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Problem 1207

# Separate 120 into two parts such that the larger exceeds three times the smaller by 12 .

Expert verified
The two numbers that fulfill the conditions are 93 (larger number) and 27 (smaller number).
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## Step 1: Declare the variables

From the problem, we understand that the total is separated into two parts, one larger and one smaller. Let's declare the larger part as x and the smaller part as y.

## Step 2: Formulate the Equations

From the problem statement, we can formulate two equations: 1. Sum of the two numbers: $$x + y = 120$$ 2. The condition given: "the larger exceeds three times the smaller by 12" can be translated to math as: $$x = 3y + 12$$

## Step 3: Substitute the value of x from the second equation into the first one

Let's substitute the second equation into the first one to solve for y. We will replace x from the first equation with the value from the second equation: Substitute x from equation (2) into equation (1): $$3y + 12 + y = 120$$ This simplifies to: $$4y + 12 = 120$$.

## Step 4: Solve for y

We can solve the equation $$4y + 12 = 120$$ for y as follows: Subtract 12 from both sides of the equation: $$4y = 120 - 12$$ $$4y = 108$$ Then, divide both sides by 4 to isolate y: $$y = 108/4$$ $$y = 27$$

## Step 5: Solve for x

We already established the formula for x: $$x = 3y + 12$$. Substituting y = 27 back into x’s equation, we get: $$x = 3*27 + 12$$ $$x = 81 + 12$$ $$x = 93$$

## Step 6: Statement of the final answer

After solving the equations, we find that the larger number is 93 and the smaller number is 27. Therefore, the two numbers that fulfill the conditions laid out in the problem are 93 and 27.

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