Two numbers are in the ratio `17 : 45`. One-third of the smaller is less than `1/5` of the bigger by 15. The smaller number is

Options

A. `25(1/2)`
B. `67(1/2)`
C. `76(1/2)`
D. `86(1/2)`

Answer

To solve the problem step-by-step, we will follow the given conditions and equations based on the ratio and the relationship between the two numbers.

### Step 1: Define the numbers based on the ratio
The two numbers are in the ratio \( 17 : 45 \). We can express these numbers in terms of a variable \( x \):
- Smaller number = \( 17x \)
- Bigger number = \( 45x \)

**Hint:** Use a variable to represent the common factor in the ratio.

### Step 2: Set up the equation based on the given condition
According to the problem, one-third of the smaller number is less than one-fifth of the bigger number by 15. We can write this as:
\[ \frac{1}{3} \times (17x) = \frac{1}{5} \times (45x) - 15 \]

**Hint:** Translate the verbal condition into a mathematical equation.

### Step 3: Simplify the equation
Now, we simplify both sides of the equation:
\[ \frac{17x}{3} = \frac{45x}{5} - 15 \]
Calculating \( \frac{45x}{5} \):
\[ \frac{45x}{5} = 9x \]
So, the equation becomes:
\[ \frac{17x}{3} = 9x - 15 \]

**Hint:** Simplify fractions and isolate terms involving \( x \).

### Step 4: Eliminate the fraction
To eliminate the fraction, multiply the entire equation by 3:
\[ 17x = 27x - 45 \]

**Hint:** Multiplying by a common denominator can help eliminate fractions.

### Step 5: Rearrange the equation
Now, rearranging the equation to isolate \( x \):
\[ 17x - 27x = -45 \]
\[ -10x = -45 \]
Dividing both sides by -10:
\[ x = \frac{45}{10} = \frac{9}{2} \]

**Hint:** Keep track of signs when rearranging equations.

### Step 6: Find the smaller number
Now that we have \( x \), we can find the smaller number:
\[ \text{Smaller number} = 17x = 17 \times \frac{9}{2} = \frac{153}{2} = 76.5 \]

**Hint:** Substitute back the value of \( x \) into the expression for the smaller number.

### Conclusion
The smaller number is \( 76.5 \).