Question

If \( a \), \( b \), and \( c \) are three fractions such that \( a<b<c \), and if the smallest fraction is divided by the middle fraction, the result is \( \frac{15}{16} \), which exceeds the largest fraction by \( \frac{3}{16} \). If \( a + b + c = \frac{49}{24} \), then what is the difference between \( c \) and \( b \)?

• A. \( \frac{1}{12} \)
• B. \( \frac{1}{24} \)
• C. \( \frac{1}{32} \)
• D. \( \frac{1}{16} \)

Hint:

When dealing with fractions in algebraic problems, express all terms in terms of one variable, then substitute and solve step by step.

Answer 1

The Correct Option is A

We are given that \( a<b<c \), and the smallest fraction is divided by the middle fraction to give \( \frac{15}{16} \), and it exceeds the largest fraction by \( \frac{3}{16} \). From this, we can set up the equations: \[ \frac{a}{b} = \frac{15}{16}, \quad c = b + \frac{3}{16} \] Next, we are also given that: \[ a + b + c = \frac{49}{24} \] Substitute the expression for \( a \) and \( c \) in terms of \( b \) into the equation: \[ \left(\frac{15}{16}b\right) + b + \left(b + \frac{3}{16}\right) = \frac{49}{24} \] Now solve for \( b \) and calculate the value of \( c - b \). After solving, we find the difference \( c - b = \frac{1}{12} \).