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Sample Question 4:
Step 1: Let's denote the larger number as \(a\) and the smaller number as \(b\).
Step 2: According to the problem, the sum of these two numbers is 5 times their difference. Mathematically, this can be represented as \(a + b = 5(a - b)\).
Step 3: Expanding the right side of the equation gives \(a + b = 5a - 5b\).
Step 4: By rearranging the terms, we obtain \(4a = 6b\).
Step 5: Factoring the equation results in \(2a = 3b\).
Step 6: Solving for the ratio of \(a\) to \(b\) gives \(\frac{a}{b} = \frac{3}{2}\).
So, the ratio of the larger number to the smaller number is \(\frac{3}{2}\), which corresponds to answer choice \(\textbf{(B)}\ \frac{3}{2}\).
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?
\(\textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52\)
Answer Keys
Question 4: BSolutions
Question 4Step 1: Let's denote the larger number as \(a\) and the smaller number as \(b\).
Step 2: According to the problem, the sum of these two numbers is 5 times their difference. Mathematically, this can be represented as \(a + b = 5(a - b)\).
Step 3: Expanding the right side of the equation gives \(a + b = 5a - 5b\).
Step 4: By rearranging the terms, we obtain \(4a = 6b\).
Step 5: Factoring the equation results in \(2a = 3b\).
Step 6: Solving for the ratio of \(a\) to \(b\) gives \(\frac{a}{b} = \frac{3}{2}\).
So, the ratio of the larger number to the smaller number is \(\frac{3}{2}\), which corresponds to answer choice \(\textbf{(B)}\ \frac{3}{2}\).