Question

You are considering learning to play the piano and check with a friend about practice time. For 3 consecutive days before a recital, she practiced $1 \frac{1}{4}\( hours, \)2 \frac{1}{2}\( hours, and \)3 \frac{2}{3}$ hours. What was her total practice time for the 3 days?

Short answer

Answer: The total practice time for the 3 days is \(7\frac{5}{12}\) hours.

Step-by-step solution

Convert mixed numbers to improper fractions

To begin, we need to convert each mixed number to an improper fraction. This can be done by multiplying the whole number part by the denominator, adding the numerator, and then placing this result over the original denominator. For \(1 \frac{1}{4}\): \((1\times4) + 1 = 5\), so the improper fraction is \(\frac{5}{4}\). For \(2 \frac{1}{2}\): \((2\times2) + 1 = 5\), so the improper fraction is \(\frac{5}{2}\). For \(3 \frac{2}{3}\): \((3\times3) + 2 = 11\), so the improper fraction is \(\frac{11}{3}\).

Find a common denominator

Now we need to find a common denominator so that we can add the fractions. In this case, the lowest common denominator is the least common multiple of 4, 2, and 3, which is 12.

Rewrite the fractions with the common denominator

Now we can rewrite each fraction using the common denominator 12: \(\frac{5}{4} = \frac{15}{12}\) \(\frac{5}{2} = \frac{30}{12}\) \(\frac{11}{3} = \frac{44}{12}\)

Add the fractions

Now that we have the fractions with the common denominator, we can add them: \(\frac{15}{12} + \frac{30}{12} + \frac{44}{12} = \frac{15+30+44}{12} = \frac{89}{12}\)

Convert the improper fraction to a mixed number

The final answer should be given as a mixed number. To do this, divide the numerator by the denominator: \(89\div12 = 7\) with a remainder of \(5\). So the mixed number is \(7\frac{5}{12}\). The total practice time for the 3 days was \(7\frac{5}{12}\) hours.