Difference of two number is 1660 . If `6(1)/(2)%` of one number is equal to `8(1)/(2)%` of the other number, the smaller number is :

Answer

To solve the problem step by step, let's follow the given information and equations.

### Step 1: Define the Variables
Let the first number be \( a \) and the second number be \( b \).

### Step 2: Set Up the Equations
According to the problem, we know:
1. The difference between the two numbers is given by:
\[ a - b = 1660 \]

2. We are also given that \( 6\frac{1}{2}\% \) of \( a \) is equal to \( 8\frac{1}{2}\% \) of \( b \). We can convert these percentages into fractions:
\[ 6\frac{1}{2}\% = \frac{13}{2}\% = \frac{13}{200} \]
\[ 8\frac{1}{2}\% = \frac{17}{2}\% = \frac{17}{200} \]
Therefore, we can write the equation:
\[ \frac{13}{200} a = \frac{17}{200} b \]

### Step 3: Simplify the Equation
We can eliminate the fraction by multiplying both sides by 200:
\[ 13a = 17b \]

### Step 4: Express \( a \) in terms of \( b \)
From the equation \( 13a = 17b \), we can express \( a \) in terms of \( b \):
\[ a = \frac{17}{13} b \]

### Step 5: Substitute into the Difference Equation
Now we substitute \( a \) into the difference equation \( a - b = 1660 \):
\[ \frac{17}{13} b - b = 1660 \]

### Step 6: Combine the Terms
To combine the terms, we need a common denominator:
\[ \frac{17b - 13b}{13} = 1660 \]
\[ \frac{4b}{13} = 1660 \]

### Step 7: Solve for \( b \)
Multiply both sides by 13:
\[ 4b = 1660 \times 13 \]
\[ 4b = 21580 \]
Now divide by 4:
\[ b = \frac{21580}{4} = 5395 \]

### Step 8: Find the Smaller Number
Since \( b \) is the second number, we need to find the smaller number. We already have \( b = 5395 \). Now we can find \( a \):
\[ a = b + 1660 = 5395 + 1660 = 7055 \]

Thus, the smaller number is:
\[ \text{Smaller number } b = 5395 \]

### Final Answer
The smaller number is **5395**.

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