For each of the following, write three fractions equal to the given fraction. $$ \begin{array}{ll} ext { a. } \frac{1}{3} & ext { b. } \frac{6}{7} \ ext { c. } \frac{-9}{10} & ext { d. } \frac{ ext { a }}{8} \ \frac{1}{3}=\frac{\square}{9} ext { (Simplify your answer.) } \ ext { Enter the correct numerator to match the new denominator. } \ \frac{1}{3}=\frac{\square}{27} ext { (Simplify your answer.) } \ ext { Enter the correct numerator to match the new denominator. } \ \frac{1}{3}=\frac{\square}{18} ext { (Simplify your answer.) }\end{array} $$

Question

For each of the following, write three fractions equal to the given fraction. $$ \begin{array}{ll}	ext { a. } \frac{1}{3} & 	ext { b. } \frac{6}{7} \ 	ext { c. } \frac{-9}{10} & 	ext { d. } \frac{	ext { a }}{8} \ \frac{1}{3}=\frac{\square}{9} 	ext { (Simplify your answer.) } \ 	ext { Enter the correct numerator to match the new denominator. } \ \frac{1}{3}=\frac{\square}{27} 	ext { (Simplify your answer.) } \ 	ext { Enter the correct numerator to match the new denominator. } \ \frac{1}{3}=\frac{\square}{18} 	ext { (Simplify your answer.) }\end{array} $$

UpStudy AI · Accepted Answer

**a. For $$ \frac{1}{3} $$:**  
One way to form equivalent fractions is by multiplying the numerator and denominator by the same nonzero number. For example:  
- Multiply by 2:  
  $$
  \frac{1}{3} = \frac{1\times2}{3\times2} = \frac{2}{6}
  $$
- Multiply by 3:  
  $$
  \frac{1}{3} = \frac{1\times3}{3\times3} = \frac{3}{9}
  $$
- Multiply by 4:  
  $$
  \frac{1}{3} = \frac{1\times4}{3\times4} = \frac{4}{12}
  $$

**b. For $$ \frac{6}{7} $$:**  
Using the same method:  
- Multiply by 2:  
  $$
  \frac{6}{7} = \frac{6\times2}{7\times2} = \frac{12}{14}
  $$
- Multiply by 3:  
  $$
  \frac{6}{7} = \frac{6\times3}{7\times3} = \frac{18}{21}
  $$
- Multiply by 4:  
  $$
  \frac{6}{7} = \frac{6\times4}{7\times4} = \frac{24}{28}
  $$

**c. For $$ \frac{-9}{10} $$:**  
Again, multiply by the same numbers:  
- Multiply by 2:  
  $$
  \frac{-9}{10} = \frac{-9\times2}{10\times2} = \frac{-18}{20}
  $$
- Multiply by 3:  
  $$
  \frac{-9}{10} = \frac{-9\times3}{10\times3} = \frac{-27}{30}
  $$
- Multiply by 4:  
  $$
  \frac{-9}{10} = \frac{-9\times4}{10\times4} = \frac{-36}{40}
  $$

**d. For $$ \frac{a}{8} $$:**  
Multiply numerator and denominator by any nonzero number:  
- Multiply by 2:  
  $$
  \frac{a}{8} = \frac{a\times2}{8\times2} = \frac{2a}{16}
  $$
- Multiply by 3:  
  $$
  \frac{a}{8} = \frac{a\times3}{8\times3} = \frac{3a}{24}
  $$
- Multiply by 4:  
  $$
  \frac{a}{8} = \frac{a\times4}{8\times4} = \frac{4a}{32}
  $$

**Additional Problems:**

1. For  
   $$
   \frac{1}{3} = \frac{\square}{9}
   $$  
   Find the numerator by determining what number multiplies with 3 to give 9. Since $$ 3 \times 3 = 9 $$, multiply numerator 1 by 3:  
   $$
   \frac{1}{3} = \frac{1\times3}{3\times3} = \frac{3}{9}
   $$  
   So, the numerator is $$3$$.

2. For  
   $$
   \frac{1}{3} = \frac{\square}{27}
   $$  
   Since $$ 3 \times 9 = 27 $$, multiply numerator 1 by 9:  
   $$
   \frac{1}{3} = \frac{1\times9}{3\times9} = \frac{9}{27}
   $$  
   Hence, the numerator is $$9$$.

3. For  
   $$
   \frac{1}{3} = \frac{\square}{18}
   $$  
   Here, $$ 3 \times 6 = 18 $$. Multiply numerator 1 by 6:  
   $$
   \frac{1}{3} = \frac{1\times6}{3\times6} = \frac{6}{18}
   $$  
   Thus, the numerator is $$6$$.
**a. $$ \frac{1}{3} = \frac{2}{6} $$, $$ \frac{3}{9} $$, $$ \frac{4}{12} $$** **b. $$ \frac{6}{7} = \frac{12}{14} $$, $$ \frac{18}{21} $$, $$ \frac{24}{28} $$** **c. $$ \frac{-9}{10} = \frac{-18}{20} $$, $$ \frac{-27}{30} $$, $$ \frac{-36}{40} $$** **d. $$ \frac{a}{8} = \frac{2a}{16} $$, $$ \frac{3a}{24} $$, $$ \frac{4a}{32} $$** **Additional Problems:** 1. $$ \frac{1}{3} = \frac{3}{9} $$ Numerator: 3 2. $$ \frac{1}{3} = \frac{9}{27} $$ Numerator: 9 3. $$ \frac{1}{3} = \frac{6}{18} $$ Numerator: 6

For each of the following, write three fractions equal to the given fraction.

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