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
\

$$\begin{tabular}{|l|c|c|c|} \\hline x (hours) & 2 & 5 & 8 \\\\ \\hline y (dollars) & 70 & 175 & 280 \\\\ \\hline \\end{tabular}$$

drive time car hire
\

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

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}{6}\\) cm = \\(\frac{1}{2}\\) meter.

Explanation:

⚡ Using what you learned: lines and angles (parallel, transversal, vertical) · 🆕 New: Unit Rates and Percent Change

Step 1: Solve Problem 1 (Marbles)

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.

Non-red marbles:

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

Therefore, the number of red marbles must be:

$$ 8 $$

Step 2: Solve Problem 2 (Value of x)

The two angles shown are vertical angles, which means they are equal to each other.

Set up the equation:

$$ 6x + 15 = 24 $$

Subtract 15 from both sides:

$$ 6x = 9 $$

Divide by 6:

$$ x = \frac{9}{6} = 1.5 $$

Step 3: Solve Problem 3 (Biking Rate)

Find the unit rate in miles per hour by dividing the total distance by the total time.

Convert the mixed number to a decimal:

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

Calculate the speed:

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

In 1 hour, he will travel:

$$ 56.5 \text{ miles} $$

Step 4: Solve Problem 4 (Rental Rates Difference)

Find the hourly rate (unit rate) for each company.

Velocity Rentals:

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

Drive Time Car Hire:

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

Find the difference:

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

Step 5: Solve Problem 5 (Percent Increase)

Find the amount of increase:

$$ \$144 - \$120 = \$24 $$

Divide the increase by the original price to find the percentage:

$$ \frac{24}{120} = 0.20 = 20\% $$

Step 6: Solve Problem 6 (Scale Model)

Find the height of the scale model using the given scale:

$$ \frac{1}{6}\text{ cm} = \frac{1}{2}\text{ meter} $$

Multiply both sides by 2 to find the scale equivalent for 1 meter:

$$ 1\text{ meter} = 2 \times \frac{1}{6}\text{ cm} = \frac{1}{3}\text{ cm} $$

Multiply by the actual height of the tree (12 meters):

$$ 12 \times \frac{1}{3}\text{ cm} = 4\text{ cm} $$

Answer:

• Problem 1: 8 red marbles
• Problem 2: \( x = 1.5 \)
• Problem 3: 56.5 miles
• Problem 4: \$10 per hour
• Problem 5: 20%
• Problem 6: 4 cm