Sister Age Problem: Solve Using Quadratic Equation

Skills This Problem Builds

Solution: Method 1 — The Algebraic Setup

The key insight is that age differences stay constant over time. If Lisa is 1 year older than Ella now, she was also 1 year older than Ella one year ago.

Step 1 — Define the variable for current ages

Let x = Ella's current age. Since Lisa is 1 year older, Lisa's current age = x + 1.

Step 2 — Express their ages one year ago

One year ago:

• Ella's age = x - 1
• Lisa's age = (x + 1) - 1 = x

Step 3 — Set up the equation using the product condition

We're told that the product of their ages one year ago was 20:

Step 4 — Expand and rearrange to standard form

Expanding the left side:

Step 5 — Factor the quadratic equation

We need two numbers that multiply to -20 and add to -1. Those numbers are -5 and 4:

So x = 5 or x = -4.

Step 6 — Interpret the solutions

Since age cannot be negative, we reject x = -4. Therefore, x = 5.

This means:

• Ella's current age = 5 years
• Lisa's current age = 5 + 1 = 6 years

Solution: Method 2 — The Factor-Pairs Approach

Instead of setting up variables immediately, let's think about what numbers could work.

Step 1 — Analyze the constraint

One year ago, their ages had a product of 20, and the ages differed by exactly 1 year. So we need two consecutive integers whose product is 20.

Step 2 — Find factor pairs of 20

The factor pairs of 20 are:

• 1 × 20 = 20 (difference: 19)
• 2 × 10 = 20 (difference: 8)
• 4 × 5 = 20 (difference: 1) ✓

Step 3 — Identify the consecutive pair

The only consecutive integers that multiply to 20 are 4 and 5. Since Lisa is older, one year ago Lisa was 5 and Ella was 4.

Step 4 — Calculate current ages

Adding one year to each:

• Ella's current age = 4 + 1 = 5 years
• Lisa's current age = 5 + 1 = 6 years

Verification

Let's check our answer by substituting back into the original conditions:

Check 1: Age difference now

Lisa's age - Ella's age = 6 - 5 = 1 year ✓

Check 2: Product of ages one year ago

One year ago, Ella was 4 and Lisa was 5:

Both conditions are satisfied, confirming our answer is correct.

Common Pitfalls

The Pattern Behind This

This problem follows the classic quadratic age problem pattern:

In our specific case: k = 1 (age difference), n = 1 (years ago), P = 20 (product), giving us x² - x - 20 = 0.

The key insight is that age differences remain constant over time. This allows us to set up equations relating ages at different points in time. When products are involved, we inevitably get quadratic equations, and we must always reject solutions that produce negative or unrealistic ages.

Recognizing This Problem Type

Watch for these telltale phrases in age problems:

• "The product of their ages..." — signals a quadratic equation setup
• "X years older/younger than..." — establishes a constant age difference
• "One year ago" or "In X years" — requires adjusting ages to a different time period
• Two people with a relationship — suggests using one variable to represent both ages

The general structure is always: establish age relationships → express ages at the specified time → use the given constraint to form an equation.

If you see "sum of ages," you'll get a linear equation. If you see "product of ages," expect a quadratic equation and the need to reject one solution.