
  2. For any x, |x| is defined as follows: | x |= x, if x > 0, and | x |= −x, if x < 0

Acute Angle
An angle whose measure is greater than 0 degrees and less than 90 degrees.

Acute Triangle
A triangle in which all three angles are acute angles.

Example of an acute angle

Addition Property of Equality
If a = b, then a + c = b + c. This property states that adding the same amount to both members of an equation preserves the equality.

Additive identity
A property that states that for any number x, x + 0 = x, zero is the additive identity.

Additive Inverse
For any number x, there exists a number −x, such that x + −x= 0. This means that there exists a pair of numbers (like 5 and –5) that are the same distance from zero on the number line, and when added together will always produce a sum of zero. These pairs of numbers are also sometimes called “opposites.”

Altitude of a Triangle
A segment drawn from a vertex of the triangle perpendicular to the opposite side of the triangle, called the base (or perpendicular to an extension of the base).

AD is an altitude of the triangle

Angle
An angle is formed when two rays share a common vertex.

Example of an angle

Area Model
A mathematical model based on the area of a rectangle, used to represent multiplication or fractional parts of a whole.

Associative Property of Addition
For any numbers x, y , and z: (x + y) + z = x + (y + z). The associative property of addition states that the order in which you group variables or numbers does not matter in determining the final sum.

Associative Property of Multiplication
For any numbers x, y , and z: (xy) z = x (yz). The associative property of multiplication states that the order in which you group variables or numbers does not matter in determining the final product.

Attribute
A distinguishing characteristic of an object. For instance, two attributes of a triangle are angles and sides.

Axis
A number line in a plane. Plural form is axes. Also see: Coordinate Plane.

Bar Graph
A graph in which rectangular bars, either vertical or horizontal, are used to display data.

Example of a bar graph
  1. If any number x is raised to the nth power, written as x^n, x is called the base of the expression
  2. Any side of a triangle
  3. Either of the parallel sides of a trapezoid
  4. Either of the parallel sides of a parallelogram.

Box and Whisker Plot
For data ordered smallest to largest the median, lower quartile and upper quartile are found and displayed in a box along a number line. Whiskers are added to the right and left and extended to the least and greatest values of the data.

Example of a box and whisker plot

Cartesian Coordinate System
See: Coordinate Plane

Center of a Circle
A point in the interior of the circle that is equidistant from all points of the circle.

A circle and its center

Chord
A segment whose endpoints are points of a circle.

An example of a chord of a circle

Circle
The set of points in a plane equidistant from a point in the plane.

Circumference
The distance around a circle. Its length is the product of the diameter of the circle and pi.

Coefficient
In the product of a constant and a variable the constant is the numerical coefficient of the variable and is frequently referred to simply as the coefficient.

Common Denominator
A common multiple of the denominators of two or more fractions. Also see: Least Common Denominator

Common Factor
A factor that two or more integers have in common. Also see: Greatest Common Factor.

Common Multiple
See: Least Common Multiple.

Complement
The complement of a set E is a set of all the elements that are not in E.

Complementary Angles
Two angles are complementary if the sum of their measures totals 90 degrees.

Example of two complementary angles, a and b

Composite Number
A prime number is an integer p greater than 1 with exactly two positive factors: 1 and p. A composite number is an integer greater than 1 that has more than two positive factors. The number 1 is the multiplicative identity that is, for any number n, n · 1 = n. The number 1 is neither a prime nor a composite number.

Compound Event
A subset of a sample space containing two or more outcomes.

Concentric circles
Circles with the same center and in the same plane that have different radii.

Cone
A three-dimensional figure with a circular base joined to a point called the apex.

Picture of a cone

Congruent
Used to refer to angles or sides having the same measure and to polygons that have the same shape and size.

Conjecture
An assumption that is thought to be true based on observations.

Constant
A fixed value.

Coordinate(s)
A number assigned to each point on the number line which shows its position or location on the line. In a coordinate plane the ordered pair, (x,y), assigned to each point of the plane, shows the point’s position in relation to the x-axis and y-axis.

Coordinate Plane
A plane that consists of a horizontal and vertical number line, intersecting at right angles at their origins. The number lines, called axes, divide the plane into four quadrants. The quadrants are numbered I, II, III, and IV beginning in the upper right quadrant and moving counterclockwise.

Counterclockwise
A circular movement opposite to the direction of the movement of the hands of a clock.

