The possible values are -
(a²,b²) = p
(a²,b) = p if p|-r and (a²,b) = p² if p|r
(a²,b³) = p² if p|-r and (a²,b³) = p³ if p|r
What is gcd?
The greatest common factor (GCF) that divides two or more numbers is known as the greatest common divisor (GCD). The highest common factor is another name for it (HCF).
Since (a, b) = p it can be written a = pr and b = ps where r and s are integers such that (r, s) = 1.
Then a² = p²r².
Then by Theorem 1.7, (a²,b) = (p²r²,ps) = p(pr²,s).
By Theorem 1.8, (r²,s) = 1 .
Thus, there are two cases to consider: if p|s then, (a²,b) = p², and if p|-s then (pr²,s) = 1 (Again by Theorem 1.8) so that (a²,b) = p in that case.
Thus, there are two cases for (a²,b): namely p or p².
The same analysis shows that (a³,b) must be either p, p² or p³ depending on the power of p that divides b.
Analyze (a²,b³) as follows.
Using the notation already introduced, and
Theorem 1.7, it is obtained that (a²,b³) = (p²r²,p³s³) = p²(r²,ps³).
Since, (r, s) = 1, Theorem 1.8 shows that (r²,s³) = 1.
Therefore, there are two possibilities for (a²,b³) -
(a²,b³) = p² if p|-r and (a²,b³) = p³ if p|r.
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4. In your own words, explain the process of simplifying square roots.
The process of simplifying square roots involves finding an equivalent expression for the square root that has smaller or more manageable numbers inside the root sign.
This can be achieved by using prime factorization, the properties of square roots, and rationalizing the denominator. The final result should be in the simplest form possible, without any radicals in the denominator, or any perfect square factors in the numerator.
What is a square root?A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 x 3 = 9. The symbol for square root is √. For example, √9 = 3.
The square root of a number can be either positive or negative, but for most purposes, the positive square root is used.
Additionally, there are also square roots of expressions, not just numbers, which can be simplified using similar methods
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Consider the cubic function g(x)=-2(x+4)^3-5. Please describe how this function has been transformed from the cubic parent function f(x)=x^3
Answer: The cubic parent function f(x)=x^3 has been transformed to form the function g(x)=-2(x+4)^3-5 in the following ways:
Vertical scaling: The function has been vertically scaled by a factor of -2, which means that the height of the function has been multiplied by -2.
Horizontal shift: The function has been horizontally shifted 4 units to the right. This is because the expression inside the parenthesis (x+4) can be seen as shifting the graph 4 units to the right.
Vertical shift: The function has been vertically shifted down by 5 units. This is because the constant term -5 has been added to the expression, which moves the graph down by 5 units.
So, the parent function f(x)=x^3 has been transformed by a vertical scaling by -2, followed by a horizontal shift 4 units to the right, and finally a vertical shift 5 units down to form the function g(x)=-2(x+4)^3-5.
Hamish has $25. 97 in his pocket to spend at the craft store. He wants to buy a paint canvas and some paint pens. How many paint pens, p, can Hamish buy?
The diagram shows a circle with centre O.
A, B, C & D lie on the circumference of this circle.
Given that AC is a diameter of the circle and ∠DCA = 33° and ∠BCA = 31°, find the size of ∠DAB as highlighted in the diagram.
Hi there, here's your answer.
Given:
A circle with center O.
A, B, C and D lie on the circumference of the circle.
AC is the diameter of the circle.
∠DCA = 33° and ∠BCA = 31°
To find:
∠DAB
Solution:
Since AC is a diameter, ∠CDA and ∠CBA will be equal to 90° (Angles in a semi-circle)
Now, ∠BCD = ∠DCA + ∠BCA
∴ ∠BCD = 33° + 31° = 64°
Now, ∠BCD + ∠CDA + ∠DAB + ∠ ABC = 360° (Angle-sum property of a quadrilateral)
Substituting the values of the angles:
64° + 90° + 90° + ∠DAB = 360°
Or
244° + ∠DAB = 360°
Therefore ∠DAB = 360° - 244° = 116°
Hope it helps!
