The product of two positive numbers is 400. The larger number is 30 more than the smaller number. What is...

1. TRANSLATE the problem information

• Given information:
  • Product of two positive numbers = 400
  • Larger number is 30 more than smaller number
  • Need to find: the larger number
• Let smaller number = \(\mathrm{x}\), then larger number = \(\mathrm{x + 30}\)

2. INFER the mathematical approach

• Since we know their product, we can write: \(\mathrm{x(x + 30) = 400}\)
• This will create a quadratic equation that we'll solve using the quadratic formula
• Strategy: Expand, rearrange to standard form, then apply quadratic formula

3. SIMPLIFY the equation setup

• Expand: \(\mathrm{x(x + 30) = x^2 + 30x = 400}\)
• Rearrange to standard form: \(\mathrm{x^2 + 30x - 400 = 0}\)
• This fits the pattern \(\mathrm{ax^2 + bx + c = 0}\) where \(\mathrm{a = 1, b = 30, c = -400}\)

4. SIMPLIFY using the quadratic formula

• Apply: \(\mathrm{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}\)
• Substitute: \(\mathrm{x = \frac{-30 \pm \sqrt{30^2 - 4(1)(-400)}}{2(1)}}\)
• Calculate discriminant: \(\mathrm{30^2 - 4(1)(-400) = 900 + 1600 = 2500}\)
• Continue: \(\mathrm{x = \frac{-30 \pm \sqrt{2500}}{2} = \frac{-30 \pm 50}{2}}\)

5. APPLY CONSTRAINTS to select valid solution

• Two solutions: \(\mathrm{x = \frac{-30 + 50}{2} = 10}\) or \(\mathrm{x = \frac{-30 - 50}{2} = -40}\)
• Since the problem specifies positive numbers, reject \(\mathrm{x = -40}\)
• Therefore: smaller number = 10, larger number = \(\mathrm{10 + 30 = 40}\)

Answer: D) 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students may incorrectly set up the relationship between the numbers, such as writing \(\mathrm{x + (x + 30) = 400}\) instead of \(\mathrm{x(x + 30) = 400}\), confusing "product" with "sum."

This fundamental translation error leads to the wrong equation (\(\mathrm{2x + 30 = 400}\)), giving \(\mathrm{x = 185}\), and the larger number as 215. Since this isn't among the answer choices, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students may make arithmetic errors when applying the quadratic formula, particularly when calculating the discriminant (\(\mathrm{30^2 - 4(1)(-400)}\)) or when taking the square root of 2500.

These calculation mistakes can lead to incorrect values for x, potentially matching one of the incorrect answer choices like Choice A (10) if they accidentally think 10 is the larger number instead of recognizing it as the smaller number.

The Bottom Line:

This problem tests whether students can correctly translate a product relationship (not sum) into equations and systematically work through quadratic solution steps while maintaining awareness of which quantity they're ultimately solving for.