Counterclockwise

Counting Numbers
The counting numbers are the numbers in the following never-ending sequence: 1, 2, 3, 4, 5, 6, 7. We can also write this as +1, +2, +3, +4, +5, +6, +7. These numbers are also called the positive integers or natural numbers.

  1. A three-dimensional shape having six congruent square faces.
  2. The third power of a number.
Cube shape

Cylinder
A three-dimensional figure with parallel circular bases of equal size joined by a lateral surface whose net is a rectangle.

Cylinder

Data
A collection of information, frequently in the form of numbers.

Data Analysis
The process of making sense of collected data.

Data Point
Each individual piece of information collected in a set of data.

  1. The circumference of a circle is divided into 360 equal parts or arcs. Radii drawnto both ends of the arc form an angle of 1 degree.
  2. The degree of a term is the sum of the exponents of the variables.
  3. A degree is also unit of measurement used for measuring temperature.

Denominator
The denominator of a fraction indicates into how many equal parts the whole is divided. The denominator appears beneath the fraction bar.

Diameter
A segment with endpoints on the circle that passes through its center.

Dividend
The quantity that is to be divided.

Divisibility
Suppose that n and d are integers, and that d is not 0. The number n is divisible by d if there is an integer q such that n = dq. Equivalently, d is a factor of n or n is a multiple of d.

Division Algorithm
Given two positive integers a and b, we can always find unique integers q and r such that a= bq + r and 0 ≤ r < b. We call a the dividend, b the divisor, q the quotient, and r the remainder.

Divisor
The quantity by which the dividend is divided.

Domain
The set of input values in a function.

Edge
A segment that joins consecutive vertices of a polygon or a polyhedron.

Elements
Members of a set.

Empirical Probability
Probability determined by real data collected from real experiments.

Equation
A math sentence using the equal sign to state that two expressions represent the same number.

Equilateral Triangle
An equilateral triangle is a triangle with three congruent sides. An equilateral triangle also has three congruent angles, which we can also call equiangular triangle.

  1. A term used to describe fractions or ratios that are equal.
  2. A term used to describe fractions, decimals, and percents that are equal.

Event
An event is any subset of the sample space. A simple event is a subset of the sample space containing only 1 possible outcome of an experiment. A compound event is a subset of the sample space containing 2 or more outcomes.

Experiment
A repeatable action with a set of outcomes.

Exponent
Suppose that n is a whole number. Then, for any number x, the nth power of x, or x to the nth power, is the product of n factors of the number x. This number is usually written x^n. The number x is usually called the base of the expression x^n, and n is called the exponent.

Exponential Notation
A notation that expresses a number in terms of a base and an exponent.

Expression
A mathematical phrase like “m + 1” used to describe quantities mathematically with numbers and variables.

Face
Each of the surface polygons that form a polyhedron.

Factor
An integer that divides evenly into a dividend. Use interchangeably with divisor except in the Division Algorithm.

Factorial
The factorial of a non-negative number n is written n! and is the product of all positive integers less than or equal to n. By definition 0!= 1!= 1.

Fraction
Numbers of the form m/n, where n is not zero.

Frequency
The number of times a data point appears in a data set.

Function
A function is a rule that assigns to each member of a set of inputs, called the domain, a member of a set of outputs, called the range.

Graph of a Function
The pictorial representation of a function.

Greater than, Less Than
Suppose that x and y are integers. We say that x is less than y, x < y, if x is to the left of y on the number line. We say that x is greater than y, x > y, if x is to the right of y on the number line.

Greatest Common Factor, GCF
Suppose m and n are positive integers. An integer d is a common factor of m and n if d is a factor of both m and n. The greatest common factor, or GCF, of m and n is the greatest positive integer that is a factor of both m and n. We write the GCF of m and n as GCF (m,n).

Height
The length of the perpendicular between the bases of a parallelogram or trapezoid also the altitude of a triangle.

Horizontal Axis
See: Coordinate Plane.

Hypotenuse
The side opposite the right angle in a right triangle.

Improper Fraction
A fraction in which the numerator is greater than or equal to the denominator.

Independent Events
If the outcome of an event does not affect the outcome of other events.

Input Values
The values of the domain of a function.

Integers
The collection of integers is composed of the counting numbers, the negatives, and zero . −4, −3, −2, −1, 0, 1, 2, 3, 4.

Isosceles Triangle
A triangle with at least two sides of equal length.

Example of isosceles triangle

Lateral Area
The surface area of any three-dimensional figure excluding the area of any surface designated as a base of the figure.

