Question

Kaya ran 1\(\frac{2}{5}\) miles on Monday and 2\(\frac{1}{3}\) miles on Tuesday. What was the total distance, in miles, Kaya ran during those 2 days? A. 3\(\frac{2}{15}\) B. 3\(\frac{3}{8}\) C. 3\(\frac{2}{5}\) D. 3\(\frac{7}{15}\) E. 3\(\frac{11}{15}\)

Short answer

Answer: The total distance Kaya ran during those 2 days is 3\(\frac{11}{15}\).

Step-by-step solution

Convert mixed numbers to improper fractions

The distances on each day are in mixed number form: 1\(\frac{2}{5}\) (Monday) and 2\(\frac{1}{3}\) (Tuesday). First, we need to convert each mixed number to an improper fraction. For 1\(\frac{2}{5}\): (1 × 5) + 2 = 7, so the improper fraction is \(\frac{7}{5}\). For 2\(\frac{1}{3}\): (2 × 3) + 1 = 7, so the improper fraction is \(\frac{7}{3}\).

Add the improper fractions

Now, we will add the improper fractions \(\frac{7}{5}\) and \(\frac{7}{3}\). We have to find a common denominator for both fractions, which is the least common multiple of 5 and 3. The LCM of 5 and 3 is 15. So, we need to convert the fractions to equivalent fractions with a denominator of 15: For \(\frac{7}{5}\): \(\frac{7}{5} \cdot \frac{3}{3} = \frac{21}{15}\) For \(\frac{7}{3}\): \(\frac{7}{3} \cdot \frac{5}{5} = \frac{35}{15}\) Now, we can add the fractions: \(\frac{21}{15} + \frac{35}{15} = \frac{21 + 35}{15} = \frac{56}{15}\)

Convert the improper fraction to a mixed number

We will now convert the improper fraction \(\frac{56}{15}\) back to a mixed number. To do this, we need to divide 56 by 15: 56 ÷ 15 = 3 with a remainder of 11, so the mixed number form will be 3\(\frac{11}{15}\). The total distance Kaya ran during those 2 days is 3\(\frac{11}{15}\). Therefore, the correct option is E.

Key concepts

Mixed Numbers

A mixed number is a whole number combined with a fraction, representing a sum of the two. For example, in the distance Kaya ran on Monday, 1\(\frac{2}{5}\) miles, the '1' represents the whole mile, and \(\frac{2}{5}\) is the fractional part of the next mile.

To work with mixed numbers in addition problems, especially when adding two mixed numbers with different fractional parts, it's crucial first to convert them to improper fractions. This simplifies the addition process because you're dealing with like terms—just fractions, rather than a combination of whole numbers and fractions.

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is larger or equal to the denominator (the bottom number), resulting in a value that is greater than or equal to one. In the exercise, we took mixed numbers and transformed them into improper fractions. This was an essential step because it made the subsequent addition possible.

An example of converting a mixed number to an improper fraction is turning 1\(\frac{2}{5}\) into \(\frac{7}{5}\). We did this by multiplying the whole number by the denominator and adding the numerator: \((1 \times 5) + 2 = 7\).

Least Common Multiple (LCM)

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. When adding fractions with different denominators, the LCM of these denominators becomes the least common denominator (LCD). It facilitates addition by converting fractions into equivalent fractions that have the same denominator.

In our problem, since the denominators were 5 and 3, we calculated the LCM of these two numbers, which is 15. This allowed us to rewrite both fractions with the same denominator of 15, turning \(\frac{7}{5}\) into \(\frac{21}{15}\) and \(\frac{7}{3}\) into \(\frac{35}{15}\), thus making their addition straightforward.

Equivalent Fractions

Equivalent fractions represent the same portion of a whole, despite having different numerators and denominators. To add fractions with different denominators, we must first convert them to equivalent fractions with a common denominator.

In the case presented, we multiplied the numerator and denominator of each fraction by a number that would give them both the same denominator of 15 while maintaining their original value. Specifically, \(\frac{7}{5}\) becomes \(\frac{21}{15}\) by multiplying by \(\frac{3}{3}\), and \(\frac{7}{3}\) becomes \(\frac{35}{15}\) by multiplying by \(\frac{5}{5}\). This ensures the fractions are equivalent to their original value but now have a common denominator, enabling us to easily add them together.