Difference between two numbers is 30. Twice the greater number is less than 7 times the smaller number by 5. Find the numbers.

Answer

Let's solve the problem step by step.

### Step 1: Define the Variables
Let the two numbers be \( X \) (the greater number) and \( Y \) (the smaller number).

### Step 2: Set Up the Equations
From the problem, we have two pieces of information:
1. The difference between the two numbers is 30:
\[ X - Y = 30 \quad \text{(Equation 1)} \]
2. Twice the greater number is less than 7 times the smaller number by 5:
\[ 2X + 5 = 7Y \quad \text{(Equation 2)} \]

### Step 3: Rearrange Equation 1
From Equation 1, we can express \( X \) in terms of \( Y \):
\[ X = Y + 30 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 3 into Equation 2
Now, substitute \( X \) from Equation 3 into Equation 2:
\[ 2(Y + 30) + 5 = 7Y \]
Expanding this gives:
\[ 2Y + 60 + 5 = 7Y \]
Simplifying further:
\[ 2Y + 65 = 7Y \]

### Step 5: Solve for \( Y \)
Rearranging the equation to isolate \( Y \):
\[ 65 = 7Y - 2Y \]
\[ 65 = 5Y \]
Dividing both sides by 5:
\[ Y = 13 \]

### Step 6: Find \( X \)
Now that we have \( Y \), we can find \( X \) using Equation 3:
\[ X = Y + 30 = 13 + 30 = 43 \]

### Step 7: Conclusion
The two numbers are:
\[ X = 43 \quad \text{(greater number)} \quad \text{and} \quad Y = 13 \quad \text{(smaller number)} \]

### Final Answer
The two required numbers are **43 and 13**.

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