Problem 49
Question
Yoshi Teramoto sells hardware tools. After driving 180 kilometers \((\mathrm{km})\), he has completed \(\frac{5}{8}\) of a sales trip. How long is the total trip? How many kilometers are left to drive?
Step-by-Step Solution
Verified Answer
The total trip is 288 km and 108 km remain.
1Step 1 - Understand the Given Fractions of the Trip
The problem states that Yoshi has completed \(\frac{5}{8}\) of his trip by driving 180 kilometers. We need to find the full length of the trip.
2Step 2 - Set up the Equation
To find the total trip length, set up an equation: \[\frac{5}{8} \times \text{Total Trip Length} = 180 \text{ km}\].
3Step 3 - Solve for the Total Trip Length
Isolate the total trip length by dividing both sides by \(\frac{5}{8}\flashmath{8kenj}\):\ \[ \text{Total Trip Length} = \frac{180}{\frac{5}{8}} = 180 \times \frac{8}{5} = 288 \text{ km}\flashmath{0}\]
4Step 4 - Calculate the Remainig Distance
Now find the remaining distance to be traveled. Subtract the distance already traveled (180 km) from the total trip length (288 km):\ \[ \text{Remaining Distance} = 288 \text{ km} - 180 \text{ km} = 108 \text{ km}\flashmath{0}\]
Key Concepts
fraction of a distancesolving equationsremaining distance calculation
fraction of a distance
Understanding fractions in real-world problems is crucial. Here, Yoshi has completed \(\frac{5}{8}\) of his trip by driving 180 kilometers.
Fractions help us represent parts of a whole. In our case, the whole trip.
To determine the full length of Yoshi's trip, we need to work out what the whole is, based on the fraction and the part he already completed.
When we know \(\frac{5}{8}\) equals 180 km, we can find the total by thinking of it as a puzzle where if we know one part, we can find the whole picture.
It's like knowing you have eaten \(\frac{5}{8}\) of a pizza and had 5 slices; you can figure out how many slices the whole pizza has.
Fractions help us represent parts of a whole. In our case, the whole trip.
To determine the full length of Yoshi's trip, we need to work out what the whole is, based on the fraction and the part he already completed.
When we know \(\frac{5}{8}\) equals 180 km, we can find the total by thinking of it as a puzzle where if we know one part, we can find the whole picture.
It's like knowing you have eaten \(\frac{5}{8}\) of a pizza and had 5 slices; you can figure out how many slices the whole pizza has.
solving equations
To find out the total distance, we need to set up an equation.
Given that \(\frac{5}{8} \times \text{Total Trip Length} = 180 \text{ km}\), we now solve for the total trip length.
First, isolate the unknown total trip length by dividing both sides of the equation by the fraction:
\[ \text{Total Trip Length} = \frac{180}{\frac{5}{8}} \]
Simplifying this, we multiply 180 by the reciprocal of \(\frac{5}{8}\), which is \(\frac{8}{5}\):
\[ \frac{180}{\frac{5}{8}} = 180 \times \frac{8}{5} = 288 \text{ km} \]
This shows us that the total trip length is 288 km. By following these steps, you can solve any similar equation by isolating the variable and performing necessary arithmetic operations.
Given that \(\frac{5}{8} \times \text{Total Trip Length} = 180 \text{ km}\), we now solve for the total trip length.
First, isolate the unknown total trip length by dividing both sides of the equation by the fraction:
\[ \text{Total Trip Length} = \frac{180}{\frac{5}{8}} \]
Simplifying this, we multiply 180 by the reciprocal of \(\frac{5}{8}\), which is \(\frac{8}{5}\):
\[ \frac{180}{\frac{5}{8}} = 180 \times \frac{8}{5} = 288 \text{ km} \]
This shows us that the total trip length is 288 km. By following these steps, you can solve any similar equation by isolating the variable and performing necessary arithmetic operations.
remaining distance calculation
After finding the total distance, calculating the remaining distance Yoshi needs to drive is straightforward.
We know Yoshi has already driven 180 km and the total distance is 288 km.
The remaining distance is found by subtracting what he has already driven from the total trip length.
\[ \text{Remaining Distance} = 288 \text{ km} - 180 \text{ km} = 108 \text{ km} \]
So, Yoshi still has 108 km left to drive.
This step is important for understanding how fractions relate to real-world distances and how you can determine how much more there is to go based on the journey already completed.
This method can be used in various scenarios, such as road trips, project progress, or any situation where you're tracking parts of a whole journey.
We know Yoshi has already driven 180 km and the total distance is 288 km.
The remaining distance is found by subtracting what he has already driven from the total trip length.
\[ \text{Remaining Distance} = 288 \text{ km} - 180 \text{ km} = 108 \text{ km} \]
So, Yoshi still has 108 km left to drive.
This step is important for understanding how fractions relate to real-world distances and how you can determine how much more there is to go based on the journey already completed.
This method can be used in various scenarios, such as road trips, project progress, or any situation where you're tracking parts of a whole journey.
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