Lattice Point
A point of the coordinate plane, (x,y), in which both x and y are integers.

Least Common Denominator
The least common denominator of the fractions p/n and k/m is the least common multiple of n and m, LCM(n, m).

Least Common Multiple, LCM
The integers a and b are positive. An integer m is a common multiple of a and b if m is a multiple of both a and b. The least common multiple, or LCM, of a and b is the smallest integer that is a common multiple of a and b. We write the LCM of a and b as LCM (a,b).

  1. The two sides of a right triangle that form the right angle.
  2. The equal sides of an isosceles triangle or the non-parallel sides of a trapezoid.

Less than
See: Greater Than.

Line graph
A graph used to display data that occurs in a sequence. Consecutive points are connected by segments.

Line Plot
A graph that shows frequency of data along a number line.

Linear Model for Multiplication
Skip counting on a number line.

Mean
The average of a set of data sum of the data divided by the number of items. Also called the arithmetic mean or average.

Measures of Central Tendency
Generally measured by the mean, median, or mode of the data set.

Median
The middle value of a set of data arranged in increasing or decreasing order. If the set has an even number of items the median is the average of the middle two items.

Missing Factor Model
A model for division in which the quotient of an indicated division is viewed as a missing factor of a related multiplication.

Mixed fraction (Numbers)
The sum of an integer and a proper fraction.

Mode
The value of the element that appears most frequently in a data set.

Multiplicative Identity
See: Composite Numbers.

Multiplicative Inverse
The number x is called the multiplicative inverse or reciprocal of n, n ≠ 0, if x · n = 1.

Natural Numbers
See: Counting Numbers.

Negative Integers
Integers less than zero.

Notation
A technical system of symbols used to convey mathematical information.

Number Line
A pictorial representation of numbers on a straight line.

Numerator
The expression written above the fraction bar in a common fraction to indicate the number of parts counted.

Obtuse Angle
An angle whose measure is greater than 90 degrees and less than 180 degrees.

Obtuse Triangle
A triangle that has one obtuse angle.

Order Of Operations
The order of mathematical operations, with computations inside parentheses to be done first, and addition and subtraction from left to right done last.

Ordered Pair
A pair of numbers that represent the coordinates of a point in the coordinate plane with the first number measured along the horizontal scale and the second along the vertical scale.

Origin
The point with coordinate 0 on a number line the point with coordinates (0,0) in the coordinate plane.

Outcomes
The set of possible results of an experiment.

Outlier
A term referring to a value that is drastically different from most of the other data values.

Output Values
The set of results obtained by applying a function rule to a set of input values.

Parallel Lines
Two lines in a plane that never intersect.

Example of two parallel lines

Parallelogram
A parallelogram is a four-sided figure with opposite sides parallel.

Percent
A way of expressing a number as parts out of 100 the numerator of a ratio with a denominator of 100.

Perfect Cube
An integer n that can be written in the form n= k³, where k is an integer.

Perfect Square
An integer n that can be written in the form n= k², where k is an integer.

Perimeter
The perimeter of a polygon is the sum of the lengths of its sides.

Perpendicular
Two lines or segments are perpendicular if they intersect to form a right angle.

Example of two perpendicular lines

Pi
The ratio of the circumference to the diameter of any circle, represented either by the symbol π, or the approximation 22/7 , or 3.1415926.

Pie (Circle) Graph
A graph using sectors of a circle that are proportional to the percent of the data represented.

Example of a pie graph

Polygon
A simple, closed, plane figure formed by three or more line segments.

Polyhedron
A three-dimensional figure with four or more faces, all of which are polygons.

Examples of polyhedrons

Positive Integers
See: Counting Numbers.

Power
See: Exponent.

Prime Number
See: Composite Number.

Prime Factorization
The process of finding the prime factors of an integer. The term is also used to refer to the result of the process.

Prism
A type of polyhedron that has two bases that are both congruent and parallel, and lateral faces which are parallelograms.

Examples of prisms

Probability
In an experiment in which each outcome is equally likely, the probability P(A) of an event A is m/n where m is the number of outcomes in the subset A and n is the total number of outcomes in the sample space S.

Proper Fraction
A fraction whose value is greater than 0 and less than 1.

Proportion
An equation of ratios in the form a/b = c/d, where b and d are not equal to zero.

Protractor
An instrument used to measure angles in degrees.

Quadrant
See: Coordinate Plane.

Quadrilateral
A plane figure with four straight edges and four angles.

Quotient
The result obtained by doing division. See the Division Algorithm for a different use of quotient.