Use the graph to solve the problem.
12) Find the equation of the line in standard and slope-intercept form from the graph given.
Make sure the standard form of the equation is written with whole numbers.
-10
K
5
10--axis.
10-
5
x-axis
Y
10
The equation of line in standard form and slope intercept form is,
⇒ y - 5 = - x
⇒ y = - x + 5
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
Two points on the line are (0, 5) and (5, 0).
Now,
Since, The equation of line passes through the points (0, 5) and (5, 0).
So, We need to find the slope of the line.
Hence, Slope of the line is,
m = (y₂ - y₁) / (x₂ - x₁)
m = (0 - 5) / (5 - 0)
m = - 5 / 5
m = - 1
Thus, The equation of line with slope - 1 is,
⇒ y - 5 = - 1 (x - 0)
⇒ y - 5 = - x
⇒ y = - x + 5
Therefore, The equation of line passes through the points (0, 5) and
(5, 0) will be;
⇒ y - 5 = - x
⇒ y = - x + 5
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at 8:00 a.m. the temperature was 3° below zero by 1:00 p.m. the temperature rose 14° and my 10:00 p.m. drop 12 degrees what was the temperature at 10:00 p.m.
Based on the information provided, it can be concluded that by 10:00 p.m. the temperature was 1°.
How did the temperature change over time?First, let's list the information provided:
8:00 a.m. = -3
1:00 p.m. = +14
10:00 p.m. = - 10
Based on this, let's calculate the temperature by 10:00 p.m.
-3 + 14 = 11° by 1:00 p.m.
This means the temperature at 1:00 p.m. was 11° grades. Now, let's calculate the temperature at 10:00 p.m.
11 -10 = 1° by 10:00 p.m
This means that by 10:00 p.m. the temperature was 1°.
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Estimate 8 1/3- 3 3/4. Tell me how you got your estimate. Susi subtracted and found the actual difference to be 5 7/1. Is her answer reasonable? Explain
After solving the fraction answer is 4 7/12 but the Susi's answer is 5 7/1. So the Susi's answer is wrong.
The given expression is 8 1/3- 3 3/4
We first solve that expression.
We convert mixed fraction into improper fraction.
After conversion we equal the denominator of of both fraction.
Then we subtract the fraction.
After solving we check Susi's answer is right or wrong.
To convert 8 1/3 into improper fraction we multiply 8 and 3 then we add 1 in the answer of multiplication of 8 and 3.
8 1/3 = 25/3
To convert 3 3/4 into improper fraction we multiply 3 and 4 then we add 3 in the answer of multiplication of 3 and 4.
3 3/4 = 15/4
Now the expression is:
= 25/3 - 15/4
Multiply and divide by 12 on both side
= 25/3 × 12/12 - 15/4 × 12/12
= (25 × 4)/12 - (15 × 3)/12
= 100/12 - 45/12
= (100 - 45)/12
= 55/12
= 4 7/12
The simplified answer of given expression is 4 7/12 but the Susi's answer is 5 7/1. So the Susi's answer is wrong.
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How do you write six times the quantity two x plus three y
Answer:
6(2x + 3y)
Step-by-step explanation:
Framing algebraic equation:Two x plus three = 2x + 3y
Six times of (2x + 3y) = 6 *(2x + 3y)
what fraction replaces these ? to make this calculation correct 5/8 + ? = 1
List five general things we get from natural resources?
Five general things we get from natural resources are food, energy, raw material, medicine, leisure, recreation.
What do you mean by resources?Resources refer to the materials, substances, and energy sources that exist in the environment and can be used to produce goods and services. They can be natural, such as minerals, water, forests, and wildlife, or human-made, such as technology, infrastructure, and knowledge. The term "resources" also encompasses the availability and accessibility of these materials, as well as the ability to use and manage them sustainably.
Food: Natural resources such as crops, fisheries, and livestock provide us with a source of nutrition.
Energy: Resources such as coal, oil, natural gas, and hydroelectric power are used to produce energy for heating, transportation, and other forms of electricity.
