💡 8th Grade Math (Algebra I): Distance, Speed, and Time Practice Questions

This problem uses the fundamental relationship between distance, speed, and time.
The formula is: Distance = Speed × Time
Here's how to solve it:

• Identify the given values:
• Speed = 60 miles per hour
• Time = 3 hours
• Apply the formula:
• Distance = 60 miles/hour × 3 hours
• Calculate the distance:
• Distance = 180 miles

So, the car will travel 180 miles in 3 hours. ✅

We need to find the speed, so we can rearrange the formula Distance = Speed × Time to solve for Speed.
The rearranged formula is: Speed = Distance / Time
Let's break it down:

• Identify the given values:
• Distance = 200 kilometers
• Time = 4 hours
• Apply the rearranged formula:
• Speed = 200 kilometers / 4 hours
• Calculate the speed:
• Speed = 50 kilometers per hour

The train's average speed is 50 kilometers per hour. 👍

We need to find the time it takes to cover a certain distance at a given speed.
We can rearrange the formula Distance = Speed × Time to solve for Time.
The rearranged formula is: Time = Distance / Speed
Here's the step-by-step solution:

• Identify the given values:
• Distance = 12 miles
• Speed = 4 miles per hour
• Apply the rearranged formula:
• Time = 12 miles / 4 miles per hour
• Calculate the time:
• Time = 3 hours

It will take you 3 hours to walk 12 miles. 💯

This problem requires us to calculate the time for each segment of the race separately and then add them together.
We'll use the formula: Time = Distance / Speed
Let's solve it step-by-step:

• Calculate time for the first segment:
• Distance1 = 10 miles
• Speed1 = 20 miles per hour
• Time1 = Distance1 / Speed1 = 10 miles / 20 miles per hour = 0.5 hours
• Calculate time for the second segment:
• Distance2 = 15 miles
• Speed2 = 15 miles per hour
• Time2 = Distance2 / Speed2 = 15 miles / 15 miles per hour = 1 hour
• Calculate the total time:
• Total Time = Time1 + Time2 = 0.5 hours + 1 hour = 1.5 hours

The total time taken for the entire race is 1.5 hours. 🏆

This problem involves a bit of geometry. The paths of the two cars form the two perpendicular sides of a right-angled triangle, and the distance between them is the hypotenuse.
We'll use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the distance between the cars.
First, let's find the distance each car travels in 2 hours:

• Car A's distance:
• SpeedA = 50 km/h
• Time = 2 hours
• DistanceA = SpeedA × Time = 50 km/h × 2 h = 100 km (North)
• Car B's distance:
• SpeedB = 40 km/h
• Time = 2 hours
• DistanceB = SpeedB × Time = 40 km/h × 2 h = 80 km (East)
• Apply the Pythagorean theorem:
• Let \(a\) be DistanceA and \(b\) be DistanceB. We want to find \(c\), the distance between them.
• \(c^2 = (100 \text{ km})^2 + (80 \text{ km})^2\)
• \(c^2 = 10000 \text{ km}^2 + 6400 \text{ km}^2\)
• \(c^2 = 16400 \text{ km}^2\)
• \(c = \sqrt{16400 \text{ km}^2}\)
• \(c \approx 128.06 \text{ km}\)

The distance between the two cars after 2 hours is approximately 128.06 kilometers. 📐

A headwind opposes the direction of travel, so it reduces the effective speed of the aircraft.
The effective speed is the plane's cruising speed minus the headwind speed.
The formula remains Time = Distance / Speed, but we need to calculate the effective speed first.
Here's the breakdown:

• Calculate the effective speed:
• Cruising Speed = 400 mph
• Headwind = 50 mph
• Effective Speed = Cruising Speed - Headwind = 400 mph - 50 mph = 350 mph
• Identify the distance:
• Distance = 1200 miles
• Apply the time formula:
• Time = Distance / Effective Speed = 1200 miles / 350 mph
• Calculate the time:
• Time ≈ 3.43 hours

The flight will take approximately 3.43 hours. 🌍

To find the total distance, we need to calculate the distance covered in each part of the journey and then add them together.
We will use the formula: Distance = Speed × Time
Let's solve it:

• Calculate distance for the first part:
• Speed1 = 50 km/h
• Time1 = 2 hours
• Distance1 = Speed1 × Time1 = 50 km/h × 2 h = 100 km
• Calculate distance for the second part:
• Speed2 = 70 km/h
• Time2 = 1.5 hours
• Distance2 = Speed2 × Time2 = 70 km/h × 1.5 h = 105 km
• Calculate the total distance:
• Total Distance = Distance1 + Distance2 = 100 km + 105 km = 205 km

The total distance covered by the car is 205 kilometers. 🗺️

This problem requires setting up equations based on the given information and solving them simultaneously.
Let \(D\) be the total distance of the journey.
We know that \(D = \text{Speed} \times \text{Time}\).
From the first statement:

• Equation 1:
• \(D = v \times t\)

From the second statement, the speed is \(v + 10\) km/h, and the time is \(t - 1\) hours. The distance \(D\) remains the same.

• Equation 2:
• \(D = (v + 10) \times (t - 1)\)

Now, we equate the two expressions for \(D\):

• \(v \times t = (v + 10) \times (t - 1)\)
• Expand the right side: \(vt = vt - v + 10t - 10\)
• Subtract \(vt\) from both sides: \(0 = -v + 10t - 10\)
• Rearrange to solve for \(v\): \(v = 10t - 10\)

Now substitute this expression for \(v\) back into Equation 1 to find \(D\) in terms of \(t\):

• \(D = v \times t\)
• \(D = (10t - 10) \times t\)
• \(D = 10t^2 - 10t\)

Alternatively, we can express \(D\) in terms of \(v\). From \(v = 10t - 10\), we get \(10t = v + 10\), so \(t = \frac{v + 10}{10}\).
Substitute this into Equation 1:

• \(D = v \times t\)
• \(D = v \times \frac{v + 10}{10}\)
• \(D = \frac{v(v + 10)}{10}\)
• \(D = \frac{v^2 + 10v}{10}\)

The total distance of the journey can be expressed as \(10t^2 - 10t\) kilometers or \(\frac{v^2 + 10v}{10}\) kilometers. 🧐