Bounds of a Product/Quotient (Guide)

Statement

When quantities with bounds are multiplied or divided, we can calculate the bounds of the result using the extreme values of each range:

Product: \[L_{xy} \approx L_x \cdot L_y, \quad U_{xy} \approx U_x \cdot U_y\] Quotient: \[L_{x/y} \approx \frac{L_x}{U_y}, \quad U_{x/y} \approx \frac{U_x}{L_y}\]

Here, \(L\) stands for lower bound and \(U\) for upper bound of a value.

Why it’s true (short reason)

• The true value of each variable lies between its lower and upper bound.
• For multiplication, the smallest possible product comes from multiplying the lower bounds, and the largest from multiplying the upper bounds (assuming positivity).
• For division, the smallest possible quotient comes from dividing the smallest numerator by the largest denominator, and the largest comes from dividing the largest numerator by the smallest denominator.

Recipe (how to use it)

1. Find the lower and upper bounds of each number (using rounding bounds formula).
2. For a product \(x \times y\):
   • Lower bound = \(L_x \times L_y\)
   • Upper bound = \(U_x \times U_y\)
3. For a quotient \(x \div y\):
   • Lower bound = \(L_x / U_y\)
   • Upper bound = \(U_x / L_y\)
4. Write the result as an interval.

Spotting it

Use this when:

• Values have been rounded to a unit, giving bounds.
• You need to calculate a product or quotient of those rounded values.
• The exam question asks for maximum or minimum possible results.

Common pairings

• Bounds from rounding (finding \(L_x, U_x\)).
• Applications in geometry (areas, densities, speeds).
• Percentage error or worst-case error analysis.

Mini examples

1. \(x = 12\) rounded to nearest 1 → bounds \([11.5, 12.5)\). \(y = 5\) rounded to nearest 1 → bounds \([4.5, 5.5)\). Product bounds: \(L=11.5 \times 4.5 = 51.75\), \(U=12.5 \times 5.5 = 68.75\).
2. \(x = 120\) rounded to nearest 10 → bounds \([115, 125)\). \(y = 6\) rounded to nearest 1 → bounds \([5.5, 6.5)\). Quotient bounds: \(L = 115/6.5 \approx 17.69\), \(U = 125/5.5 \approx 22.73\).

Pitfalls

• Forgetting to calculate the original bounds before applying the product/quotient rule.
• Mixing up which way to divide in quotients (lower uses denominator upper, upper uses denominator lower).
• Including the upper bound when it should be excluded.
• Not checking that values are positive (rule changes with negatives).

Exam strategy

• Always find the lower and upper bound first.
• Write working clearly for product and quotient cases.
• Check answers are reasonable compared to rounded values.
• If negative numbers are involved, consider all extreme combinations.

Summary

Bounds of a product or quotient are found by combining the extreme values of the factors. Multiply or divide bounds carefully, and always base your work on the ranges from rounding. This gives maximum and minimum possible results.