Raw Materials: Natural resources such as minerals, metals, and forests provide the raw materials needed to manufacture a wide range of products, including construction materials, electronics, and consumer goods.
Medicines: Many natural resources, such as plants and minerals, have medicinal properties that can be used to treat and prevent diseases.
Leisure and recreation: Natural resources like parks, forests, lakes, and beaches provide opportunities for outdoor recreation and leisure activities, contributing to our mental and physical well-being.
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I need help somewhat quickly please! || one half ÷ 3 = ___.
6
one and one-half
two-thirds
one-sixth
Answer:
One-SixthStep-by-step explanation:
One half ÷ 3 = 1/2 ÷ 3
To divide fractions, we need to multiply the first fraction by the reciprocal of the second fraction. So,
1/2 ÷ 3 = 1/2 x 1/3 = 1/6
Therefore, one half ÷ 3 is equal to one-sixth.
Hope it helps!if jack has three packs of gum and sam has 4 packs of gum. how much do they have in total
If jack has three packs of gum and sam has 4 packs of gum, then the total amount of gum they have together is 7 packs.
What is the addition?
The addition is putting numbers together to make a sum or total. The plus sign + tells you to add two numbers. = is an equals sign. It shows that things on either side of it are equal.
We have given,
Jack has 3 packs of gum and Sam has 4 packs of gum.
To find the total number of packs of gum they have together, we simply add their individual amounts:
3 packs (Jack) + 4 packs (Sam) = 7 packs
Therefore, the total amount of gum they have together is 7 packs.
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Sin Cos Tan
Geometry
The trigonometric ratios for each triangles are:
Triangle 1:Please note that all angles B are the right angle, then:
cos B = 1; sin B = 0; tan B = -
The trigonometric ratio of sin, cos, tan, can be formulated as:
(please refer to the attached triangle below for reference)
cos A = side adjacent to A / Hypotenuse
Cos A = b/c
Sin A = side opposite to A / Hypotenuse
Sin A = a/b
Tan A = Side opposite to A / Side adjacent to A
Tan A = a/b
From these formulas, we can find that:
Triangle 1
Cos A = 40/50 = 4/5
Sin A = 30/50 = 3/5
Tan A = 30/40 = 3/4
Cos C = 30/50 = 3/5
Sin C = 40/50 = 4/5
Tan C = 40/30 = 4/3
Triangle 2
Cos A = 27/45 = 3/5
Sin A = 36/45 = 4/5
Tan A = 36/27 = 4/3
Cos C = 36/45 = 4/5
Sin C = 27/45 = 3/5
Tan C= 27/36 = 3/4
Triangle 3
Cos A = 15/17
Sin A = 8/15
Tan A = 8/15
Cos C = 8/17
Sin C = 15/17
Tan C = 15/8
Please note that all angle B on the three triangles are all a right angle, then:
Cos B = 1
Sin B = 0
Tan B = -
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(a)(i) which of the following is the most appropriate measure for the typical value of the average salary paid to nfl players? --- what is the value of this salary? (use jmp to compute it, the closest integer is sufficient.) $ (ii) which is the most appropriate measure for the spread of these salaries?
The most appropriate measure for the typical value of the average salary paid to nfl players is mean.
The most appropriate measure for the spread of these salaries is IQR
In statistics, in addition to the mode and median, the mean is one of the measures of central tendency. Simply said, the mean is the average of the values in the given collection. It indicates that values in a certain data collection are distributed equally.
The three most often employed measures of central tendency are the mean, median, and mode. The total values provided in a datasheet must be added, and the sum must be divided by the total number of values in order to get the mean.
The difference between the third and the first quartile is defined by the interquartile range. The partitioned values known as quartiles divide the whole series into four equally sized segments. There are so three quartiles.
The first quartile, also known as the lower quartile, is marked by Q1, the second quartile by Q2, and the third quartile, sometimes known as the upper quartile, is denoted by Q3. The lower quartile less the higher quartile equals the interquartile range.
