Question

If \( a \) and \( b \) are positive integers such that \( a \times b = 36 \), and \( a \) is 3 less than twice \( b \), what is the value of \( a + b \)?

• A. \( 11 \)
• B. \( 13 \)
• C. \( 15 \)
• D. \( 17 \)
• E. \( 19 \)

Hint:

When solving problems with two variables, express one variable in terms of the other and substitute it into the given equation to solve.

Answer 1

The Correct Option is C

Step 1: We are given that \( a \times b = 36 \), and \( a = 2b - 3 \). Substitute \( a = 2b - 3 \) into the equation \( a \times b = 36 \): \[ (2b - 3) \times b = 36. \] Step 2: Expand the equation: \[ 2b^2 - 3b = 36. \] Step 3: Rearrange the equation: \[ 2b^2 - 3b - 36 = 0. \] Now solve for \( b \) using the quadratic formula: \[ b = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-36)}}{2(2)} = \frac{3 \pm \sqrt{9 + 288}}{4} = \frac{3 \pm \sqrt{297}}{4}. \] \[ \sqrt{297} \approx 17.2,
b \approx \frac{3 + 17.2}{4} = \frac{20.2}{4} \approx 5.05. \] Since \( b \) is a positive integer, the value of \( b \) is 5. Step 4: Substitute \( b = 5 \) into \( a = 2b - 3 \): \[ a = 2(5) - 3 = 10 - 3 = 7. \] Thus, \( a + b = 7 + 5 = 15 \).