Question

divide.
$1\frac{3}{5}\div\frac{9}{10} = $

Explanation:

Step1: Convert mixed number to improper fraction

The mixed number \(1\frac{3}{5}\) can be converted 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 \(1\frac{3}{5}\), we have \(a = 1\), \(b=3\), \(c = 5\). Then \(\frac{1\times5+3}{5}=\frac{5 + 3}{5}=\frac{8}{5}\). So the expression becomes \(\frac{8}{5}\div\frac{9}{10}\).

Step2: Divide fractions (multiply by reciprocal)

To divide two fractions \(\frac{a}{b}\div\frac{c}{d}\), we multiply the first fraction by the reciprocal of the second fraction, i.e., \(\frac{a}{b}\times\frac{d}{c}\). So \(\frac{8}{5}\div\frac{9}{10}=\frac{8}{5}\times\frac{10}{9}\).

Step3: Simplify and multiply

Simplify the fractions before multiplying. The 10 in the numerator and 5 in the denominator have a common factor of 5. So \(\frac{8}{5}\times\frac{10}{9}=\frac{8\times2}{1\times9}\) (since \(10\div5 = 2\) and \(5\div5=1\)). Then multiply the numerators and denominators: \(8\times2 = 16\) and \(1\times9=9\)? Wait, no, wait: \(\frac{8}{5}\times\frac{10}{9}=\frac{8\times10}{5\times9}=\frac{80}{45}\). Then simplify \(\frac{80}{45}\) by dividing numerator and denominator by 5: \(\frac{80\div5}{45\div5}=\frac{16}{9}\)? Wait, no, wait, 10 divided by 5 is 2, so 8 times 2 is 16, 5 times 1 is 5? No, I made a mistake. Let's do it again. \(\frac{8}{5}\times\frac{10}{9}\): 10 and 5, 10 is 25, so cancel 5: \(\frac{8}{1}\times\frac{2}{9}=\frac{16}{9}\)? Wait, no, 810 = 80, 59=45, 80/45 simplifies to 16/9? Wait, 80 divided by 5 is 16, 45 divided by 5 is 9. Yes. Wait, but let's check again. \(1\frac{3}{5}\) is \(1 + \frac{3}{5}=\frac{5 + 3}{5}=\frac{8}{5}\). Then dividing by \(\frac{9}{10}\) is multiplying by \(\frac{10}{9}\). So \(\frac{8}{5}\times\frac{10}{9}=\frac{8\times10}{5\times9}=\frac{80}{45}=\frac{16}{9}\approx1.777...\)? Wait, no, that can't be right. Wait, maybe I messed up the reciprocal. Wait, no, the reciprocal of \(\frac{9}{10}\) is \(\frac{10}{9}\), that's correct. Wait, let's do the calculation again: \(\frac{8}{5}\times\frac{10}{9}\). 10 divided by 5 is 2, so we have 82 = 16, and 19 = 9? Wait, no, 5 is in the denominator of the first fraction, 10 is in the numerator of the second. So \(\frac{8}{5}\times\frac{10}{9}=\frac{8\times(10)}{5\times9}=\frac{8\times2}{1\times9}\) (because 10/5 = 2). So 82=16, 1*9=9. So \(\frac{16}{9}\) is \(\1\frac{7}{9}\)? Wait, no, wait, maybe I made a mistake. Wait, let's use decimals to check. \(1\frac{3}{5}=1.6\), \(\frac{9}{10}=0.9\). Then 1.6 / 0.9 = 16/9 ≈1.777..., which is \(1\frac{7}{9}\). Wait, but let's do the fraction division again. \(1\frac{3}{5}=\frac{8}{5}\), \(\frac{8}{5}\div\frac{9}{10}=\frac{8}{5}\times\frac{10}{9}=\frac{8\times10}{5\times9}=\frac{80}{45}=\frac{16}{9}=1\frac{7}{9}\). Wait, but maybe I made a mistake in the simplification. Wait, 80 divided by 5 is 16, 45 divided by 5 is 9. Yes. So the result is \(\frac{16}{9}\) or \(1\frac{7}{9}\).

Answer:

\(\frac{16}{9}\) (or \(1\frac{7}{9}\))