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i need to know every one of these questions
Which option is correct? Explain your working after choosing your answer
18. 2x3 + 4x2 + 2x
A. This polynomial could be factored by finding the GCF, then by grouping or using the perfect squares method.
B. This polynomial could be factored by using the difference of squares method, perfect squares method, or grouping.
C. This polynomial could be factored only by using the perfect squares method.
D. This polynomial could be factored by using grouping or the perfect squares methods.
E. This polynomial could be factored only by using the difference of squares method.
F. This polynomial cannot be factored by any of the methods used in this lesson.
The option that is correct is ' This polynomial could be factored by finding the GCF, then by grouping or using the perfect squares method. (optionA)
What is factorization of polynomial?Factoring a polynomial is expressing the polynomial as a product of two or more factors; it is somewhat the reverse process of multiplying. To factor polynomials, we generally make use of the following properties or identities; along with other more techniques. Distributive Property: ab+ac=a(b+c)
To factorize the expression 2x3 + 4x2 + 2x
Let's factor out the Greatest common factor which is 2x
2x( x²+2x+1)
2x( x² +2x + 1)
Therefore the factors of 2x³ + 4x²+ 2x are 2x and x²+2x+1)
Therefore the expression can be factored by finding the GCF and then grouping or using perfect square method.
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3/4 Divided by 6 2/3 as a fraction in simplest form
The fraction in simplest form for 3/4 divided by 6 2/3 is 5/9. To get this fraction, the numerator 3/4 is multiplied by the reciprocal of 6 2/3 which is 9/6. This results in 5/9.
To solve 3/4 divided by 6 2/3, the first step is to convert 6 2/3 into an equivalent fraction with a denominator of 4. To do this, the numerator of 6 2/3 is multiplied by 4/4 which results in 25/4. The next step is to take the reciprocal of the denominator 25/4. The reciprocal is 4/25. This can be done by flipping the numerator and denominator of 25/4. The final step is to multiply 3/4 by 4/25. The fraction in simplest form for 3/4 divided by 6 2/3 is 5/9. This results in the fraction 5/9. Therefore 3/4 divided by 6 2/3 is equal to 5/9 in simplest form.
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A picture frame is 112 inches wider than the picture
it holds. Let f represent the width of the frame. Let p represent the width of the picture. Complete the table.
Answer: (1,2.5), (2, 3.5), (3, 4.5)
Step-by-step explanation: The question states that the frame is one and a half inches wider than the picture, so the equation would be f = p+ 1.5. Add 1.5 to each value of p to find the value of f
What is the Reciprocal of 1/10
Answer: 10
Step-by-step explanation:
ex: reciprocal of 2 would be 1/2 because 2 can be written as 2/1; you just switch the demoninator and numerator
explain and solve part a and b please, do not understand
The graph of the first function is y = √(x - 5).
The graph of the second function is y = |x - 1| + 4.
What are some rules of function transformations?Suppose we have a function f(x).
f(x) ± d = Vertical upshift/downshift by d units (x, y ±d).
f(x ± c) = Horizontal left/right shift by c units (x - + c, y).
(a)f(x) = Vertical stretch for a > 0, vertical shrink a < 0. (x, ay).
f(bx) = Horizonatal compression b > 0, horizontal stretch for b < 0. (bx , y).
f(-x) = Reflection over y-axis, (-x, y).
-f(x) = Reflection over x-axis, (x, -y).
The graph of the first function is f(x) = y = √x.
Now, f(x - 5) would be the same function shifter 5 units to the right,
f(x) = √(x - 5).
The graph of the second function is a modulus function shifted up 2 units and to right by 1 unit g(x) = |x - 1| + 2 and again g(x) + 2 = |x - 1| + 4.
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find the area of the region bounded by the line y=3x-6, the line 2x+8, and the line y=6
The area of the region bounded by the lines y = 3x - 6 and y = -2x + 8 between the x-axis is 12/5 units
How to calculate the areaWe can find the intersection point between these two lines;
y = 3x - 6
y = -2x + 8
Set these two equations equal to each other.
3x - 6 = -2x + 8
5x - 6 = 8
Add 6 to both sides of the equation.
