Question

The difference between two numbers is 16. If one-third of the smaller number is greater than one-seventh of the larger number by 4, then what is the larger number?

• A. 33
• B. 46
• C. 49
• D. 54

Hint:

After solving, always plug your results back into the original problem statement to verify that they meet all the given conditions. This helps catch any calculation errors.

Answer 1

The Correct Option is C

Step 1: Problem Analysis:
This problem involves establishing and resolving a system of two linear equations based on the provided criteria.
Step 2: Method and Variables:
Designate the larger number as L and the smaller number as S.
The problem statement yields two equations:
1. \( L - S = 16 \)
2. \( \frac{1}{3}S = \frac{1}{7}L + 4 \)
Step 3: Solution Procedure:
From Equation 1, isolate L:
\[ L = S + 16 \] Substitute this expression for L into Equation 2:
\[ \frac{1}{3}S = \frac{1}{7}(S + 16) + 4 \] To clear the fractions, multiply the entire equation by the least common multiple of 3 and 7, which is 21:
\[ 21 \times \left(\frac{1}{3}S\right) = 21 \times \left(\frac{1}{7}(S + 16)\right) + 21 \times 4 \] \[ 7S = 3(S + 16) + 84 \] \[ 7S = 3S + 48 + 84 \] \[ 7S = 3S + 132 \] Solve for S:
\[ 7S - 3S = 132 \] \[ 4S = 132 \] \[ S = \frac{132}{4} = 33 \] The smaller number is 33.
Calculate the larger number (L) using Equation 1:
\[ L = S + 16 = 33 + 16 = 49 \] Step 4: Conclusion:
The larger number is 49. Verification: The difference is \( 49 - 33 = 16 \). One-third of the smaller is \( \frac{33}{3} = 11 \). One-seventh of the larger is \( \frac{49}{7} = 7 \). The difference is \( 11 - 7 = 4 \). Both conditions are met.