Math Masters: Grade 9 Showdown!

Number Ninjas

100

3/4 is an example of which type of number?

Answer = rational

A rational number is any number that can be expressed as a fraction, where the numerator and denominator are integers and the denominator is not zero.

200

The expression (x⁵)(x^-4)(x^-3) is simplified to:

Answer = x^-2

(x⁵)(x^-4)(x^-3)

= x^{5 + (-4) +(-3)}

Simplifying the exponents:

5 + (-4) + (-3) = 5 - 4 - 3 = -2

Therefor, the expression simplifies to:

x^-2

300

Solve for x.

2/5 = x/40

Answer x=16

First, cross multiply

5 x X = 5x

2 x 40 = 80

5x=80

Then, divide so x with be by it self

5/x = x

80/5 = 16

So, the final answer is:

X = 16

400

Evaluate: 2(3 - 5)² + 4(-2) 

                                              

Answer = 0

First, inside the brackets, preform the subtraction:

2(-2)² + 4(-2)

Next, square the value inside the brackets:

2*4 + 4(-2)

Now, multiply and then add:

8 - 8

Finally, Subtract:

= 0

500

Evaluate and simplify if possible: (x⁵)³ / x³(x⁶)

Answer = x⁶

First, apply the exponent rule for powers of powers by multiplying the exponents inside the brackets:

x¹⁵ / x³(x⁶) 

Next, apply the rule for multiplying powers with the same base by adding the exponents:

x¹⁵ / x^{3 + 6}

x¹⁵ / x⁹

Finally, subtract the exponents in the denominator from the exponents in the numerator:

x^15-9

= x⁶

Algebra Adventures

100

Simplify. 

8x + 4 - 6x + 7

Answer = 2x + 11

Combine the x terms:

8x - 6x = 2x

Combine the constant terms:

4 + 7 = 11

So, the expression simplifies to:

2x + 11

200

Simplify.

2x + 5x - 3x + 5x - 6x

Answer = 3x

Add the coefficients of x:

2 + 5 - 3 + 5 - 6 = 3

So, the expression simplifies to:

3x

Therefor, the answer is:

3x

300

Solve. 

4m + 2 = 18

Answer m=4

Subtract 2 from both sides:

4m + 2 - 2 = 18 - 2

4m = 16

Divide both sides by 4:

4m / 4 = 16 / 4

m = 4 

So, the solution is: 

m = 4

400

Solve.

6x + 12 = 4x - 8

Answer = - 10

Subtract 4x from both sides to get all the x-terms on one side:

6x - 4x + 12 = -8

2x + 12 = -8

Subtract 12 from both sides to have the x-terms by itselve:

2x + 12 - 12 = -8 - 12

2x + 12 = -8

Divide both sides by 2 to solve for x:

2x/2 = -20/2

x=-10 

The solution is:

x = -10

500

Expand and simplify. 

2(x + 6) + 4(3x - 2) - 2(3x - 1)

Answer = 8x + 6

Distribute the coefficients:

2(x + 6) = 2x + 12

4(3x - 2) = 12x - 8

-2(3x - 1) = 16x + 2

Combine the distributed terms:

2x + 12 + 12x - 8 - 6x + 2

Combine the terms:

(2x + 12x - 6x) + (12 - 8 + 2)

8x + 6

The simplified expression is:

8x + 6

Linear Legends

100

What is the rate and initial value for C = 25n + 100

Rate = 25

Initial Value = 100

200

What is the rate and initial value for the graph shown below. 

Rate = 10

Initial Value = 20

300

Find the slope of the line segment shown on the graph below. 

Answer - 5/5 = 1

The slope of a line is calculated using the formula:

slope = rise/run

Given that the rise is 5 and the run is 5:

slope = 5/5 = 1

Therefor, the slope of the line segment would be 1.

400

Determine the equation. Passing through the points (2,3) and (4, -5).