5x = 14
Divide both sides of the equation by 5.
x = 14/5
Find the y-value where these points intersect by plugging this x-value back into either equation.
y = 3(14/5) - 6
Multiply and simplify.
y = 42/5 - 6
Multiply 6 by (5/5) to get common denominators.
y = 42/5 - 30/5
Subtract and simplify.
y = 12/5
These two lines intersect at the point 12/5. This is the height of the triangle formed by these two lines and the x-axis.
Now let's find the roots of these equations (where they touch the x-axis) so we can determine the base of the triangle.
Set both equations equal to 0.
(I) 0 = 3x - 6
Add 6 both sides of the equation.
6 = 3x
x = 2
Set the second equation equal to 0.
(II) 0 = -2x + 8
Add 2x to both sides of the equation.
2x = 8
Divide both sides of the equation by 2.
x = 4
The base of the triangle is from (2,0) to (4,0), making it a length of 2 units.
The height of the triangle is 12/5 units.
Formula for the Area of a Triangle:
A = 1/2bh
Substitute 2 for b and 14/5 for h.
A = (1/2) · (2) · (12/5)
Multiply and simplify.
A = 12/5
The area of the region is 12/5 units
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x =
1
2
cos²x how does this equal? Please help
Answer:
Step-by-step explanation:
1/2*2cosx*sinx
= sin2x/2
suppose 20 students are chosen from a group of 50 juniors and 60 seniors to receive an award. if all students are equally likely to be chosen for the award, find the expected value and variance of the number of seniors who receive the award.
The variance is 4.93 and the expected value of the number of seniors receiving the award is 10.91.
Let X represent the total number of seniors who will be honored. Using parameters n = 20 and p = 60/110, where n is the number of trials and p is the likelihood that a trial would succeed, we can represent X as a binomial random variable (i.e., a senior receiving the award).
X's anticipated value is determined by:
E(X) = np = 20 * 60/110 = 10.91
The formula for X's variance is:
Var(X) = np(1 - p) = 20 x 60 x 50 x 110 = 4.93
As a result, the variance is 4.93 and the expected value of the number of seniors receiving the award is 10.91.
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A line with a slope of 8 passes through the point (4, 5). What is its equation in point-slop form?
y= (1/9)x + b, plug in the point to solve for b
5= (1/9)(-4) + b
b = 5 + 4/9 = 49/9
y= (1/9)x + 49/9
9y = x + 49
Find the value of the variable in the following figures.Round answers to the nearest hundredth.
Answer:
42.9
Step-by-step explanation:
tan 40 = 36/x
so x = 36/tan 40 =36/0.84 =42.9
PLEASE HELPPP
ABCD is a rhombus in which AC = 16cm and BD = 12cm.
i) What is the magnitude of AÔB?
ii) Find the length of AB
Answer:
(i) 90°
(ii) 10 cm
Step-by-step explanation:
What is a Rhombus?
A rhombus is a special parallelogram that has all four sides equal. But unlike a square, the angles are not right angles
Properties of a rhombus
Opposite angles are equal.The opposite sides are equal and parallel.Diagonals bisect each other at 90°The sum of any two adjacent or consecutive angles is 180°.We are given AC = 16 cm, BD = 12 cm
Using property 3, we see that
AO = CO = AB/2 = 16/2 = 8 cm
BO = DO =12/2 = 6 cm
m∠AOB = 90° Answer (i)
(ii) Since m∠AOB = 90°, ΔAOB is a right triangle with AB as the hypotenuse and AO, BO as the legs of the right triangle
By the Pythagorean Theorem
AB² = AO² + BO²
AB² = 8² + 6²
AB² = 64 + 36
AB² = 100
AB = √100 = 10 cm Answer (ii)
Answer:
∠AOB = 90°AB = 10 cmStep-by-step explanation:
You have a rhombus ABCD with diagonals AC = 16 cm and BD = 12 cm. You want the magnitude of central angle AOB, and the length of one side, AB.
RhombusThe diagonals of a rhombus bisect each other and cross at right angle. They divide the figure into four (4) congruent right triangles.
The central angle AOB has magnitude 90°.
