Question

learning goal 2.7: students will multiply positive fractions. find each product. rewrite each product in simplest form. show your work. 4. \\(\frac{3}{16} \times \frac{4}{5}\\) 5. \\(\frac{7}{11} \times 3\frac{2}{3}\\) 6. \\(4\frac{2}{5} \times 4\frac{4}{9}\\) 7. \\(\frac{37}{100} \times 3\frac{1}{2}\\)

Explanation:

Problem 4: $\boldsymbol{\frac{3}{16} \times \frac{4}{5}}$

Step1: Multiply numerators and denominators

To multiply fractions, we use the rule $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. So, for $\frac{3}{16} \times \frac{4}{5}$, we multiply the numerators ($3 \times 4$) and the denominators ($16 \times 5$).
$$\frac{3 \times 4}{16 \times 5} = \frac{12}{80}$$

Step2: Simplify the fraction

We simplify $\frac{12}{80}$ by finding the greatest common divisor (GCD) of 12 and 80, which is 4. Then we divide both the numerator and the denominator by 4.
$$\frac{12 \div 4}{80 \div 4} = \frac{3}{20}$$

Step1: Convert mixed number to improper fraction

First, we convert the mixed number $3\frac{2}{3}$ 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 $3\frac{2}{3}$, we have:
$$3\frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$$

Step2: Multiply the fractions

Now we multiply $\frac{7}{11}$ by $\frac{11}{3}$ using the rule $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.
$$\frac{7}{11} \times \frac{11}{3} = \frac{7 \times 11}{11 \times 3}$$

Step3: Simplify the fraction

We can cancel out the common factor of 11 in the numerator and the denominator.
$$\frac{7 \times \cancel{11}}{\cancel{11} \times 3} = \frac{7}{3}$$
(If we want a mixed number, $\frac{7}{3} = 2\frac{1}{3}$)

Step1: Convert mixed numbers to improper fractions

First, convert $4\frac{2}{5}$ to an improper fraction:
$$4\frac{2}{5} = \frac{4 \times 5 + 2}{5} = \frac{20 + 2}{5} = \frac{22}{5}$$
Then, convert $4\frac{4}{9}$ to an improper fraction:
$$4\frac{4}{9} = \frac{4 \times 9 + 4}{9} = \frac{36 + 4}{9} = \frac{40}{9}$$

Step2: Multiply the improper fractions

Now multiply $\frac{22}{5}$ by $\frac{40}{9}$ using the rule $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.
$$\frac{22}{5} \times \frac{40}{9} = \frac{22 \times 40}{5 \times 9}$$

Step3: Simplify before multiplying (optional but easier)

We can simplify the fraction before multiplying. Notice that 40 and 5 have a common factor of 5. So we divide 40 by 5 and 5 by 5:
$$\frac{22 \times (40 \div 5)}{(5 \div 5) \times 9} = \frac{22 \times 8}{1 \times 9} = \frac{176}{9}$$
(If we want a mixed number, $\frac{176}{9} = 19\frac{5}{9}$)

Answer:

$\boldsymbol{\frac{3}{20}}$

Problem 5: $\boldsymbol{\frac{7}{11} \times 3\frac{2}{3}}$