Solving for the Ratio of Two Numbers

We are given a problem involving two numbers, let's call them m and n. We know their sum and their difference, and we need to find their ratio (m : n).

Information Given:

• The sum of the two numbers is 84. This can be written as the equation:

  m + n = 84

• The difference between the two numbers is 6. Since the problem implies m is the larger number (as m > n), we write this as:

  m - n = 6

• We are also told that m > n.
• The goal is to find the ratio m : n.

Step-by-Step Calculation

We can solve this problem using a system of linear equations.

Step 1: Set up the Equations

We have the two equations:

1. m + n = 84
2. m - n = 6

Step 2: Solve for 'm'

To find the value of m, we can add the two equations together. Notice that the '+ n' and '- n' terms will cancel each other out:

(m + n) + (m - n) = 84 + 6

m + n + m - n = 90

2m = 90

Now, divide by 2 to find m:

m = \frac{90}{2}

m = 45

Step 3: Solve for 'n'

Now that we know m = 45, we can substitute this value back into either of the original equations to find n. Let's use the first equation (m + n = 84):

45 + n = 84

To find n, subtract 45 from both sides:

n = 84 - 45

n = 39

We can check our work using the second equation: m - n = 45 - 39 = 6. This matches the given information, and 45 > 39, so our values for m and n are correct.

Step 4: Calculate the Ratio

The question asks for the ratio of the two numbers, which is m : n.

m : n = 45 : 39

Step 5: Simplify the Ratio

To express the ratio in its simplest form, we need to find the greatest common divisor (GCD) of 45 and 39 and divide both numbers by it.

The divisors of 45 are 1, 3, 5, 9, 15, 45.

The divisors of 39 are 1, 3, 13, 39.

The greatest common divisor (GCD) of 45 and 39 is 3.

Now, divide both parts of the ratio by 3:

\frac{45}{3} = 15

\frac{39}{3} = 13

So, the simplified ratio m : n is 15 : 13.