SidesEach of these right triangles will have leg lengths of 6 cm and 8 cm, which you recognize as having the ratio 3 : 4. These are the leg lengths of a 3:4:5 right triangle. In this rhombus, those right triangles have lengths 6 cm : 8 cm : 10 cm. The side AB of the rhombus is 10 cm.
__
Additional comment
If you don't remember the 3-4-5 right triangle, you can use the Pythagorean theorem to find the hypotenuse of the triangle with legs 6 cm and 8 cm:
c² = a² +b²
c² = 6² +8² = 36 +64 = 100
c = √100 = 10
The side AB is 10 cm long.
HELP HELP PLS LOOK AT IMAGEE
Answer:
4 hours
Step-by-step explanation:
By adding the x's on the graph, they practiced 2/4 + 3/4 + 3/4 + 1 + 1 hours, which is 4 hours
The Top-Notch Middle School had athletes from all the grades playing on a sports team.
6th grade 7th grade 8th grade
Basketball 2 4 7
Baseball 1 6 5
Soccer 7 4 4
Football 8 15 10
Explain what the ratio 15:73 represents. pls help
Based on the information presented, the ratio represents 8th-grade baseball players to total athletes.
What is a ratio?A ratio is a mathematical expression that shows the proportion or the number of a specific element in a category compared to the number of another element in the same category. For example, if you count the animals on a farm the ratio of cows to chickens can be 5:7, which means that for every 5 cows there are 7 chickens.
here, we have,
What does the ratio 5:73 mean?
5 : This is the number of baseball players who are in 8th-grade
73: This is the total of players. You get this number if you add all the numbers given.
Based on this, this ratio is comparing the number of baseball players and total players, which means the meaning of this ratio is " the ratio of 8th grade baseball players to total athletes".
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If 6 times the 6th term of an A.P., is equal to 9 times the 9th term, find its 15th term
If 6 times the 6th term of an A.P is equal to 9 times the 9th term, then 15th term of the A.P. is -2d with d as the common difference.
Let's call the common difference of the arithmetic progression (A.P.) "d". The nth term of an A.P. is given by the formula T = a + (n - 1)d, where a is the first term and n is the term number.
Using this formula, we can write two expressions for the 6th and 9th terms:
T₆ = a + (6 - 1)d = a + 5d
T₉ = a + (9 - 1)d = a + 8d
According to the problem, 6 times the 6th term of the A.P. is equal to 9 times the 9th term, so we can write the following equation:
6T₆ = 9T₉
Substituting the expressions for T₆ and T₉, we get:
6(a + 5d) = 9(a + 8d)
Expanding and solving for a, we get:
6a + 30d = 9a + 72d
Subtracting 6a from both sides:
24d = 3a + 72d
Subtracting 72d from both sides:
-48d = 3a
Dividing both sides by 3:
-16d = a
So, the first term of the A.P. is -16d.
To find the 15th term, we can use the formula for the nth term:
T₁₅ = a + (15 - 1)d = -16d + 14d = -2d
So, the 15th term of the A.P. is -2d.
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56 boys and 34 girls visit a zoo.
46 of these children have a voucher.
Three times as many boys as girls do not have a voucher.
Complete the frequency tree
The frequency tree will be:
Total Visitor of Zoo: 90Boys = 56Girls = 34Voucher = 46No voucher = 44Tree imagination attached as image below.
The Frequency Tree and RatioThis question can be answered by calculating the ratio of boys to girls and the number of people with and without vouchers.
To find the total number of people visiting the zoo, simply add the number of boys (56) to the number of girls (34) to get 90. To find the ratio of boys to girls, divide the number of boys (56) by the total number of people (90) to get 62%. Divide the number of girls (34) by the total number of people (90) to get 38%. To find the number of people with and without vouchers, divide the number of people with vouchers (46) by the total number of people (90) to get 51%. Divide the number of people without vouchers (44) by the total number of people (90) to get 49%. This information can be used to determine the total number of people visiting the zoo, the ratio of boys to girls, and the number of people with and without vouchers.Learn more about The Frequency Tree and Ratio here:
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