Two brothers were talking about how the older they get, the less significant the difference in their ages seems to...

Phase 1: Owning the Dataset

Timeline Visualization

This timeline helps us see that we're tracking two people's ages at three different time points.

Phase 2: Understanding the Question

We need to find:

1. Difference in ages: The age gap between the brothers (this stays constant over time)
2. Sum of ages now: Their combined ages at the present time

Setting Up Equations

Let's denote:

• Younger brother's current age = x
• Older brother's current age = y

From the given ratios:

• 20 years ago: \((\mathrm{x}-20):(\mathrm{y}-20) = 1:2\)
• Now: \(\mathrm{x}:\mathrm{y} = 5:6\)
• 20 years from now: \((\mathrm{x}+20):(\mathrm{y}+20) = 9:10\)

Phase 3: Finding the Answer

From the past ratio (20 years ago):
\(\frac{\mathrm{x}-20}{\mathrm{y}-20} = \frac{1}{2}\)

Cross-multiplying:
\(2(\mathrm{x}-20) = \mathrm{y}-20\)
\(2\mathrm{x} - 40 = \mathrm{y} - 20\)
\(\mathrm{y} = 2\mathrm{x} - 20\)

From the current ratio:
\(\frac{\mathrm{x}}{\mathrm{y}} = \frac{5}{6}\)

Cross-multiplying:
\(6\mathrm{x} = 5\mathrm{y}\)
\(\mathrm{y} = \frac{6\mathrm{x}}{5}\)

Solving for x:
Setting our two expressions for y equal:
\(2\mathrm{x} - 20 = \frac{6\mathrm{x}}{5}\)

Multiplying by 5:
\(5(2\mathrm{x} - 20) = 6\mathrm{x}\)
\(10\mathrm{x} - 100 = 6\mathrm{x}\)
\(4\mathrm{x} = 100\)
\(\mathrm{x} = 25\)

Finding y:
\(\mathrm{y} = 2\mathrm{x} - 20 = 2(25) - 20 = 30\)

Verification:
Let's verify all three ratios work:

• 20 years ago: \((25-20):(30-20) = 5:10 = 1:2\) (tick)
• Now: \(25:30 = 5:6\) (tick)
• 20 years from now: \((25+20):(30+20) = 45:50 = 9:10\) (tick)

Phase 4: Solution

Difference in ages: \(30 - 25 = 5\) years

Sum of ages now: \(25 + 30 = 55\) years

Therefore:

• For "Difference in ages", we select: 5
• For "Sum of ages now", we select: 55