Radius
The distance from the center of a circle to a point on the circle. Plural form is radii.

Range
The difference between the largest and smallest values of a data set. See Function for another meaning of range.

Rate
A rate is a division comparison between two quantities with different units. Also see Unit Rate.

Ratio
A division comparison of two quantities with or without the same units. If the units are different they must be expressed to make the ratio meaningful.

Rational Number
A number that can be written as a/b where a is an integer and b is a natural number.

Ray
Part of a line that has a starting point and continues forever in only one direction.

Reciprocal
See: Multiplicative Inverse.

Regular Polygon
A polygon with equal side lengths and equal angle measures.

Relatively Prime
Two integers m and n are relatively prime if the GCF of m and n is 1.

Remainder
See: Division Algorithm.

Repeating Decimal
A decimal in which a cycle of one or more digits is repeated infinitely.

Right Angle
An angle formed by the intersection of perpendicular lines an angle whose measure is 90º.

Right Triangle
A triangle that contains a right angle.

Sample Space
The set of all possible outcomes of an experiment.

Scaffolding
A method of division in which partial quotients are computed, stacked, and then combined.

Scalene Triangle
A triangle with all three sides of different lengths is called a scalene triangle.

Examples of scalene triangles
  1. A process by which a shape is reduced or expanded proportionally.
  2. Choosing the unit of measure to be used on a number line.

Sector
A part of a circle that represents the interior portion of the circle between two radii.

Sequence
A list of terms ordered by the natural numbers.

Set
A collection of objects or elements.

Simple Event
See: Event

Simplest Form of a Fraction
A form of a fraction in which the greatest common factor of the numerator and denominator is 1.

Simplifying
The process of finding equivalent fractions to obtain the simplest form.

Skewed
An uneven representation of a set of data.

Slant Height
An altitude of a face of a pyramid or a cone.

Slant Height of a pyramid

Square Root
For non-negative numbers x and y, y= x , read “y is equal to the square root of x,” means y²= x.

Stem and Leaf Plot
A method of showing the frequency of a certain data by sorting and ordering the values.

Straight Angle
An angle with a measure of 180 degrees formed by opposite rays.

Subset
Set B is a subset of set A if every element of set B is also an element of set A.

Supplementary
Two angles are supplementary if the sum of their measures totals 180º.

Example of two supplementary angles, x and y

Surface Area
The surface area of a three-dimensional figure is the area needed to form its exterior.

Terminating Decimal
If the quotient of a division problem contains a remainder of zero, the quotient is said to be a terminating decimal.

Tessellation
Tiling of a plane with one or more shapes as a way of covering the plane with the shape(s) with no gaps or overlaps.

Theoretical Probability
Probability based on thought experiments rather than a collection of data.

Translation
A transformation that slides a figure a certain distance along a line in a specified direction.

Trapezoid
A four sided plane figure with exactly one set of parallel sides.

Tree Diagram

  1. A process used to find the prime factors of an integer.
  2. A method to organize the sample space of compound events.

Triangle
A plane figure with three straight edges and three angles.

Trichotomy
A property stating that exactly one of these statements is true for each real number: it is positive, negative, or zero.

Unit Fraction
For an integer n, the multiplicative inverse or reciprocal of n is the unit fraction 1/n. 1/n is said to be a unit fraction because its numerator is 1.

Unit Rate
A ratio of two unlike quantities that has a denominator of 1 unit.

Variable
A letter or symbol that represents an unknown quantity.

Venn Diagram
A diagram involving two or more overlapping circles that aids in organizing data.

  1. The common endpoint of two rays forming an angle.
  2. A point of a polygon or polyhedron where edges meet.

Vertical Angles
A pair of angles of equal measure less than 180° that are formed by opposite rays of a pair of intersecting lines.

Vertical Axis
See: Coordinate Plane.

Volume
A measure of space the number of unit cubes needed to fill a three-dimensional shape.

Whole Numbers
The whole numbers are the numbers in the following never-ending sequence: 0, 1, 2, 3, 4, 5, . These numbers are also called the non-negative integers.

x-axis
The horizontal axis of a coordinate plane.

y-axis
The vertical axis of a coordinate plane.

x-coordinate
The first number provided in an ordered pair (a, b).

y-coordinate
The second number provided in an ordered pair (a, b).


Watch the video: 6th Grade, Unit 4, Lesson 1 Size of Divisor and Size of Quotient Open Up Resources - Tutorial (December 2021).