Two numbers are in the ratio \(5:6\). If 8 is subtracted from each, the ratio becomes \(4:5\). Find the numbers.
\(40\) and \(48\).
Step 1: Suppose the two numbers are in the form of \(5k\) and \(6k\).
Step 2: According to the question, if we subtract 8 from each number, the new ratio becomes \(4:5\).
This means:
\(\dfrac{5k - 8}{6k - 8} = \dfrac{4}{5}\)
Step 3: Now we cross multiply:
\(5 \times (5k - 8) = 4 \times (6k - 8)\)
Step 4: Multiply both sides:
\(25k - 40 = 24k - 32\)
Step 5: Bring like terms together:
\(25k - 24k = -32 + 40\)
Step 6: Simplify:
\(k = 8\)
Step 7: Put the value of \(k\) back into the numbers:
First number = \(5k = 5 \times 8 = 40\)
Second number = \(6k = 6 \times 8 = 48\)
Final Answer: The two numbers are \(40\) and \(48\).