Answer - Y = -4x + 11

First, calculate the slope (m) using the formula: 

m = y² - y¹ / x² - x¹

m = -5 - 3 / 4 - 2

m = -8 / 2

m = -4

Now that you have the slope (m = −4), you can use one of the points to find the y-intercept (b). Let's use the point (2, 3):

3 = (−4)(2) + b

3 = −8 + b

b = 3 + 8

b = 11  

So, the equation of the line passing through the points (2, 3) and (4, -5) is: 

Y = -4x + 11 

500

Determine the equation of the line that has the same slope as the line y = 2x - 4 and the same x-intercept as 4x + 3y + 12 = 0.

Answer - y = 2x + 6

The first line: y = 2x - 4, has a slope of 2. 

The second line: 4x+3y+12 = 0, crosses the x-axis at -3. 

Therefore, the equation of the line with the same slope and x-intercept is y=2x + 6

Geometric Geniuses

100

if 1 inch is equivalent to 2.6 cm, calculate the following conversion: 52 cm to inches. 

Answer = 52cm is equivalent to 20 inches

200

Determine the hypotenuse of the right triangle shown below.

 Answer - x = 9.22

a² + b² = c²

7² + 6² = c²

49 + 36 = c²

85 = c²

9.22 = c

9.22 = x

300

Determine the unknown side length in the triangle below. 

Answer - b = 12

9² + b² = 15²

81 + b² = 225

b² = 225 - 81

b² = 144

b = 12

400

Calculate the perimeter and area of the shape shown below. Round to two decimal places.

Area = 2231.52 in²

Perimeter = 189.98 in

500

Determine the value of y if x is 126.6 degrees. 

Answer - y = 63.3 degrees 

(central angle = double of inscribed)

Data Detectives

100

What is the mean of the following set of numbers: 10, 15, 20, 25?

Answer = 17.5

To find the mean of a set of numbers, you add up all the numbers and then divide the sum by the total count of numbers.

For the set of numbers 10, 15, 20, and 25:

Mean = 10 + 15 + 20 + 25 / 4

Mean = 70 / 4

Mean = 17.5

200

Define the term "median" in statistics and provide an example of how to calculate it.

Answer: The median is the middle value in a data set when arranged in order. If the number of values is odd, the median is the middle one. If even, it's the average of the two middle values.

Example (odd number of values):

Data set: 3, 5, 7, 9, 11
Median: 7

Example (even number of values):

Data set: 4, 8, 15, 16, 23, 42
Median: (15 + 16) / 2 = 15.5

300

Explain the difference between a box-and-whisker plot and a scatterplot, providing examples of when each would be most appropriate to use.

Box-and-Whisker Plot:

- Description: Shows distribution with median, quartiles, and extremes.

- Use: Summarize a single variable and compare groups.

- Example: Compare test scores of different classes.

Scatterplot:

- Description: Shows relationship between two variables with points on a graph.

- Use: Analyze correlation between two variables.

- Example: Examine relationship between hours studied and test scores.

400

Use the following key words to describe the scatter plot below:

linear, non-linear, relationship, no relationship, positive, negative, strong, weak

Answer: This graph is negative and strong, with a relationship and is linear!

500

You have completed five chapters of a book and received ratings of 4 stars, 5 stars, 3 stars, 4 stars, and 2 stars. What rating do you need to give the sixth chapter to maintain an average rating of 4 stars for the entire book?

Answer = 5 Stars

Calculate current average:

4 + 5 + 3 + 4 + 2= 18

Average Rating = 18/5 = 3.6 stars

Determine Target Total:

Total sum for 6 chapters 

= average rating X number of chapters

= 4 x 6

= 24 stars 

Find Required Rating: 

Stars needed for sixth chapter

= Total sum for 6 chapters - stars earned from first five chapters

= 24 − 18 = 6 stars

Maximum Rating: 

Since the maximum rating is 5 stars, you would need to give the sixth chapter a rating of 5 stars to maintain an average of 4 stars!