Hussain15
The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k ?
A. 1
B. 13
C. 14
D. 18
E. 21
A student ask me whether we're obligated to use the "multiplier strategy" (e.g., ratio = 3x/4x) for this question.
The answer is "No," and what follows is a solution that uses three variables.
Let A = the smaller of the two numbers
Let B = the bigger number
The ratio of two positive numbers is 3 to 4.We can write: A/B = 3/4
Cross multiply: 4A = 3B
Rewrite as:
4A - 3B = 0If k is added to each number the new ratio will be 4 to 5...We can write: (A + k)/(B + k) = 4/5
Cross multiply: 5(A + k) = 4(B + k)
Expand: 5A + 5k = 4B + 4k
Rewrite as:
5A - 4B = -k...the sum of the numbers will be 117We can write: (A + k) + (B + k) = 117
Simplify: A + B + 2k = 117
Rewrite as:
A + B = 117 - 2kWe now have the following system:
4A - 3B = 05A - 4B = -kA + B = 117 - 2kSolve, solve, solve....
A = 39, B = 52 and k = 13
Answer: B