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 \( a + b \)?
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 \( a + b \)?
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When solving problems with two variables, express one variable in terms of the other and substitute it into the given equation to solve.
Updated On: Oct 6, 2025
- \( 11 \)
- \( 13 \)
- \( 15 \)
- \( 17 \)
- \( 19 \)
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The Correct Option is C
Solution and Explanation
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, \( a + b = 7 + 5 = 15 \).
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, \( a + b = 7 + 5 = 15 \).
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