If the numerator of a fraction is increased by 3 , the fraction becomes \( 3 / 4 \). If the denominator is decreased by 7 . the fraction becomes 1 . Determine the original fraction. Which of the following equations represents "If the numerator of a fraction is increased by 3 , the fraction becomes \( 3 / 4 \) "? (Hint cross products) \( 4 n+3=3 d \) \( 4 n+12=3 d \) \( 3 n+9=4 d \)

To solve the problem, let's denote the original fraction as \( \frac{n}{d} \), where \( n \) is the numerator and \( d \) is the denominator. When we increase the numerator by 3, we have: \[ \frac{n + 3}{d} = \frac{3}{4} \] Cross multiplying gives us: \[ 4(n + 3) = 3d \] Which simplifies to: \[ 4n + 12 = 3d \] Now let’s check the second condition: if the denominator is decreased by 7, it becomes 1. So: \[ \frac{n}{d - 7} = 1 \] This implies: \[ n = d - 7 \] Now we have two equations: 1. \( 4n + 12 = 3d \) 2. \( n = d - 7 \) Substituting equation 2 into equation 1 allows us to solve for \( n \) and \( d \). In essence, the best representation of the equation is \( 4n + 12 = 3d \) as it directly relates to the first condition of the numerator being increased by 3. For a fun fact, understanding fractions dates back thousands of years! The Egyptians were marvelous in the use of fractions, employing them extensively in trade and construction around 3000 BC. They innovatively represented them using unit fractions (like 1/2, 1/3) making mathematics truly fascinating! When tackling equations, a common mistake is neglecting to simplify before solving. Always check your equations—getting \( d \) isolated on one side can help avoid errors! For fractions, balancing changes in numerators and denominators can really pay off when you remember to keep your eyes peeled for simplifications along the way.