Two numbers are in the ratio 1: 3. If their sum is 240, then their difference is

Options

A. 120
B. 108
C. 100
D. 96

Answer

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

### Step 1: Define the Variables
Let the first number be \( a \) and the second number be \( b \). According to the problem, the ratio of these two numbers is given as \( 1:3 \).

**Hint:** When dealing with ratios, you can express the numbers in terms of a common variable.

### Step 2: Express the Numbers in Terms of a Variable
From the ratio \( 1:3 \), we can express the two numbers as:
- \( a = k \)
- \( b = 3k \)

**Hint:** Use a variable (like \( k \)) to represent the common factor in the ratio.

### Step 3: Set Up the Equation for Their Sum
According to the problem, the sum of the two numbers is 240. Therefore, we can write the equation:
\[ a + b = 240 \]
Substituting the expressions for \( a \) and \( b \):
\[ k + 3k = 240 \]

**Hint:** Always substitute the expressions you have derived into the equation to simplify it.

### Step 4: Simplify the Equation
Combine like terms:
\[ 4k = 240 \]

**Hint:** Look for ways to simplify equations by combining similar terms.

### Step 5: Solve for \( k \)
Now, divide both sides of the equation by 4 to find \( k \):
\[ k = \frac{240}{4} = 60 \]

**Hint:** When isolating a variable, perform the same operation on both sides of the equation.

### Step 6: Find the Values of \( a \) and \( b \)
Now that we have \( k \), we can find the values of \( a \) and \( b \):
- \( a = k = 60 \)
- \( b = 3k = 3 \times 60 = 180 \)

**Hint:** Substitute back the value of \( k \) to find the actual numbers.

### Step 7: Calculate the Difference
Now, we need to find the difference between the two numbers:
\[ b - a = 180 - 60 = 120 \]

**Hint:** The difference can be found by subtracting the smaller number from the larger number.

### Final Answer
The difference between the two numbers is \( 120 \).