Solve this problem. Reduce to lowest terms.
Josh spent 6 1/2 hours at school, 3 2/3 hours at work, 2 3/4 hours doing homework, and 1 1/3 hours watching television. How much time did Josh spend doing these activities?
To find the total time Josh spent doing these activities, we need to add the time spent at school, work, doing homework, and watching television.
First, let's change all the mixed numbers to improper fractions.
6 1/2 = 13/2
3 2/3 = 11/3
2 3/4 = 11/4
1 1/3 = 4/3
Now, let's add these fractions together:
13/2 + 11/3 + 11/4 + 4/3
To add fractions with different denominators, we need to find a common denominator. In this case, the least common multiple (LCM) of 2, 3, and 4 is 12.
Converting all the fractions into twelfths:
(13/2) * (6/6) = 78/12
(11/3) * (4/4) = 44/12
(11/4) * (3/3) = 33/12
(4/3) * (4/4) = 16/12
Now, we can add the fractions together:
78/12 + 44/12 + 33/12 + 16/12
Adding the numerators:
78 + 44 + 33 + 16 = 171
The denominator remains the same:
171/12
Reducing this fraction to its lowest terms:
The greatest common divisor (GCD) of 171 and 12 is 3.
Dividing the numerator and denominator by 3:
171/3 รท 12/3 = 57/4
Therefore, Josh spent a total of 57/4 hours doing these activities.
To find the total time Josh spent doing these activities, we need to add the hours together.
Let's convert each mixed number to an improper fraction in order to simplify the addition:
6 1/2 hours = 6 + 1/2 = 6 + 2/4 = 6 + (2/4) = 6 + (1/2) = 6 2/4 = 6 1/2
3 2/3 hours = 3 + 2/3 = 3 + (2/3) = 3 + (2/3) = 3 2/3
2 3/4 hours = 2 + 3/4 = 2 + (3/4) = 2 + (3/4) = 2 3/4
1 1/3 hours = 1 + 1/3 = 1 + (1/3) = 1 + (1/3) = 1 1/3
Now, let's add the fractions together:
6 1/2 + 3 2/3 + 2 3/4 + 1 1/3
To combine the whole numbers, we have 6 + 3 + 2 + 1 = 12
Now let's add the fractions:
1/2 + 2/3 + 3/4 + 1/3
To add the fractions, we need a common denominator.
The least common multiple (LCM) of 2, 3, and 4 is 12.
Now, we convert each fraction to have a denominator of 12:
1/2 = (1/2) * (6/6) = 6/12
2/3 = (2/3) * (4/4) = 8/12
3/4 = (3/4) * (3/3) = 9/12
1/3 = (1/3) * (4/4) = 4/12
Now we can add the fractions:
6/12 + 8/12 + 9/12 + 4/12 = (6 + 8 + 9 + 4)/12 = 27/12
Next, we simplify the fraction to lowest terms:
27/12 = (9/3) / (4/3) = 9/4
So, Josh spent a total of 12 hours and 9/4 hours on these activities.
To find the total time Josh spent doing these activities, we need to add up the times spent at school, work, doing homework, and watching television.
Given:
Josh spent 6 1/2 hours at school,
Josh spent 3 2/3 hours at work,
Josh spent 2 3/4 hours doing homework,
Josh spent 1 1/3 hours watching television.
To add these times, we need to convert all the mixed fractions to improper fractions. Here's how we do that:
For 6 1/2 hours:
To convert the mixed fraction 6 1/2 to an improper fraction, we multiply the whole number (6) by the denominator of the fraction (2), and then add the numerator (1). This gives us (6 x 2) + 1 = 13.
So, 6 1/2 = 13/2 hours.
For 3 2/3 hours:
To convert the mixed fraction 3 2/3 to an improper fraction, we multiply the whole number (3) by the denominator of the fraction (3), and then add the numerator (2). This gives us (3 x 3) + 2 = 11.
So, 3 2/3 = 11/3 hours.
For 2 3/4 hours:
To convert the mixed fraction 2 3/4 to an improper fraction, we multiply the whole number (2) by the denominator of the fraction (4), and then add the numerator (3). This gives us (2 x 4) + 3 = 11.
So, 2 3/4 = 11/4 hours.
For 1 1/3 hours:
To convert the mixed fraction 1 1/3 to an improper fraction, we multiply the whole number (1) by the denominator of the fraction (3), and then add the numerator (1). This gives us (1 x 3) + 1 = 4.
So, 1 1/3 = 4/3 hours.
Now, let's add up the times in the improper fraction form:
13/2 + 11/3 + 11/4 + 4/3
To add these fractions, we need to find a common denominator. In this case, the least common multiple (LCM) of 2, 3, and 4 is 12.
So, let's convert the fractions to have a common denominator of 12:
13/2 = (13/2) * (6/6) = 78/12
11/3 = (11/3) * (4/4) = 44/12
11/4 = (11/4) * (3/3) = 33/12
4/3 = (4/3) * (4/4) = 16/12
Now that we have all the fractions with a common denominator, we can add them together:
78/12 + 44/12 + 33/12 + 16/12 = (78 + 44 + 33 + 16)/12 = 171/12
Finally, let's reduce the resulting fraction to its lowest terms:
The greatest common divisor (GCD) of 171 and 12 is 3.
Dividing both the numerator and denominator by 3, we get:
(171/3) / (12/3) = 57/4
Therefore, Josh spent a total of 57/4 hours doing these activities.