Question
Jessica and her friends have a half-day at school, so they plan to spend the afternoon paddleboarding at Lake Evergreen. They want to spend the rest of the day on the water, but it starts to rain after paddleboarding for only 3 4 of an hour! Jessica is disappointed when she realizes the sun doesn't go down for another 3 1 2 hours. Use an equation to find the amount of time Jessica planned to spend paddleboarding.
Answers
Answered by
GPT-4o mini
To find the total amount of time Jessica planned to spend paddleboarding, we can set up an equation based on the information given.
Let \( x \) represent the total amount of time (in hours) that Jessica planned to spend paddleboarding. According to the problem, they paddleboarded for \( \frac{3}{4} \) of an hour before it started to rain, and after that, there are still \( 3 \frac{1}{2} \) hours left until the sun goes down.
We can express \( 3 \frac{1}{2} \) hours as a fraction:
\[
3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}
\]
Now we can create the equation based on the total planned paddleboarding time \( x \):
\[
x - \frac{3}{4} = \frac{7}{2}
\]
This equation states that the total planned time minus the time already spent paddleboarding gives us the time left until sundown.
Next, we need to solve for \( x \):
1. Add \( \frac{3}{4} \) to both sides:
\[
x = \frac{7}{2} + \frac{3}{4}
\]
2. To add these fractions, we need a common denominator. The least common multiple of 2 and 4 is 4. We can convert \( \frac{7}{2} \) to have a denominator of 4:
\[
\frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4}
\]
3. Now, we can add the two fractions:
\[
x = \frac{14}{4} + \frac{3}{4} = \frac{14 + 3}{4} = \frac{17}{4}
\]
4. Finally, we can convert \( \frac{17}{4} \) back to a mixed number:
\[
\frac{17}{4} = 4 \frac{1}{4}
\]
Thus, the amount of time Jessica originally planned to spend paddleboarding is \( 4 \frac{1}{4} \) hours, or \( 4.25 \) hours.
Let \( x \) represent the total amount of time (in hours) that Jessica planned to spend paddleboarding. According to the problem, they paddleboarded for \( \frac{3}{4} \) of an hour before it started to rain, and after that, there are still \( 3 \frac{1}{2} \) hours left until the sun goes down.
We can express \( 3 \frac{1}{2} \) hours as a fraction:
\[
3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}
\]
Now we can create the equation based on the total planned paddleboarding time \( x \):
\[
x - \frac{3}{4} = \frac{7}{2}
\]
This equation states that the total planned time minus the time already spent paddleboarding gives us the time left until sundown.
Next, we need to solve for \( x \):
1. Add \( \frac{3}{4} \) to both sides:
\[
x = \frac{7}{2} + \frac{3}{4}
\]
2. To add these fractions, we need a common denominator. The least common multiple of 2 and 4 is 4. We can convert \( \frac{7}{2} \) to have a denominator of 4:
\[
\frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4}
\]
3. Now, we can add the two fractions:
\[
x = \frac{14}{4} + \frac{3}{4} = \frac{14 + 3}{4} = \frac{17}{4}
\]
4. Finally, we can convert \( \frac{17}{4} \) back to a mixed number:
\[
\frac{17}{4} = 4 \frac{1}{4}
\]
Thus, the amount of time Jessica originally planned to spend paddleboarding is \( 4 \frac{1}{4} \) hours, or \( 4.25 \) hours.