4. Write each of the rational numbers below as a common fraction in its simplest form (without the use of a calculator): $$ \begin{array}{lllll} ext { a. } \frac{20}{24} & ext { b. }-\frac{48}{80} & ext { c. } 2 \frac{5}{40} & ext { d. }-7 \frac{25}{200} & ext { e. } 5 \frac{6}{8}\end{array} $$ f. $$ \frac{85}{170} $$ 5. Write the following mixed numbers as improper fractions. $$ \begin{array}{llll} ext { a. } 6 \frac{4}{9} & ext { b. }-4 \frac{5}{6} & ext { c. }-11 \frac{2}{3} & ext { d. } 13 \frac{7}{15}\end{array} $$

Question

4. Write each of the rational numbers below as a common fraction in its simplest form (without the use of a calculator): $$ \begin{array}{lllll}	ext { a. } \frac{20}{24} & 	ext { b. }-\frac{48}{80} & 	ext { c. } 2 \frac{5}{40} & 	ext { d. }-7 \frac{25}{200} & 	ext { e. } 5 \frac{6}{8}\end{array} $$ f. $$ \frac{85}{170} $$ 5. Write the following mixed numbers as improper fractions. $$ \begin{array}{llll}	ext { a. } 6 \frac{4}{9} & 	ext { b. }-4 \frac{5}{6} & 	ext { c. }-11 \frac{2}{3} & 	ext { d. } 13 \frac{7}{15}\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's work through each part of questions 4 and 5 step-by-step.

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### **Question 4**
**Write each of the rational numbers below as a common fraction in its simplest form (without using a calculator):**

#### **a. $$ \frac{20}{24} $$**

1. **Find the Greatest Common Divisor (GCD) of 20 and 24:**
   - **Prime factors of 20:** $$2^2 \times 5$$
   - **Prime factors of 24:** $$2^3 \times 3$$
   - **GCD:** $$2^2 = 4$$

2. **Simplify the Fraction:**
   $$
   \frac{20}{24} = \frac{20 \div 4}{24 \div 4} = \frac{5}{6}
   $$

**Answer:** $$ \frac{5}{6} $$

---

#### **b. $$ -\frac{48}{80} $$**

1. **Find the GCD of 48 and 80:**
   - **Prime factors of 48:** $$2^4 \times 3$$
   - **Prime factors of 80:** $$2^4 \times 5$$
   - **GCD:** $$2^4 = 16$$

2. **Simplify the Fraction:**
   $$
   -\frac{48}{80} = -\frac{48 \div 16}{80 \div 16} = -\frac{3}{5}
   $$

**Answer:** $$ -\frac{3}{5} $$

---

#### **c. $$ 2 \frac{5}{40} $$**

1. **Convert Mixed Number to Improper Fraction:**
   $$
   2 \frac{5}{40} = 2 + \frac{5}{40} = \frac{80}{40} + \frac{5}{40} = \frac{85}{40}
   $$

2. **Simplify the Fraction:**
   - **GCD of 85 and 40:** 5
   $$
   \frac{85}{40} = \frac{85 \div 5}{40 \div 5} = \frac{17}{8}
   $$

**Answer:** $$ \frac{17}{8} $$

---

#### **d. $$ -7 \frac{25}{200} $$**

1. **Convert Mixed Number to Improper Fraction:**
   $$
   -7 \frac{25}{200} = -\left(7 + \frac{25}{200}\right) = -\frac{1400}{200} - \frac{25}{200} = -\frac{1425}{200}
   $$

2. **Simplify the Fraction:**
   - **GCD of 1425 and 200:** 25
   $$
   -\frac{1425}{200} = -\frac{1425 \div 25}{200 \div 25} = -\frac{57}{8}
   $$

**Answer:** $$ -\frac{57}{8} $$

---

#### **e. $$ 5 \frac{6}{8} $$**

1. **Convert Mixed Number to Improper Fraction:**
   $$
   5 \frac{6}{8} = 5 + \frac{6}{8} = \frac{40}{8} + \frac{6}{8} = \frac{46}{8}
   $$

2. **Simplify the Fraction:**
   - **GCD of 46 and 8:** 2
   $$
   \frac{46}{8} = \frac{46 \div 2}{8 \div 2} = \frac{23}{4}
   $$

**Answer:** $$ \frac{23}{4} $$

---

#### **f. $$ \frac{85}{170} $$**

1. **Find the GCD of 85 and 170:**
   - **85 is a factor of 170 (since $$85 \times 2 = 170$$)**
   - **GCD:** 85

2. **Simplify the Fraction:**
   $$
   \frac{85}{170} = \frac{85 \div 85}{170 \div 85} = \frac{1}{2}
   $$

**Answer:** $$ \frac{1}{2} $$

---

### **Question 5**
**Write the following mixed numbers as improper fractions:**

#### **a. $$ 6 \frac{4}{9} $$**

1. **Convert to Improper Fraction:**
   $$
   6 \frac{4}{9} = 6 \times 9 + 4 = 54 + 4 = 58
   $$
   $$
   = \frac{58}{9}
   $$

**Answer:** $$ \frac{58}{9} $$

---

#### **b. $$ -4 \frac{5}{6} $$**

1. **Convert to Improper Fraction:**
   $$
   -4 \frac{5}{6} = -\left(4 \times 6 + 5\right) = -\left(24 + 5\right) = -29
   $$
   $$
   = -\frac{29}{6}
   $$

**Answer:** $$ -\frac{29}{6} $$

---

#### **c. $$ -11 \frac{2}{3} $$**

1. **Convert to Improper Fraction:**
   $$
   -11 \frac{2}{3} = -\left(11 \times 3 + 2\right) = -\left(33 + 2\right) = -35
   $$
   $$
   = -\frac{35}{3}
   $$

**Answer:** $$ -\frac{35}{3} $$

---

#### **d. $$ 13 \frac{7}{15} $$**

1. **Convert to Improper Fraction:**
   $$
   13 \frac{7}{15} = 13 \times 15 + 7 = 195 + 7 = 202
   $$
   $$
   = \frac{202}{15}
   $$

**Answer:** $$ \frac{202}{15} $$

---

I hope this helps! Let me know if you have any further questions.
**Simplified Answers:** 1. **Question 4:** - **a.** $$ \frac{5}{6} $$ - **b.** $$ -\frac{3}{5} $$ - **c.** $$ \frac{17}{8} $$ - **d.** $$ -\frac{57}{8} $$ - **e.** $$ \frac{23}{4} $$ - **f.** $$ \frac{1}{2} $$ 2. **Question 5:** - **a.** $$ \frac{58}{9} $$ - **b.** $$ -\frac{29}{6} $$ - **c.** $$ -\frac{35}{3} $$ - **d.** $$ \frac{202}{15} $$

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