Question

question id: 706417
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
multiply. simplify and write as a mixed number.
\\(\frac{3}{8} \cdot 5\frac{5}{7}\\)
\\(\circ\\) \\(2\frac{1}{7}\\)
\\(\circ\\) 15
\\(\circ\\) \\(5\frac{15}{56}\\)
\\(\circ\\) \\(1\frac{19}{56}\\)

Explanation:

Step1: Convert mixed number to improper fraction

To multiply a fraction by a mixed number, first convert the mixed number \( 5\frac{5}{7} \) to an improper fraction. The formula for converting a mixed number \( a\frac{b}{c} \) to an improper fraction is \( \frac{a\times c + b}{c} \). So for \( 5\frac{5}{7} \), we have \( a = 5 \), \( b = 5 \), \( c = 7 \). Then \( 5\times7 + 5 = 35 + 5 = 40 \), so \( 5\frac{5}{7}=\frac{40}{7} \).

Step2: Multiply the fractions

Now we multiply \( \frac{3}{8} \) by \( \frac{40}{7} \). When multiplying fractions, we multiply the numerators together and the denominators together: \( \frac{3\times40}{8\times7}=\frac{120}{56} \).

Step3: Simplify the fraction

Simplify \( \frac{120}{56} \). First, find the greatest common divisor (GCD) of 120 and 56. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56. The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. The GCD of 120 and 56 is 8. Divide both the numerator and the denominator by 8: \( \frac{120\div8}{56\div8}=\frac{15}{7} \).

Step4: Convert improper fraction to mixed number

Now convert \( \frac{15}{7} \) to a mixed number. Divide 15 by 7: \( 15\div7 = 2 \) with a remainder of 1 (because \( 7\times2 = 14 \) and \( 15 - 14 = 1 \)). So \( \frac{15}{7}=2\frac{1}{7} \). Wait, wait, no, wait. Wait, in step 2, \( \frac{3}{8}\times\frac{40}{7}=\frac{120}{56} \), simplifying \( \frac{120}{56} \): 120 divided by 8 is 15, 56 divided by 8 is 7? Wait, no, 56 divided by 8 is 7? Wait, 8*7=56, yes. 120 divided by 8 is 15. So \( \frac{120}{56}=\frac{15}{7} \), and \( 15\div7 = 2 \) with remainder 1, so \( 2\frac{1}{7} \). Wait, but let's check again. Wait, \( \frac{3}{8}\times\frac{40}{7} \): 40 and 8 can be simplified before multiplying. 40 divided by 8 is 5, so \( \frac{3}{8}\times\frac{40}{7}=\frac{3\times5}{7}=\frac{15}{7}=2\frac{1}{7} \). Oh, I see, I made a mistake in step 3 earlier when simplifying. Let's correct that. When multiplying \( \frac{3}{8} \) and \( \frac{40}{7} \), we can simplify the 40 and 8 first. 40 ÷ 8 = 5, so then we have \( 3\times5 = 15 \) in the numerator and 7 in the denominator, so \( \frac{15}{7} \), which is \( 2\frac{1}{7} \).

Answer:

\( 2\frac{1}{7} \) (corresponding to option \( 2\frac{1}{7} \))