Math: Calculate Speed In A Word Problem

Hey guys, welcome back to Plastik Magazine! Today, we're diving into a classic math challenge that's all about figuring out speed. You know, that thing that tells us how fast something is moving. We've got a question that asks us to complete a table for an object that travels a certain distance in a specific amount of time. It's a super common type of problem, and once you get the hang of it, you'll be a speed demon yourself! Let's break it down.

Understanding the Core Concept: Speed, Distance, and Time

The fundamental relationship we're dealing with here is between speed, distance, and time. You've probably heard the formula: Speed = Distance / Time. This is the golden rule, the key to unlocking this problem and many others like it. Think about it: if you know how far you've gone and how long it took you, you can easily calculate how fast you were moving. Conversely, if you know your speed and how long you traveled, you can figure out the distance. And if you know your speed and the distance, you can calculate the time.

In our specific problem, we're given a starting point: an object travels rac{3}{4} mile in 6 minutes. This is our baseline information. The table then asks us to fill in the blanks for other distances and times, assuming the object maintains the same speed. This is a crucial assumption, guys. We're not dealing with a car that's speeding up or slowing down; we're talking about a constant velocity. So, the first step, the most important step, is to calculate this constant speed using the initial information provided. Once we have that speed, we can use it to solve for the missing values in the table. Remember, math problems often build on each other, and this one is no exception. Mastering that initial calculation is the gateway to solving the rest.

Step 1: Calculate the Object's Speed

Alright, let's get down to business. The problem states the object goes rac{3}{4} mile in 6 minutes. Our speed formula is Speed = Distance / Time. However, we have a slight issue: the distance is in miles, but the time is in minutes. To get a consistent speed (usually expressed in miles per hour), we need to convert the time to hours. Remember, there are 60 minutes in 1 hour. So, 6 minutes is rac{6}{60} of an hour, which simplifies to rac{1}{10} of an hour. This conversion is absolutely vital! If you forget this, all your calculations will be off, and you'll end up with a speed in miles per minute, which isn't what we typically use in these kinds of problems.

Now, let's plug our values into the formula:

Speed = Distance / Time Speed = rac{3}{4} mile / rac{1}{10} hour

Dividing by a fraction is the same as multiplying by its reciprocal. So, we flip rac{1}{10} to get rac{10}{1}.

Speed = rac{3}{4} * rac{10}{1} miles per hour Speed = rac{30}{4} miles per hour

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

Speed = rac{15}{2} miles per hour

Or, as a decimal, that's 7.5 miles per hour. So, our object is cruising along at a steady 7.5 mph. Keep this number handy, guys, because it's our magic number for the rest of the problem. This calculation is the cornerstone of solving the entire table.

Filling in the Table: Applying the Calculated Speed

Now that we've got our object's speed (7.5 mph or rac{15}{2} mph), we can tackle the rest of the table. Remember, the table has columns for Distance (mi) and Time (h). We're given some values and need to find the missing ones.

Let's look at the table structure:

We already know the first column: rac{3}{4} mile takes rac{1}{10} hour, which gave us our speed of rac{15}{2} mph.

Second Column: Finding the Missing Time

The second column asks us to find the time it takes to travel a distance of 1 rac{1}{2} miles. We know our speed is rac{15}{2} mph. We need to rearrange our speed formula to solve for time: Time = Distance / Speed.

First, let's convert the mixed number 1 rac{1}{2} to an improper fraction. That's rac{2*1 + 1}{2} = rac{3}{2} miles.

Now, plug in the values:

Time = rac{3}{2} miles / rac{15}{2} mph

Again, dividing by a fraction means multiplying by its reciprocal:

Time = rac{3}{2} * rac{2}{15} hours

Time = rac{3 * 2}{2 * 15} hours

Time = rac{6}{30} hours

Simplify the fraction by dividing both numerator and denominator by 6:

Time = rac{1}{5} hours

So, it takes rac{1}{5} of an hour to travel 1 rac{1}{2} miles at a speed of 7.5 mph. Pretty neat, right? You can also think of rac{1}{5} of an hour as 12 minutes (since rac{1}{5} * 60 = 12). If you want to double-check, does going 1 rac{1}{2} miles in 12 minutes match our speed? Distance = 1.5 miles, Time = 12 minutes = 0.2 hours. Speed = 1.5 / 0.2 = 7.5 mph. Yep, it checks out!

Third Column: Finding the Missing Time

Finally, let's look at the third column. Here, we need to find the time it takes to travel a distance of 2 rac{1}{4} miles. Our speed remains constant at rac{15}{2} mph. We use the same formula: Time = Distance / Speed.

First, convert the mixed number 2 rac{1}{4} to an improper fraction. That's rac{4*2 + 1}{4} = rac{9}{4} miles.

Now, plug in the values:

Time = rac{9}{4} miles / rac{15}{2} mph

Multiply by the reciprocal of the speed:

Time = rac{9}{4} * rac{2}{15} hours

Time = rac{9 * 2}{4 * 15} hours

Time = rac{18}{60} hours

We can simplify this fraction. Both 18 and 60 are divisible by 6:

Time = rac{3}{10} hours

And there you have it! It takes rac{3}{10} of an hour to travel 2 rac{1}{4} miles at 7.5 mph. To put that in minutes, rac{3}{10} * 60 = 18 minutes. So, the object travels 2 rac{1}{4} miles in 18 minutes. Another check! Does 2.25 miles in 18 minutes give us 7.5 mph? Distance = 2.25 miles, Time = 18 minutes = 0.3 hours. Speed = 2.25 / 0.3 = 7.5 mph. Perfect!

The Completed Table

So, putting it all together, here's the completed table:

And if you prefer those times in minutes:

See, guys? It's all about breaking down the problem into small, manageable steps. First, understand the relationship between speed, distance, and time. Second, use the given information to calculate the constant speed. Third, use that speed to find the missing values. Always remember to keep your units consistent – that's where a lot of mistakes happen! Practice these kinds of problems, and you'll be acing your math tests in no time. Keep those calculators handy and your thinking caps on! Catch you in the next article!