The sum of two numbers is 184. If one-third of the one exceeds one-seventh of the other by 8, find the smaller number.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Variables
Let the two numbers be \( a \) and \( b \).

### Step 2: Set Up the Equations
According to the problem, we have two pieces of information:
1. The sum of the two numbers is 184:
\[ a + b = 184 \quad \text{(Equation 1)} \]
2. One-third of one number exceeds one-seventh of the other by 8:
\[ \frac{1}{3}a - \frac{1}{7}b = 8 \quad \text{(Equation 2)} \]

### Step 3: Eliminate the Fractions in Equation 2
To eliminate the fractions in Equation 2, we can multiply the entire equation by 21 (the least common multiple of 3 and 7):
\[ 21 \left( \frac{1}{3}a \right) - 21 \left( \frac{1}{7}b \right) = 21 \cdot 8 \]
This simplifies to:
\[ 7a - 3b = 168 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( a + b = 184 \) (Equation 1)
2. \( 7a - 3b = 168 \) (Equation 3)

We can solve these equations simultaneously. First, we can express \( b \) from Equation 1:
\[ b = 184 - a \]

### Step 5: Substitute \( b \) in Equation 3
Substituting \( b \) in Equation 3:
\[ 7a - 3(184 - a) = 168 \]
Expanding this gives:
\[ 7a - 552 + 3a = 168 \]
Combining like terms:
\[ 10a - 552 = 168 \]

### Step 6: Isolate \( a \)
Now, add 552 to both sides:
\[ 10a = 720 \]
Dividing both sides by 10:
\[ a = 72 \]

### Step 7: Find \( b \)
Now that we have \( a \), we can find \( b \) using Equation 1:
\[ b = 184 - a = 184 - 72 = 112 \]

### Step 8: Identify the Smaller Number
The smaller number is:
\[ \text{Smaller number} = a = 72 \]

### Final Answer
The smaller number is **72**.

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