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.

To express the given fraction equation, we write it as \( 5f = \frac{1}{8} \). To solve for \( f \), we divide both sides by 5, giving us \( f = \frac{1}{8} \div 5 = \frac{1}{8} \times \frac{1}{5} = \frac{1}{40} \). So, the solution tells us that the fraction \( f \) is \( \frac{1}{40} \). Now, regarding the equation involving River, we start with \( 5 \frac{3}{8} + m = 8 \). First, convert \( 5 \frac{3}{8} \) to an improper fraction: \( 5 \frac{3}{8} = \frac{43}{8} \). To isolate \( m \), change the equation to \( m = 8 - \frac{43}{8} \). Converting 8 to eighths gives us \( \frac{64}{8} \), leading to \( m = \frac{64}{8} - \frac{43}{8} = \frac{21}{8} \), or \( 2 \frac{5}{8} \) miles remaining for River to swim. As for Oscar's equation of \( a \div 2 \) representing his sister's age, it doesn't hold consistently. Sibling ages change over time; for instance, if Oscar is currently 12, dividing by 2 gives 6, which is correct now. But next year, if Oscar is 13, then his sister will be 7, and \( 13 \div 2 \approx 6.5 \), which no longer works. Thus, it’s not a reliable method for all future ages.