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The questions given below contain two st...

The questions given below contain two statement numbered I and II giving certain data. You have to decide whether the data given in the statements are sufficient to answer the question. Give answer.
What is the value of `a/b` ?
I. The value of a-b is 3. The value of `a^2-b^2` is 45.
II. The value of a+b is 15.

A

statement I alone is sufficient but statement II alone is not sufficient.

B

statement II alone is sufficient but statement I alone is not sufficient.

C

either statement I or II is sufficient

D

both statements I and II together are not sufficient.

Text Solution

AI Generated Solution

The correct Answer is:
To find the value of \( \frac{a}{b} \), we will analyze the information given in the two statements. ### Step 1: Analyze Statement I Statement I provides: 1. \( a - b = 3 \) 2. \( a^2 - b^2 = 45 \) We can use the identity \( a^2 - b^2 = (a - b)(a + b) \). Substituting the first equation into this identity, we get: \[ a^2 - b^2 = (3)(a + b) = 45 \] ### Step 2: Solve for \( a + b \) From the equation derived from Statement I: \[ 3(a + b) = 45 \] Dividing both sides by 3: \[ a + b = 15 \] ### Step 3: Set Up the System of Equations Now we have two equations: 1. \( a - b = 3 \) (from Statement I) 2. \( a + b = 15 \) (derived from Statement I) ### Step 4: Solve the System of Equations We can add these two equations: \[ (a - b) + (a + b) = 3 + 15 \] This simplifies to: \[ 2a = 18 \implies a = 9 \] Now substitute \( a = 9 \) back into one of the equations to find \( b \): \[ 9 + b = 15 \implies b = 6 \] ### Step 5: Calculate \( \frac{a}{b} \) Now that we have \( a = 9 \) and \( b = 6 \): \[ \frac{a}{b} = \frac{9}{6} = \frac{3}{2} \] ### Conclusion Thus, the value of \( \frac{a}{b} \) is \( \frac{3}{2} \).
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