Question

1: in a bag of marbles, there are 3 green marbles, 5 white marbles, and some red marbles. how many red marbles must be in the bag for the likelihood of drawing a red to be equal to the likelihood of drawing a marble that is not red?

2: find the value of x.

3: john bikes 84.75 miles in 1\frac{1}{2} hours on thursday. if he plans to bike at the same rate on friday, how many miles will he travel in 1 hour?

4: the tables shows the cost to rent a car from two different companies. what is the difference in their hourly rates?
velocity rentals
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$$\begin{array}{|c|c|c|c|} \\hline x (hours) & 2 & 5 & 8 \\\\ \\hline y (dollars) & 70 & 175 & 280 \\\\ \\hline \\end{array}$$

drive time car hire
\

$$\begin{array}{|c|c|c|c|} \\hline x (hours) & 3 & 6 & 9 \\\\ \\hline y (dollars) & 135 & 270 & 405 \\\\ \\hline \\end{array}$$

5: the price of a jacket was increased from \\$120 to \\$144. by what percentage was the price of the jacket increased?

6: cristobal is drawing a scale model of a giant redwood tree.

• the actual height of the tree is 12 meters.
• the scale of the model is \frac{1}{4}\text{ cm} = \frac{1}{2}\text{ meter}.

what is the height, in centimeters, of the model tree?

7: determine the value of: -2 + 0.75(3^2) + (0.25)^2 \times 4^2

8: a gift box has dimensions of 8 inches in width and 4 inches in length. if the volume of the gift box is 192 cubic inches, what is the height, in inches, of the gift box?

9: in a sports center, there are 30 people. \frac{1}{3} of them are swimming, \frac{2}{5} are playing basketball, and the rest are doing yoga. what fraction of the people are doing yoga?

10: the record low temperature in denver, co is -14^{\circ}\text{f}, and the record high is 105^{\circ}\text{f}. what is the difference between the record high and the record low temperatures?

11: a bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. two marbles are drawn from the bag one after the other without replacement. what is the probability of drawing a red marble first and then a blue marble? write your answer as a fraction in simplest form.

12: solve for x: -3x + 12 = -15

Explanation:

Here are the step-by-step solutions for all 12 problems shown in the image.

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🆕 New Concept Discovered: Scale Drawings and Unit Rates
Converting scales and finding rates of change

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Problem 1: Marble Likelihood

Step 1: Understand the condition for equal likelihood

For the likelihood of drawing a red marble to be equal to the likelihood of drawing a marble that is not red, the number of red marbles must equal the total number of non-red marbles.

Step 2: Calculate the number of non-red marbles

The non-red marbles are the green and white marbles:

$$ \text{Non-red marbles} = 3 \text{ (green)} + 5 \text{ (white)} = 8 $$

Step 3: Set the number of red marbles

$$ \text{Red marbles} = \text{Non-red marbles} = 8 $$

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Problem 2: Find the value of \(x\)

Step 1: Identify the angle relationship

The two angles shown in the diagram are vertical angles formed by intersecting lines. Vertical angles are equal in measure:

$$ 2x = 40^\circ $$

Step 2: Solve for \(x\)

$$ x = \frac{40}{2} = 20 $$

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Problem 3: John's Biking Rate

Step 1: Convert the mixed number to an improper fraction

John bikes \(84.75\) miles in \(1\frac{1}{2}\) hours.

$$ 1\frac{1}{2} = 1.5 \text{ hours} $$

Step 2: Calculate the unit rate (miles per hour)

$$ \text{Rate} = \frac{84.75 \text{ miles}}{1.5 \text{ hours}} = 56.5 \text{ miles per hour} $$

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Problem 4: Difference in Hourly Rates

Step 1: Find the hourly rate for Velocity Rentals

Using the table for Velocity Rentals:

$$ \text{Rate} = \frac{\$70}{2 \text{ hours}} = \$35 \text{ per hour} $$

Step 2: Find the hourly rate for Drive Time Car Hire

Using the table for Drive Time Car Hire:

$$ \text{Rate} = \frac{\$135}{3 \text{ hours}} = \$45 \text{ per hour} $$

Step 3: Calculate the difference

$$ \text{Difference} = \$45 - \$35 = \$10 \text{ per hour} $$

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Problem 5: Percentage Increase of Jacket Price

Step 1: Find the amount of increase

$$ \text{Increase} = \$144 - \$120 = \$24 $$

Step 2: Calculate the percentage increase relative to the original price

$$ \text{Percentage Increase} = \frac{24}{120} \times 100\% = 0.2 \times 100\% = 20\% $$

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Problem 6: Scale Model of a Redwood Tree

Step 1: Identify the scale ratio

The scale is given as:

$$ \frac{1}{5} \text{ cm} = \frac{1}{2} \text{ meter} $$

Multiply both sides by 2 to find how many centimeters represent 1 meter:

$$ 1 \text{ meter} = \frac{2}{5} \text{ cm} $$

Step 2: Calculate the model height for a 12-meter tree

$$ \text{Model height} = 12 \times \frac{2}{5} \text{ cm} = \frac{24}{5} \text{ cm} = 4.8 \text{ cm} $$

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Problem 7: Order of Operations Evaluation

Step 1: Write down the expression

$$ -2 + 0.75(3^2) + (0.25)^2 \times 4^2 $$

Step 2: Evaluate exponents

$$ 3^2 = 9 $$

$$ (0.25)^2 = 0.0625 $$

$$ 4^2 = 16 $$

Substitute these back into the expression:

$$ -2 + 0.75(9) + 0.0625 \times 16 $$

Step 3: Perform multiplications

$$ 0.75 \times 9 = 6.75 $$

$$ 0.0625 \times 16 = 1 $$

Substitute these values:

$$ -2 + 6.75 + 1 $$

Step 4: Perform addition and subtraction

$$ -2 + 6.75 + 1 = 4.75 + 1 = 5.75 $$

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Problem 8: Height of the Gift Box

Step 1: Use the volume formula for a rectangular prism

$$ \text{Volume} = \text{width} \times \text{length} \times \text{height} $$

$$ 192 = 8 \times 4 \times \text{height} $$

Step 2: Solve for height

$$ 192 = 32 \times \text{height} $$

$$ \text{height} = \frac{192}{32} = 6 \text{ inches} $$

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Problem 9: Fraction of People Doing Yoga

Step 1: Find the sum of the fractions of people swimming and playing basketball

\[ \text{Swimming} = \frac{1}{3}, \quad \text{Basketb…

Answer:

• Problem 1: \(8\) red marbles
• Problem 2: \(x = 20\)
• Problem 3: \(56.5\) miles
• Problem 4: \(\$10\) per hour
• Problem 5: \(20\%\)
• Problem 6: \(4.8\text{ cm}\) (or \(\frac{24}{5}\text{ cm}\))
• Problem 7: \(5.75\) (or \(\frac{23}{4}\))
• Problem 8: \(6\) inches
• Problem 9: \(\frac{4}{15}\)
• Problem 10: \(119^\circ\text{F}\)
• Problem 11: \(\frac{1}{6}\)
• Problem 12: \(x = 9\)