Use Patterns and Structure A fraction, $$ f $$. multiplied by 5 equals $$ \frac{1}{8} $$. Write an algebraic sentence to show the equation. Then solve the equation and explain how you solved it. Ian any equation that is written using addition e written as an equivalent equation using hbtraction? Explain your reasoning and give an rample containing decimals that shows your asoning. 35. River needs to swim a total of 8 miles this week So far, River has swum $$ 5 \frac{3}{3} $$ miles. Use the equati $$ 5 \frac{3}{8}+m=8 $$ to find the number of miles River $$ s $$ needs to swim. 37. Communicate and Justify Oscar is 12 yea and his little sister is 6 . Oscar uses a to repr his age. He says that he can use the expres $$ a \div 2 $$ to always know his sister's age. Do y agree? Explain.

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UpStudy AI · Accepted Answer

Sure, let's address each of your questions step by step.

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### **1. Writing and Solving the Equation**

**Problem:**  
A fraction, $$ f $$, multiplied by 5 equals $$ \frac{1}{8} $$. Write an algebraic sentence to show the equation. Then solve the equation and explain how you solved it.

**Algebraic Sentence:**  
$$ 5f = \frac{1}{8} $$

**Solving the Equation:**  
To find the value of $$ f $$, divide both sides of the equation by 5:

$$
f = \frac{1}{8} \div 5 = \frac{1}{8} \times \frac{1}{5} = \frac{1}{40}
$$

**Explanation:**  
- **Step 1:** Start with the equation $$ 5f = \frac{1}{8} $$.
- **Step 2:** To isolate $$ f $$, divide both sides by 5.
- **Step 3:** Dividing by 5 is the same as multiplying by $$ \frac{1}{5} $$.
- **Step 4:** Multiply $$ \frac{1}{8} $$ by $$ \frac{1}{5} $$ to get $$ \frac{1}{40} $$.

So, the fraction $$ f $$ is $$ \frac{1}{40} $$.

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### **2. Converting Addition Equations to Subtraction Equations**

**Problem:**  
Explain why any equation written using addition can be written as an equivalent equation using subtraction. Provide reasoning and give an example containing decimals.

**Explanation:**  
Yes, any equation that uses addition can be rewritten using subtraction by performing the inverse operation to isolate the variable.

**Reasoning:**  
In an equation, whatever operation you perform on one side must also be performed on the other side to maintain equality. Addition and subtraction are inverse operations, meaning one can always convert between them to solve for a variable.

**Example with Decimals:**

**Original Addition Equation:**  
$$ x + 3.5 = 7 $$

**Equivalent Subtraction Equation:**  
$$ x = 7 - 3.5 $$

**Solving:**  
$$ x = 3.5 $$

**Explanation:**  
To isolate $$ x $$, subtract 3.5 from both sides of the original equation. This converts the addition into a subtraction, allowing you to solve for $$ x $$.

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### **3. Calculating Remaining Miles for River**

**Problem:**  
River needs to swim a total of 8 miles this week. So far, River has swum $$ 5 \frac{3}{8} $$ miles. Use the equation $$ 5 \frac{3}{8} + m = 8 $$ to find the number of miles River needs to swim.

**Given Equation:**  
$$ 5 \frac{3}{8} + m = 8 $$

**Converting Mixed Number to Decimal:**
$$ 5 \frac{3}{8} = 5 + 0.375 = 5.375 $$

**Solving for $$ m $$:**
$$
m = 8 - 5.375 = 2.625
$$

**Converting Decimal Back to Fraction:**  
$$ 2.625 = 2 \frac{5}{8} $$

**Answer:**  
River needs to swim **2 and 5/8 miles** more to reach a total of 8 miles for the week.

**Explanation:**  
Subtract the miles already swum (5.375 miles) from the total required miles (8 miles) to find the remaining distance River needs to swim.

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### **4. Evaluating Oscar's Age Representation**

**Problem:**  
Oscar is 12 years old and his little sister is 6. Oscar uses a variable $$ a $$ to represent his age. He says that he can use the expression $$ a \div 2 $$ to always know his sister's age. Do you agree? Explain.

**Answer:**  
No, I do not agree.

**Explanation:**  
- **Current Ages:**  
  - Oscar ($$ a $$) = 12 years  
  - Sister = 6 years  
  - $$ a \div 2 = 12 \div 2 = 6 $$ (This accurately represents his sister's age now.)

- **In Future Years:**  
  - Next year, Oscar will be 13, sibling will be 7.  
  - $$ a \div 2 = 13 \div 2 = 6.5 $$ (Does not equal 7.)

As both Oscar and his sister age by the same number of years each year, the ratio of their ages changes over time. Therefore, the expression $$ a \div 2 $$ only accurately represents his sister's age at the specific moment when Oscar is 12 and his sister is 6. It does not hold true **always** as they grow older.

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Feel free to ask if you have any more questions or need further clarification!
1. **Fraction Equation:** - **Equation:** $$ 5f = \frac{1}{8} $$ - **Solution:** $$ f = \frac{1}{40} $$ 2. **Converting Addition to Subtraction:** - **Reasoning:** Addition and subtraction are inverse operations, so any addition equation can be rewritten using subtraction to solve for a variable. - **Example:** - **Addition:** $$ x + 3.5 = 7 $$ - **Subtraction:** $$ x = 7 - 3.5 = 3.5 $$ 3. **River's Remaining Miles:** - **Total Miles:** 8 - **Miles Swum:** $$ 5 \frac{3}{8} = 5.375 $$ miles - **Miles Needed:** $$ 8 - 5.375 = 2.625 $$ miles ($$ 2 \frac{5}{8} $$ miles) 4. **Oscar's Age Representation:** - **Agreement:** No, because $$ a \div 2 $$ only accurately represents his sister's age when Oscar is 12. As they age, the ratio changes, making the expression inaccurate over time. **Summary:** - **Fraction:** $$ f = \frac{1}{40} $$ - **Addition to Subtraction:** $$ x = 7 - 3.5 = 3.5 $$ - **River's Miles:** $$ 2 \frac{5}{8} $$ miles - **Oscar's Age:** Disagree with the expression $$ a \div 2 $$ for representing his sister's age always.

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